int_rules_1_3_3 = @theory begin

    #= ::Subsection::Closed:: =#
    #= 1.3.3*Miscellaneous*algebraic*functions =#
    @apply_utils Antiderivative(~u / (~(e') * sqrt(~(a') + ~(b') * ~x) + ~(f') * sqrt(~(c') + ~(d') * ~x)), ~x) => (~c / (~e * (~b * ~c - ~a * ~d))) * Antiderivative((~u * sqrt(~a + ~b * ~x)) / ~x, ~x) - (~a / (~f * (~b * ~c - ~a * ~d))) * Antiderivative((~u * sqrt(~c + ~d * ~x)) / ~x, ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f], ~x) && (NeQ(~b * ~c - ~a * ~d, 0) && EqQ(~a * (~e) ^ 2 - ~c * (~f) ^ 2, 0))
    @apply_utils Antiderivative(~u / (~(e') * sqrt(~(a') + ~(b') * ~x) + ~(f') * sqrt(~(c') + ~(d') * ~x)), ~x) => (-(~d) / (~e * (~b * ~c - ~a * ~d))) * Antiderivative(~u * sqrt(~a + ~b * ~x), ~x) + (~b / (~f * (~b * ~c - ~a * ~d))) * Antiderivative(~u * sqrt(~c + ~d * ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f], ~x) && (NeQ(~b * ~c - ~a * ~d, 0) && EqQ(~b * (~e) ^ 2 - ~d * (~f) ^ 2, 0))
    @apply_utils Antiderivative(~u / (~(e') * sqrt(~(a') + ~(b') * ~x) + ~(f') * sqrt(~(c') + ~(d') * ~x)), ~x) => ~e * Antiderivative((~u * sqrt(~a + ~b * ~x)) / ((~a * (~e) ^ 2 - ~c * (~f) ^ 2) + (~b * (~e) ^ 2 - ~d * (~f) ^ 2) * ~x), ~x) - ~f * Antiderivative((~u * sqrt(~c + ~d * ~x)) / ((~a * (~e) ^ 2 - ~c * (~f) ^ 2) + (~b * (~e) ^ 2 - ~d * (~f) ^ 2) * ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f], ~x) && (NeQ(~a * (~e) ^ 2 - ~c * (~f) ^ 2, 0) && NeQ(~b * (~e) ^ 2 - ~d * (~f) ^ 2, 0))
    @apply_utils Antiderivative(~(u') / (~(d') * (~x) ^ ~(n') + ~(c') * sqrt(~(a') + ~(b') * (~x) ^ ~(p'))), ~x) => (-(~b) / (~a * ~d)) * Antiderivative(~u * (~x) ^ ~n, ~x) + (1 / (~a * ~c)) * Antiderivative(~u * sqrt(~a + ~b * (~x) ^ (2 * ~n)), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~n], ~x) && (EqQ(~p, 2 * ~n) && EqQ(~b * (~c) ^ 2 - (~d) ^ 2, 0))
    @apply_utils Antiderivative((~x) ^ ~(m') / (~(d') * (~x) ^ ~(n') + ~(c') * sqrt(~(a') + ~(b') * (~x) ^ ~(p'))), ~x) => -(~d) * Antiderivative((~x) ^ (~m + ~n) / (~a * (~c) ^ 2 + (~b * (~c) ^ 2 - (~d) ^ 2) * (~x) ^ (2 * ~n)), ~x) + ~c * Antiderivative(((~x) ^ ~m * sqrt(~a + ~b * (~x) ^ (2 * ~n))) / (~a * (~c) ^ 2 + (~b * (~c) ^ 2 - (~d) ^ 2) * (~x) ^ (2 * ~n)), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~m, ~n], ~x) && (EqQ(~p, 2 * ~n) && NeQ(~b * (~c) ^ 2 - (~d) ^ 2, 0))
    @apply_utils Antiderivative(1 / ((~a + ~(b') * (~x) ^ 3) * sqrt(~(d') + ~(e') * ~x + ~(f') * (~x) ^ 2)), ~x) => With([r = Numerator(Rt(~a / ~b, 3)), s = Denominator(Rt(~a / ~b, 3))], (r / (3 * ~a)) * Antiderivative(1 / ((r + s * ~x) * sqrt(~d + ~e * ~x + ~f * (~x) ^ 2)), ~x) + (r / (3 * ~a)) * Antiderivative((2r - s * ~x) / (((r ^ 2 - r * s * ~x) + s ^ 2 * (~x) ^ 2) * sqrt(~d + ~e * ~x + ~f * (~x) ^ 2)), ~x)) <-- FreeQ([~a, ~b, ~d, ~e, ~f], ~x) && PosQ(~a / ~b)
    @apply_utils Antiderivative(1 / ((~a + ~(b') * (~x) ^ 3) * sqrt(~(d') + ~(f') * (~x) ^ 2)), ~x) => With([r = Numerator(Rt(~a / ~b, 3)), s = Denominator(Rt(~a / ~b, 3))], (r / (3 * ~a)) * Antiderivative(1 / ((r + s * ~x) * sqrt(~d + ~f * (~x) ^ 2)), ~x) + (r / (3 * ~a)) * Antiderivative((2r - s * ~x) / (((r ^ 2 - r * s * ~x) + s ^ 2 * (~x) ^ 2) * sqrt(~d + ~f * (~x) ^ 2)), ~x)) <-- FreeQ([~a, ~b, ~d, ~f], ~x) && PosQ(~a / ~b)
    @apply_utils Antiderivative(1 / ((~a + ~(b') * (~x) ^ 3) * sqrt(~(d') + ~(e') * ~x + ~(f') * (~x) ^ 2)), ~x) => With([r = Numerator(Rt(-(~a) / ~b, 3)), s = Denominator(Rt(-(~a) / ~b, 3))], (r / (3 * ~a)) * Antiderivative(1 / ((r - s * ~x) * sqrt(~d + ~e * ~x + ~f * (~x) ^ 2)), ~x) + (r / (3 * ~a)) * Antiderivative((2r + s * ~x) / ((r ^ 2 + r * s * ~x + s ^ 2 * (~x) ^ 2) * sqrt(~d + ~e * ~x + ~f * (~x) ^ 2)), ~x)) <-- FreeQ([~a, ~b, ~d, ~e, ~f], ~x) && NegQ(~a / ~b)
    @apply_utils Antiderivative(1 / ((~a + ~(b') * (~x) ^ 3) * sqrt(~(d') + ~(f') * (~x) ^ 2)), ~x) => With([r = Numerator(Rt(-(~a) / ~b, 3)), s = Denominator(Rt(-(~a) / ~b, 3))], (r / (3 * ~a)) * Antiderivative(1 / ((r - s * ~x) * sqrt(~d + ~f * (~x) ^ 2)), ~x) + (r / (3 * ~a)) * Antiderivative((2r + s * ~x) / ((r ^ 2 + r * s * ~x + s ^ 2 * (~x) ^ 2) * sqrt(~d + ~f * (~x) ^ 2)), ~x)) <-- FreeQ([~a, ~b, ~d, ~f], ~x) && NegQ(~a / ~b)
    @apply_utils Antiderivative((~u * (~A + ~(B') * (~x) ^ 4)) / sqrt(~v), ~x) => With([a = Coeff(~v, ~x, 0), b = Coeff(~v, ~x, 2), c = Coeff(~v, ~x, 4), d = Coeff(1 / ~u, ~x, 0), e = Coeff(1 / ~u, ~x, 2), f = Coeff(1 / ~u, ~x, 4)], ~A * Subst(Antiderivative(1 / (d - (b * d - a * e) * (~x) ^ 2), ~x), ~x, ~x / sqrt(~v)) <-- EqQ(a * ~B + ~A * c, 0) && EqQ(c * d - a * f, 0)) <-- FreeQ([~A, ~B], ~x) && (PolyQ(~v, (~x) ^ 2, 2) && PolyQ(1 / ~u, (~x) ^ 2, 2))
    @apply_utils Antiderivative(1 / ((~a + ~(b') * ~x) * sqrt(~c + ~(d') * (~x) ^ 2) * sqrt(~e + ~(f') * (~x) ^ 2)), ~x) => ~a * Antiderivative(1 / (((~a) ^ 2 - (~b) ^ 2 * (~x) ^ 2) * sqrt(~c + ~d * (~x) ^ 2) * sqrt(~e + ~f * (~x) ^ 2)), ~x) - ~b * Antiderivative(~x / (((~a) ^ 2 - (~b) ^ 2 * (~x) ^ 2) * sqrt(~c + ~d * (~x) ^ 2) * sqrt(~e + ~f * (~x) ^ 2)), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f], ~x)
    @apply_utils Antiderivative((~(g') + ~(h') * ~x) * sqrt(~(d') + ~(e') * ~x + ~(f') * sqrt(~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2)), ~x) => ((2 * ((~f * (((5 * ~b * ~c * (~g) ^ 2 - 2 * (~b) ^ 2 * ~g * ~h) - 3 * ~a * ~c * ~g * ~h) + 2 * ~a * ~b * (~h) ^ 2) + ~c * ~f * ((10 * ~c * (~g) ^ 2 - ~b * ~g * ~h) + ~a * (~h) ^ 2) * ~x + 9 * (~c) ^ 2 * ~f * ~g * ~h * (~x) ^ 2 + 3 * (~c) ^ 2 * ~f * (~h) ^ 2 * (~x) ^ 3) - (~e * ~g - ~d * ~h) * ((5 * ~c * ~g - 2 * ~b * ~h) + ~c * ~h * ~x) * sqrt(~a + ~b * ~x + ~c * (~x) ^ 2))) / (15 * (~c) ^ 2 * ~f * (~g + ~h * ~x))) * sqrt(~d + ~e * ~x + ~f * sqrt(~a + ~b * ~x + ~c * (~x) ^ 2)) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~h], ~x) && (EqQ((~e * ~g - ~d * ~h) ^ 2 - (~f) ^ 2 * ((~c * (~g) ^ 2 - ~b * ~g * ~h) + ~a * (~h) ^ 2), 0) && EqQ((2 * (~e) ^ 2 * ~g - 2 * ~d * ~e * ~h) - (~f) ^ 2 * (2 * ~c * ~g - ~b * ~h), 0))
    @apply_utils Antiderivative((~(g') + ~(h') * ~x) ^ ~(m') * (~u + ~(f') * (~(j') + ~(k') * sqrt(~v))) ^ ~(n'), ~x) => Antiderivative((~g + ~h * ~x) ^ ~m * (ExpandToSum(~u + ~f * ~j, ~x) + ~f * ~k * sqrt(ExpandToSum(~v, ~x))) ^ ~n, ~x) <-- FreeQ([~f, ~g, ~h, ~j, ~k, ~m, ~n], ~x) && (LinearQ(~u, ~x) && (QuadraticQ(~v, ~x) && (Not(LinearMatchQ(~u, ~x) && (QuadraticMatchQ(~v, ~x) && (EqQ(~j, 0) || EqQ(~f, 1)))) && EqQ((Coefficient(~u, ~x, 1) * ~g - ~h * (Coefficient(~u, ~x, 0) + ~f * ~j)) ^ 2 - (~f) ^ 2 * (~k) ^ 2 * ((Coefficient(~v, ~x, 2) * (~g) ^ 2 - Coefficient(~v, ~x, 1) * ~g * ~h) + Coefficient(~v, ~x, 0) * (~h) ^ 2), 0))))
    #= Antiderivative(1/((~d')+(~e')*(~x)+(~f')*sqrt((~a')+(~b')*(~x)+(~c')*(~x)^2)),~x) :=  Antiderivative((d+e*x)/(d^2-a*f^2+(2*d*e-b*f^2)*x),x) - f*Antiderivative(sqrt(a+b*x+c*x^2)/(d^2-a*f^2+(2*d*e-b*f^2)*x),x) <--  FreeQ([a,b,c,d,e,f],x) && EqQ(e^2-c*f^2,0) =#
    #= Antiderivative(1/((~d')+(~e')*(~x)+(~f')*sqrt((~a')+(~c')*(~x)^2)),~x) :=  Antiderivative((d+e*x)/(d^2-a*f^2+2*d*e*x),x) - f*Antiderivative(sqrt(a+c*x^2)/(d^2-a*f^2+2*d*e*x),x) <-- FreeQ([a,c,d,e,f],x) &&  EqQ(e^2-c*f^2,0) =#
    @apply_utils Antiderivative((~(g') + ~(h') * (~(d') + ~(e') * ~x + ~(f') * sqrt(~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2)) ^ ~n) ^ ~(p'), ~x) => 2 * Subst(Antiderivative(((~g + ~h * (~x) ^ ~n) ^ ~p * ((((~d) ^ 2 * ~e - (~b * ~d - ~a * ~e) * (~f) ^ 2) - (2 * ~d * ~e - ~b * (~f) ^ 2) * ~x) + ~e * (~x) ^ 2)) / (-2 * ~d * ~e + ~b * (~f) ^ 2 + 2 * ~e * ~x) ^ 2, ~x), ~x, ~d + ~e * ~x + ~f * sqrt(~a + ~b * ~x + ~c * (~x) ^ 2)) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~h, ~n], ~x) && (EqQ((~e) ^ 2 - ~c * (~f) ^ 2, 0) && IntegerQ(~p))
    @apply_utils Antiderivative((~(g') + ~(h') * (~(d') + ~(e') * ~x + ~(f') * sqrt(~a + ~(c') * (~x) ^ 2)) ^ ~n) ^ ~(p'), ~x) => (1 / (2 * ~e)) * Subst(Antiderivative(((~g + ~h * (~x) ^ ~n) ^ ~p * ((((~d) ^ 2 + ~a * (~f) ^ 2) - 2 * ~d * ~x) + (~x) ^ 2)) / (~d - ~x) ^ 2, ~x), ~x, ~d + ~e * ~x + ~f * sqrt(~a + ~c * (~x) ^ 2)) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~h, ~n], ~x) && (EqQ((~e) ^ 2 - ~c * (~f) ^ 2, 0) && IntegerQ(~p))
    @apply_utils Antiderivative((~(g') + ~(h') * (~u + ~(f') * sqrt(~v)) ^ ~n) ^ ~(p'), ~x) => Antiderivative((~g + ~h * (ExpandToSum(~u, ~x) + ~f * sqrt(ExpandToSum(~v, ~x))) ^ ~n) ^ ~p, ~x) <-- FreeQ([~f, ~g, ~h, ~n], ~x) && (LinearQ(~u, ~x) && (QuadraticQ(~v, ~x) && (Not(LinearMatchQ(~u, ~x) && QuadraticMatchQ(~v, ~x)) && (EqQ(Coefficient(~u, ~x, 1) ^ 2 - Coefficient(~v, ~x, 2) * (~f) ^ 2, 0) && IntegerQ(~p)))))
    @apply_utils Antiderivative((~(g') + ~(h') * ~x) ^ ~(m') * (~(e') * ~x + ~(f') * sqrt(~(a') + ~(c') * (~x) ^ 2)) ^ ~(n'), ~x) => (1 / (2 ^ (~m + 1) * (~e) ^ (~m + 1))) * Subst(Antiderivative((~x) ^ ((~n - ~m) - 2) * (~a * (~f) ^ 2 + (~x) ^ 2) * (-(~a) * (~f) ^ 2 * ~h + 2 * ~e * ~g * ~x + ~h * (~x) ^ 2) ^ ~m, ~x), ~x, ~e * ~x + ~f * sqrt(~a + ~c * (~x) ^ 2)) <-- FreeQ([~a, ~c, ~e, ~f, ~g, ~h, ~n], ~x) && (EqQ((~e) ^ 2 - ~c * (~f) ^ 2, 0) && IntegerQ(~m))
    @apply_utils Antiderivative((~x) ^ ~(p') * (~g + ~(i') * (~x) ^ 2) ^ ~(m') * (~(e') * ~x + ~(f') * sqrt(~a + ~(c') * (~x) ^ 2)) ^ ~(n'), ~x) => (1 / (2 ^ (2 * ~m + ~p + 1) * (~e) ^ (~p + 1) * (~f) ^ (2 * ~m))) * (~i / ~c) ^ ~m * Subst(Antiderivative((~x) ^ (((~n - 2 * ~m) - ~p) - 2) * (-(~a) * (~f) ^ 2 + (~x) ^ 2) ^ ~p * (~a * (~f) ^ 2 + (~x) ^ 2) ^ (2 * ~m + 1), ~x), ~x, ~e * ~x + ~f * sqrt(~a + ~c * (~x) ^ 2)) <-- FreeQ([~a, ~c, ~e, ~f, ~g, ~i, ~n], ~x) && (EqQ((~e) ^ 2 - ~c * (~f) ^ 2, 0) && (EqQ(~c * ~g - ~a * ~i, 0) && (IntegersQ(~p, 2 * ~m) && (IntegerQ(~m) || GtQ(~i / ~c, 0)))))
    @apply_utils Antiderivative((~(g') + ~(h') * ~x + ~(i') * (~x) ^ 2) ^ ~(m') * (~(d') + ~(e') * ~x + ~(f') * sqrt(~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2)) ^ ~(n'), ~x) => (2 / (~f) ^ (2 * ~m)) * (~i / ~c) ^ ~m * Subst(Antiderivative(((~x) ^ ~n * ((((~d) ^ 2 * ~e - (~b * ~d - ~a * ~e) * (~f) ^ 2) - (2 * ~d * ~e - ~b * (~f) ^ 2) * ~x) + ~e * (~x) ^ 2) ^ (2 * ~m + 1)) / (-2 * ~d * ~e + ~b * (~f) ^ 2 + 2 * ~e * ~x) ^ (2 * (~m + 1)), ~x), ~x, ~d + ~e * ~x + ~f * sqrt(~a + ~b * ~x + ~c * (~x) ^ 2)) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~h, ~i, ~n], ~x) && (EqQ((~e) ^ 2 - ~c * (~f) ^ 2, 0) && (EqQ(~c * ~g - ~a * ~i, 0) && (EqQ(~c * ~h - ~b * ~i, 0) && (IntegerQ(2 * ~m) && (IntegerQ(~m) || GtQ(~i / ~c, 0))))))
    @apply_utils Antiderivative((~g + ~(i') * (~x) ^ 2) ^ ~(m') * (~(d') + ~(e') * ~x + ~(f') * sqrt(~a + ~(c') * (~x) ^ 2)) ^ ~(n'), ~x) => (1 / (2 ^ (2 * ~m + 1) * ~e * (~f) ^ (2 * ~m))) * (~i / ~c) ^ ~m * Subst(Antiderivative(((~x) ^ ~n * ((((~d) ^ 2 + ~a * (~f) ^ 2) - 2 * ~d * ~x) + (~x) ^ 2) ^ (2 * ~m + 1)) / (-(~d) + ~x) ^ (2 * (~m + 1)), ~x), ~x, ~d + ~e * ~x + ~f * sqrt(~a + ~c * (~x) ^ 2)) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~i, ~n], ~x) && (EqQ((~e) ^ 2 - ~c * (~f) ^ 2, 0) && (EqQ(~c * ~g - ~a * ~i, 0) && (IntegerQ(2 * ~m) && (IntegerQ(~m) || GtQ(~i / ~c, 0)))))
    @apply_utils Antiderivative((~(g') + ~(h') * ~x + ~(i') * (~x) ^ 2) ^ ~(m') * (~(d') + ~(e') * ~x + ~(f') * sqrt(~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2)) ^ ~(n'), ~x) => (((~i / ~c) ^ (~m - 1 / 2) * sqrt(~g + ~h * ~x + ~i * (~x) ^ 2)) / sqrt(~a + ~b * ~x + ~c * (~x) ^ 2)) * Antiderivative((~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~m * (~d + ~e * ~x + ~f * sqrt(~a + ~b * ~x + ~c * (~x) ^ 2)) ^ ~n, ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~h, ~i, ~n], ~x) && (EqQ((~e) ^ 2 - ~c * (~f) ^ 2, 0) && (EqQ(~c * ~g - ~a * ~i, 0) && (EqQ(~c * ~h - ~b * ~i, 0) && (IGtQ(~m + 1 / 2, 0) && Not(GtQ(~i / ~c, 0))))))
    @apply_utils Antiderivative((~g + ~(i') * (~x) ^ 2) ^ ~(m') * (~(d') + ~(e') * ~x + ~(f') * sqrt(~a + ~(c') * (~x) ^ 2)) ^ ~(n'), ~x) => (((~i / ~c) ^ (~m - 1 / 2) * sqrt(~g + ~i * (~x) ^ 2)) / sqrt(~a + ~c * (~x) ^ 2)) * Antiderivative((~a + ~c * (~x) ^ 2) ^ ~m * (~d + ~e * ~x + ~f * sqrt(~a + ~c * (~x) ^ 2)) ^ ~n, ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~i, ~n], ~x) && (EqQ((~e) ^ 2 - ~c * (~f) ^ 2, 0) && (EqQ(~c * ~g - ~a * ~i, 0) && (IGtQ(~m + 1 / 2, 0) && Not(GtQ(~i / ~c, 0)))))
    @apply_utils Antiderivative((~(g') + ~(h') * ~x + ~(i') * (~x) ^ 2) ^ ~(m') * (~(d') + ~(e') * ~x + ~(f') * sqrt(~(a') + ~(b') * ~x + ~(c') * (~x) ^ 2)) ^ ~(n'), ~x) => (((~i / ~c) ^ (~m + 1 / 2) * sqrt(~a + ~b * ~x + ~c * (~x) ^ 2)) / sqrt(~g + ~h * ~x + ~i * (~x) ^ 2)) * Antiderivative((~a + ~b * ~x + ~c * (~x) ^ 2) ^ ~m * (~d + ~e * ~x + ~f * sqrt(~a + ~b * ~x + ~c * (~x) ^ 2)) ^ ~n, ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~h, ~i, ~n], ~x) && (EqQ((~e) ^ 2 - ~c * (~f) ^ 2, 0) && (EqQ(~c * ~g - ~a * ~i, 0) && (EqQ(~c * ~h - ~b * ~i, 0) && (ILtQ(~m - 1 / 2, 0) && Not(GtQ(~i / ~c, 0))))))
    @apply_utils Antiderivative((~g + ~(i') * (~x) ^ 2) ^ ~(m') * (~(d') + ~(e') * ~x + ~(f') * sqrt(~a + ~(c') * (~x) ^ 2)) ^ ~(n'), ~x) => (((~i / ~c) ^ (~m + 1 / 2) * sqrt(~a + ~c * (~x) ^ 2)) / sqrt(~g + ~i * (~x) ^ 2)) * Antiderivative((~a + ~c * (~x) ^ 2) ^ ~m * (~d + ~e * ~x + ~f * sqrt(~a + ~c * (~x) ^ 2)) ^ ~n, ~x) <-- FreeQ([~a, ~c, ~d, ~e, ~f, ~g, ~i, ~n], ~x) && (EqQ((~e) ^ 2 - ~c * (~f) ^ 2, 0) && (EqQ(~c * ~g - ~a * ~i, 0) && (ILtQ(~m - 1 / 2, 0) && Not(GtQ(~i / ~c, 0)))))
    @apply_utils Antiderivative((~w) ^ ~(m') * (~u + ~(f') * (~(j') + ~(k') * sqrt(~v))) ^ ~(n'), ~x) => Antiderivative(ExpandToSum(~w, ~x) ^ ~m * (ExpandToSum(~u + ~f * ~j, ~x) + ~f * ~k * sqrt(ExpandToSum(~v, ~x))) ^ ~n, ~x) <-- FreeQ([~f, ~j, ~k, ~m, ~n], ~x) && (LinearQ(~u, ~x) && (QuadraticQ([~v, ~w], ~x) && (Not(LinearMatchQ(~u, ~x) && (QuadraticMatchQ([~v, ~w], ~x) && (EqQ(~j, 0) || EqQ(~f, 1)))) && EqQ(Coefficient(~u, ~x, 1) ^ 2 - Coefficient(~v, ~x, 2) * (~f) ^ 2 * (~k) ^ 2, 0))))
    @apply_utils Antiderivative(1 / ((~a + ~(b') * (~x) ^ ~(n')) * sqrt(~(c') * (~x) ^ 2 + ~(d') * (~a + ~(b') * (~x) ^ ~(n')) ^ ~(p'))), ~x) => (1 / ~a) * Subst(Antiderivative(1 / (1 - ~c * (~x) ^ 2), ~x), ~x, ~x / sqrt(~c * (~x) ^ 2 + ~d * (~a + ~b * (~x) ^ ~n) ^ (2 / ~n))) <-- FreeQ([~a, ~b, ~c, ~d, ~n], ~x) && EqQ(~p, 2 / ~n)
    @apply_utils Antiderivative(sqrt(~a + ~(b') * sqrt(~c + ~(d') * (~x) ^ 2)), ~x) => (2 * (~b) ^ 2 * ~d * (~x) ^ 3) / (3 * (~a + ~b * sqrt(~c + ~d * (~x) ^ 2)) ^ (3 / 2)) + (2 * ~a * ~x) / sqrt(~a + ~b * sqrt(~c + ~d * (~x) ^ 2)) <-- FreeQ([~a, ~b, ~c, ~d], ~x) && EqQ((~a) ^ 2 - (~b) ^ 2 * ~c, 0)
    @apply_utils Antiderivative(sqrt(~(a') * (~x) ^ 2 + ~(b') * ~x * sqrt(~c + ~(d') * (~x) ^ 2)) / (~x * sqrt(~c + ~(d') * (~x) ^ 2)), ~x) => ((sqrt(2) * ~b) / ~a) * Subst(Antiderivative(1 / sqrt(1 + (~x) ^ 2 / ~a), ~x), ~x, ~a * ~x + ~b * sqrt(~c + ~d * (~x) ^ 2)) <-- FreeQ([~a, ~b, ~c, ~d], ~x) && (EqQ((~a) ^ 2 - (~b) ^ 2 * ~d, 0) && EqQ((~b) ^ 2 * ~c + ~a, 0))
    @apply_utils Antiderivative(sqrt(~(e') * ~x * (~(a') * ~x + ~(b') * sqrt(~c + ~(d') * (~x) ^ 2))) / (~x * sqrt(~c + ~(d') * (~x) ^ 2)), ~x) => Antiderivative(sqrt(~a * ~e * (~x) ^ 2 + ~b * ~e * ~x * sqrt(~c + ~d * (~x) ^ 2)) / (~x * sqrt(~c + ~d * (~x) ^ 2)), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e], ~x) && (EqQ((~a) ^ 2 - (~b) ^ 2 * ~d, 0) && EqQ((~b) ^ 2 * ~c * ~e + ~a, 0))
    @apply_utils Antiderivative(sqrt(~(c') * (~x) ^ 2 + ~(d') * sqrt(~a + ~(b') * (~x) ^ 4)) / sqrt(~a + ~(b') * (~x) ^ 4), ~x) => ~d * Subst(Antiderivative(1 / (1 - 2 * ~c * (~x) ^ 2), ~x), ~x, ~x / sqrt(~c * (~x) ^ 2 + ~d * sqrt(~a + ~b * (~x) ^ 4))) <-- FreeQ([~a, ~b, ~c, ~d], ~x) && EqQ((~c) ^ 2 - ~b * (~d) ^ 2, 0)
    @apply_utils Antiderivative(((~(c') + ~(d') * ~x) ^ ~(m') * sqrt(~(b') * (~x) ^ 2 + sqrt(~a + ~(e') * (~x) ^ 4))) / sqrt(~a + ~(e') * (~x) ^ 4), ~x) => ((1 - im) / 2) * Antiderivative((~c + ~d * ~x) ^ ~m / sqrt(sqrt(~a) - im * ~b * (~x) ^ 2), ~x) + ((1 + im) / 2) * Antiderivative((~c + ~d * ~x) ^ ~m / sqrt(sqrt(~a) + im * ~b * (~x) ^ 2), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~m], ~x) && (EqQ(~e, (~b) ^ 2) && GtQ(~a, 0))
    @apply_utils Antiderivative(1 / ((~c + ~(d') * ~x) * sqrt(~a + ~(b') * (~x) ^ 3)), ~x) => (2 / (3 * ~c)) * Antiderivative(1 / sqrt(~a + ~b * (~x) ^ 3), ~x) + (1 / (3 * ~c)) * Antiderivative((~c - 2 * ~d * ~x) / ((~c + ~d * ~x) * sqrt(~a + ~b * (~x) ^ 3)), ~x) <-- FreeQ([~a, ~b, ~c, ~d], ~x) && EqQ(~b * (~c) ^ 3 - 4 * ~a * (~d) ^ 3, 0)
    @apply_utils Antiderivative(1 / ((~c + ~(d') * ~x) * sqrt(~a + ~(b') * (~x) ^ 3)), ~x) => ((-6 * ~a * (~d) ^ 3) / (~c * (~b * (~c) ^ 3 - 28 * ~a * (~d) ^ 3))) * Antiderivative(1 / sqrt(~a + ~b * (~x) ^ 3), ~x) + (1 / (~c * (~b * (~c) ^ 3 - 28 * ~a * (~d) ^ 3))) * Antiderivative(Simp(~c * (~b * (~c) ^ 3 - 22 * ~a * (~d) ^ 3) + 6 * ~a * (~d) ^ 4 * ~x, ~x) / ((~c + ~d * ~x) * sqrt(~a + ~b * (~x) ^ 3)), ~x) <-- FreeQ([~a, ~b, ~c, ~d], ~x) && EqQ(((~b) ^ 2 * (~c) ^ 6 - 20 * ~a * ~b * (~c) ^ 3 * (~d) ^ 3) - 8 * (~a) ^ 2 * (~d) ^ 6, 0)
    @apply_utils Antiderivative(1 / ((~c + ~(d') * ~x) * sqrt(~a + ~(b') * (~x) ^ 3)), ~x) => With([q = Rt(~b / ~a, 3)], (-q / ((1 + sqrt(3)) * ~d - ~c * q)) * Antiderivative(1 / sqrt(~a + ~b * (~x) ^ 3), ~x) + (~d / ((1 + sqrt(3)) * ~d - ~c * q)) * Antiderivative((1 + sqrt(3) + q * ~x) / ((~c + ~d * ~x) * sqrt(~a + ~b * (~x) ^ 3)), ~x)) <-- FreeQ([~a, ~b, ~c, ~d], ~x) && NeQ(((~b) ^ 2 * (~c) ^ 6 - 20 * ~a * ~b * (~c) ^ 3 * (~d) ^ 3) - 8 * (~a) ^ 2 * (~d) ^ 6, 0)
    @apply_utils Antiderivative((~e + ~(f') * ~x) / ((~c + ~(d') * ~x) * sqrt(~a + ~(b') * (~x) ^ 3)), ~x) => ((2 * ~e) / ~d) * Subst(Antiderivative(1 / (1 + 3 * ~a * (~x) ^ 2), ~x), ~x, (1 + (2 * ~d * ~x) / ~c) / sqrt(~a + ~b * (~x) ^ 3)) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f], ~x) && (NeQ(~d * ~e - ~c * ~f, 0) && (EqQ(~b * (~c) ^ 3 - 4 * ~a * (~d) ^ 3, 0) && EqQ(2 * ~d * ~e + ~c * ~f, 0)))
    @apply_utils Antiderivative((~e + ~(f') * ~x) / ((~c + ~(d') * ~x) * sqrt(~a + ~(b') * (~x) ^ 3)), ~x) => ((-2 * ~e) / ~d) * Subst(Antiderivative(1 / (9 - ~a * (~x) ^ 2), ~x), ~x, (1 + (~f * ~x) / ~e) ^ 2 / sqrt(~a + ~b * (~x) ^ 3)) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f], ~x) && (NeQ(~d * ~e - ~c * ~f, 0) && (EqQ(~b * (~c) ^ 3 + 8 * ~a * (~d) ^ 3, 0) && EqQ(2 * ~d * ~e + ~c * ~f, 0)))
    @apply_utils Antiderivative((~(e') + ~(f') * ~x) / ((~c + ~(d') * ~x) * sqrt(~a + ~(b') * (~x) ^ 3)), ~x) => ((2 * ~d * ~e + ~c * ~f) / (3 * ~c * ~d)) * Antiderivative(1 / sqrt(~a + ~b * (~x) ^ 3), ~x) + ((~d * ~e - ~c * ~f) / (3 * ~c * ~d)) * Antiderivative((~c - 2 * ~d * ~x) / ((~c + ~d * ~x) * sqrt(~a + ~b * (~x) ^ 3)), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f], ~x) && (NeQ(~d * ~e - ~c * ~f, 0) && ((EqQ(~b * (~c) ^ 3 - 4 * ~a * (~d) ^ 3, 0) || EqQ(~b * (~c) ^ 3 + 8 * ~a * (~d) ^ 3, 0)) && NeQ(2 * ~d * ~e + ~c * ~f, 0)))
    @apply_utils Antiderivative((~e + ~(f') * ~x) / ((~c + ~(d') * ~x) * sqrt(~a + ~(b') * (~x) ^ 3)), ~x) => With([k = Simplify((~d * ~e + 2 * ~c * ~f) / (~c * ~f))], (((1 + k) * ~e) / ~d) * Subst(Antiderivative(1 / (1 + (3 + 2k) * ~a * (~x) ^ 2), ~x), ~x, (1 + ((1 + k) * ~d * ~x) / ~c) / sqrt(~a + ~b * (~x) ^ 3))) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f], ~x) && (NeQ(~d * ~e - ~c * ~f, 0) && (EqQ(((~b) ^ 2 * (~c) ^ 6 - 20 * ~a * ~b * (~c) ^ 3 * (~d) ^ 3) - 8 * (~a) ^ 2 * (~d) ^ 6, 0) && EqQ(6 * ~a * (~d) ^ 4 * ~e - ~c * ~f * (~b * (~c) ^ 3 - 22 * ~a * (~d) ^ 3), 0)))
    @apply_utils Antiderivative((~(e') + ~(f') * ~x) / ((~c + ~(d') * ~x) * sqrt(~a + ~(b') * (~x) ^ 3)), ~x) => (-((6 * ~a * (~d) ^ 4 * ~e - ~c * ~f * (~b * (~c) ^ 3 - 22 * ~a * (~d) ^ 3))) / (~c * ~d * (~b * (~c) ^ 3 - 28 * ~a * (~d) ^ 3))) * Antiderivative(1 / sqrt(~a + ~b * (~x) ^ 3), ~x) + ((~d * ~e - ~c * ~f) / (~c * ~d * (~b * (~c) ^ 3 - 28 * ~a * (~d) ^ 3))) * Antiderivative((~c * (~b * (~c) ^ 3 - 22 * ~a * (~d) ^ 3) + 6 * ~a * (~d) ^ 4 * ~x) / ((~c + ~d * ~x) * sqrt(~a + ~b * (~x) ^ 3)), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f], ~x) && (NeQ(~d * ~e - ~c * ~f, 0) && (EqQ(((~b) ^ 2 * (~c) ^ 6 - 20 * ~a * ~b * (~c) ^ 3 * (~d) ^ 3) - 8 * (~a) ^ 2 * (~d) ^ 6, 0) && NeQ(6 * ~a * (~d) ^ 4 * ~e - ~c * ~f * (~b * (~c) ^ 3 - 22 * ~a * (~d) ^ 3), 0)))
    #= Antiderivative(((~e)+(~f')*(~x))/(((~c)+(~d')*(~x))*sqrt((~a)+(~b')*(~x)^3)),~x) :=  With([q=(1+sqrt(3))*f/e], 4*3^(1/4)*sqrt(2-sqrt(3))*f*(1+sqrt(3)+q*x)^2*sqrt((1+q^3*x^3)/(1+ sqrt(3)+q*x)^4)/(q*sqrt(a+b*x^3))* Subst(Antiderivative(1/(((1-sqrt(3))*d-c*q+((1+sqrt(3))*d-c*q)*x)* sqrt(7-4*sqrt(3)-2*(3-2*sqrt(3))*x^2-x^4)),x),x,(-1+sqrt(3)-q*x) /(1+sqrt(3)+q*x))) <-- FreeQ([a,b,c,d,e,f],x) && NeQ(d*e-c*f,0) &&  EqQ(b*e^3-2*(5+3*sqrt(3))*a*f^3,0) &&  NeQ(b*c^3-2*(5-3*sqrt(3))*a*d^3,0) =#
    @apply_utils Antiderivative((~e + ~(f') * ~x) / ((~c + ~(d') * ~x) * sqrt(~a + ~(b') * (~x) ^ 3)), ~x) => With([q = Simplify(((1 + sqrt(3)) * ~f) / ~e)], ((4 * 3 ^ (1 / 4) * sqrt(2 - sqrt(3)) * ~f * (1 + q * ~x) * sqrt(((1 - q * ~x) + q ^ 2 * (~x) ^ 2) / (1 + sqrt(3) + q * ~x) ^ 2)) / (q * sqrt(~a + ~b * (~x) ^ 3) * sqrt((1 + q * ~x) / (1 + sqrt(3) + q * ~x) ^ 2))) * Subst(Antiderivative(1 / ((((1 - sqrt(3)) * ~d - ~c * q) + ((1 + sqrt(3)) * ~d - ~c * q) * ~x) * sqrt(1 - (~x) ^ 2) * sqrt((7 - 4 * sqrt(3)) + (~x) ^ 2)), ~x), ~x, ((-1 + sqrt(3)) - q * ~x) / (1 + sqrt(3) + q * ~x))) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f], ~x) && (NeQ(~d * ~e - ~c * ~f, 0) && (EqQ(~b * (~e) ^ 3 - 2 * (5 + 3 * sqrt(3)) * ~a * (~f) ^ 3, 0) && NeQ(~b * (~c) ^ 3 - 2 * (5 - 3 * sqrt(3)) * ~a * (~d) ^ 3, 0)))
    @apply_utils Antiderivative((~e + ~(f') * ~x) / ((~c + ~(d') * ~x) * sqrt(~a + ~(b') * (~x) ^ 3)), ~x) => With([q = Simplify(((-1 + sqrt(3)) * ~f) / ~e)], ((4 * 3 ^ (1 / 4) * sqrt(2 + sqrt(3)) * ~f * (1 - q * ~x) * sqrt((1 + q * ~x + q ^ 2 * (~x) ^ 2) / ((1 - sqrt(3)) - q * ~x) ^ 2)) / (q * sqrt(~a + ~b * (~x) ^ 3) * sqrt(-((1 - q * ~x)) / ((1 - sqrt(3)) - q * ~x) ^ 2))) * Subst(Antiderivative(1 / (((1 + sqrt(3)) * ~d + ~c * q + ((1 - sqrt(3)) * ~d + ~c * q) * ~x) * sqrt(1 - (~x) ^ 2) * sqrt(7 + 4 * sqrt(3) + (~x) ^ 2)), ~x), ~x, ((1 + sqrt(3)) - q * ~x) / (-1 + sqrt(3) + q * ~x))) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f], ~x) && (NeQ(~d * ~e - ~c * ~f, 0) && (EqQ(~b * (~e) ^ 3 - 2 * (5 - 3 * sqrt(3)) * ~a * (~f) ^ 3, 0) && NeQ(~b * (~c) ^ 3 - 2 * (5 + 3 * sqrt(3)) * ~a * (~d) ^ 3, 0)))
    @apply_utils Antiderivative((~(e') + ~(f') * ~x) / ((~c + ~(d') * ~x) * sqrt(~a + ~(b') * (~x) ^ 3)), ~x) => With([q = Rt(~b / ~a, 3)], (((1 + sqrt(3)) * ~f - ~e * q) / ((1 + sqrt(3)) * ~d - ~c * q)) * Antiderivative(1 / sqrt(~a + ~b * (~x) ^ 3), ~x) + ((~d * ~e - ~c * ~f) / ((1 + sqrt(3)) * ~d - ~c * q)) * Antiderivative((1 + sqrt(3) + q * ~x) / ((~c + ~d * ~x) * sqrt(~a + ~b * (~x) ^ 3)), ~x)) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f], ~x) && (NeQ(~d * ~e - ~c * ~f, 0) && (NeQ(((~b) ^ 2 * (~c) ^ 6 - 20 * ~a * ~b * (~c) ^ 3 * (~d) ^ 3) - 8 * (~a) ^ 2 * (~d) ^ 6, 0) && NeQ(((~b) ^ 2 * (~e) ^ 6 - 20 * ~a * ~b * (~e) ^ 3 * (~f) ^ 3) - 8 * (~a) ^ 2 * (~f) ^ 6, 0)))
    @apply_utils Antiderivative((~f + ~(g') * ~x + ~(h') * (~x) ^ 2) / ((~c + ~(d') * ~x + ~(e') * (~x) ^ 2) * sqrt(~a + ~(b') * (~x) ^ 3)), ~x) => -2 * ~g * ~h * Subst(Antiderivative(1 / (2 * ~e * ~h - (~b * ~d * ~f - 2 * ~a * ~e * ~h) * (~x) ^ 2), ~x), ~x, (1 + (2 * ~h * ~x) / ~g) / sqrt(~a + ~b * (~x) ^ 3)) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f, ~g, ~h], ~x) && (NeQ(~b * ~d * ~f - 2 * ~a * ~e * ~h, 0) && (EqQ(~b * (~g) ^ 3 - 8 * ~a * (~h) ^ 3, 0) && (EqQ((~g) ^ 2 + 2 * ~f * ~h, 0) && EqQ((~b * ~d * ~f + ~b * ~c * ~g) - 4 * ~a * ~e * ~h, 0))))
    @apply_utils Antiderivative((~f + ~(g') * ~x + ~(h') * (~x) ^ 2) / ((~c + ~(e') * (~x) ^ 2) * sqrt(~a + ~(b') * (~x) ^ 3)), ~x) => (-(~g) / ~e) * Subst(Antiderivative(1 / (1 + ~a * (~x) ^ 2), ~x), ~x, (1 + (2 * ~h * ~x) / ~g) / sqrt(~a + ~b * (~x) ^ 3)) <-- FreeQ([~a, ~b, ~c, ~e, ~f, ~g, ~h], ~x) && (EqQ(~b * (~g) ^ 3 - 8 * ~a * (~h) ^ 3, 0) && (EqQ((~g) ^ 2 + 2 * ~f * ~h, 0) && EqQ(~b * ~c * ~g - 4 * ~a * ~e * ~h, 0)))
    @apply_utils Antiderivative(sqrt(~a + ~(b') * (~x) ^ 3) / (~c + ~(d') * ~x), ~x) => (~b / ~d) * Antiderivative((~x) ^ 2 / sqrt(~a + ~b * (~x) ^ 3), ~x) + ((~b * ~c) / (~d) ^ 3) * Antiderivative((~c - ~d * ~x) / sqrt(~a + ~b * (~x) ^ 3), ~x) <-- FreeQ([~a, ~b, ~c, ~d], ~x) && EqQ(~b * (~c) ^ 3 - ~a * (~d) ^ 3, 0)
    @apply_utils Antiderivative(sqrt(~a + ~(b') * (~x) ^ 3) / (~c + ~(d') * ~x), ~x) => ((~b / ~d) * Antiderivative((~x) ^ 2 / sqrt(~a + ~b * (~x) ^ 3), ~x) + ((~b * ~c) / (~d) ^ 3) * Antiderivative((~c - ~d * ~x) / sqrt(~a + ~b * (~x) ^ 3), ~x)) - ((~b * (~c) ^ 3 - ~a * (~d) ^ 3) / (~d) ^ 3) * Antiderivative(1 / ((~c + ~d * ~x) * sqrt(~a + ~b * (~x) ^ 3)), ~x) <-- FreeQ([~a, ~b, ~c, ~d], ~x) && NeQ(~b * (~c) ^ 3 - ~a * (~d) ^ 3, 0)
    @apply_utils Antiderivative(1 / ((~c + ~(d') * ~x) * (~a + ~(b') * (~x) ^ 3) ^ (1 / 3)), ~x) => ((sqrt(3) * atan((1 - (2 ^ (1 / 3) * Rt(~b, 3) * (~c - ~d * ~x)) / (~d * (~a + ~b * (~x) ^ 3) ^ (1 / 3))) / sqrt(3))) / (2 ^ (4 / 3) * Rt(~b, 3) * ~c) + log((~c + ~d * ~x) ^ 2 * (~c - ~d * ~x)) / (2 ^ (7 / 3) * Rt(~b, 3) * ~c)) - (3 * log(Rt(~b, 3) * (~c - ~d * ~x) + 2 ^ (2 / 3) * ~d * (~a + ~b * (~x) ^ 3) ^ (1 / 3))) / (2 ^ (7 / 3) * Rt(~b, 3) * ~c) <-- FreeQ([~a, ~b, ~c, ~d], ~x) && EqQ(~b * (~c) ^ 3 + ~a * (~d) ^ 3, 0)
    @apply_utils Antiderivative(1 / ((~c + ~(d') * ~x) * (~a + ~(b') * (~x) ^ 3) ^ (1 / 3)), ~x) => (1 / (2 * ~c)) * Antiderivative(1 / (~a + ~b * (~x) ^ 3) ^ (1 / 3), ~x) + (1 / (2 * ~c)) * Antiderivative((~c - ~d * ~x) / ((~c + ~d * ~x) * (~a + ~b * (~x) ^ 3) ^ (1 / 3)), ~x) <-- FreeQ([~a, ~b, ~c, ~d], ~x) && EqQ(2 * ~b * (~c) ^ 3 - ~a * (~d) ^ 3, 0)
    @apply_utils Antiderivative((~e + ~(f') * ~x) / ((~c + ~(d') * ~x) * (~a + ~(b') * (~x) ^ 3) ^ (1 / 3)), ~x) => ((sqrt(3) * ~f * atan((1 + (2 * Rt(~b, 3) * (2 * ~c + ~d * ~x)) / (~d * (~a + ~b * (~x) ^ 3) ^ (1 / 3))) / sqrt(3))) / (Rt(~b, 3) * ~d) + (~f * log(~c + ~d * ~x)) / (Rt(~b, 3) * ~d)) - (3 * ~f * log(Rt(~b, 3) * (2 * ~c + ~d * ~x) - ~d * (~a + ~b * (~x) ^ 3) ^ (1 / 3))) / (2 * Rt(~b, 3) * ~d) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f], ~x) && (EqQ(~d * ~e + ~c * ~f, 0) && EqQ(2 * ~b * (~c) ^ 3 - ~a * (~d) ^ 3, 0))
    @apply_utils Antiderivative((~(e') + ~(f') * ~x) / ((~(c') + ~(d') * ~x) * (~a + ~(b') * (~x) ^ 3) ^ (1 / 3)), ~x) => (~f / ~d) * Antiderivative(1 / (~a + ~b * (~x) ^ 3) ^ (1 / 3), ~x) + ((~d * ~e - ~c * ~f) / ~d) * Antiderivative(1 / ((~c + ~d * ~x) * (~a + ~b * (~x) ^ 3) ^ (1 / 3)), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~f], ~x)
    @apply_utils Antiderivative((~a + ~(b') * (~x) ^ 3) ^ (2 / 3) / (~c + ~(d') * ~x), ~x) => ((~a + ~b * (~x) ^ 3) ^ (2 / 3) / (2 * ~d) - ((~b * ~c) / (~d) ^ 2) * Antiderivative(~x / (~a + ~b * (~x) ^ 3) ^ (1 / 3), ~x)) + (1 / (~d) ^ 2) * Antiderivative((~a * (~d) ^ 2 + ~b * (~c) ^ 2 * ~x) / ((~c + ~d * ~x) * (~a + ~b * (~x) ^ 3) ^ (1 / 3)), ~x) <-- FreeQ([~a, ~b, ~c, ~d], ~x)
    @apply_utils Antiderivative(1 / ((~c + ~(d') * ~x) * (~a + ~(b') * (~x) ^ 3) ^ (2 / 3)), ~x) => With([q = Rt(~b, 3)], ((((-(~d) * atan((1 + (2 * q * ~x) / (~a + ~b * (~x) ^ 3) ^ (1 / 3)) / sqrt(3))) / (2 * sqrt(3) * q ^ 2 * (~c) ^ 2) + (sqrt(3) * ~d * atan((1 + (2 * q * (2 * ~c + ~d * ~x)) / (~d * (~a + ~b * (~x) ^ 3) ^ (1 / 3))) / sqrt(3))) / (2 * q ^ 2 * (~c) ^ 2)) - (~d * log(~c + ~d * ~x)) / (2 * q ^ 2 * (~c) ^ 2)) - (~d * log(q * ~x - (~a + ~b * (~x) ^ 3) ^ (1 / 3))) / (4 * q ^ 2 * (~c) ^ 2)) + (3 * ~d * log(q * (2 * ~c + ~d * ~x) - ~d * (~a + ~b * (~x) ^ 3) ^ (1 / 3))) / (4 * q ^ 2 * (~c) ^ 2)) <-- FreeQ([~a, ~b, ~c, ~d], ~x) && EqQ(2 * ~b * (~c) ^ 3 - ~a * (~d) ^ 3, 0)
    @apply_utils Antiderivative((~x) ^ ~(m') * ~Px * (~c + ~(d') * ~x) ^ ~q * (~a + ~(b') * (~x) ^ 3) ^ ~(p'), ~x) => Antiderivative(ExpandIntegrand(((~c) ^ 3 + (~d) ^ 3 * (~x) ^ 3) ^ ~q * (~a + ~b * (~x) ^ 3) ^ ~p, ((~x) ^ ~m * ~Px) / (((~c) ^ 2 - ~c * ~d * ~x) + (~d) ^ 2 * (~x) ^ 2) ^ ~q, ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~m, ~p], ~x) && (PolyQ(~Px, ~x) && (ILtQ(~q, 0) && (IntegerQ(~m) && (RationalQ(~p) && EqQ(Denominator(~p), 3)))))
    @apply_utils Antiderivative(~(Px') * (~c + ~(d') * ~x) ^ ~q * (~a + ~(b') * (~x) ^ 3) ^ ~(p'), ~x) => Antiderivative(ExpandIntegrand(((~c) ^ 3 + (~d) ^ 3 * (~x) ^ 3) ^ ~q * (~a + ~b * (~x) ^ 3) ^ ~p, ~Px / (((~c) ^ 2 - ~c * ~d * ~x) + (~d) ^ 2 * (~x) ^ 2) ^ ~q, ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~p], ~x) && (PolyQ(~Px, ~x) && (ILtQ(~q, 0) && (RationalQ(~p) && EqQ(Denominator(~p), 3))))
    @apply_utils Antiderivative((~x) ^ ~(m') * ~Px * (~c + ~(d') * ~x + ~(e') * (~x) ^ 2) ^ ~q * (~a + ~(b') * (~x) ^ 3) ^ ~(p'), ~x) => (1 / (~c) ^ ~q) * Antiderivative(ExpandIntegrand(((~c) ^ 3 - (~d) ^ 3 * (~x) ^ 3) ^ ~q * (~a + ~b * (~x) ^ 3) ^ ~p, ((~x) ^ ~m * ~Px) / (~c - ~d * ~x) ^ ~q, ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~m, ~p], ~x) && (PolyQ(~Px, ~x) && (EqQ((~d) ^ 2 - ~c * ~e, 0) && (ILtQ(~q, 0) && (IntegerQ(~m) && (RationalQ(~p) && EqQ(Denominator(~p), 3))))))
    @apply_utils Antiderivative(~(Px') * (~c + ~(d') * ~x + ~(e') * (~x) ^ 2) ^ ~q * (~a + ~(b') * (~x) ^ 3) ^ ~(p'), ~x) => (1 / (~c) ^ ~q) * Antiderivative(ExpandIntegrand(((~c) ^ 3 - (~d) ^ 3 * (~x) ^ 3) ^ ~q * (~a + ~b * (~x) ^ 3) ^ ~p, ~Px / (~c - ~d * ~x) ^ ~q, ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~p], ~x) && (PolyQ(~Px, ~x) && (EqQ((~d) ^ 2 - ~c * ~e, 0) && (ILtQ(~q, 0) && (RationalQ(~p) && EqQ(Denominator(~p), 3)))))
    @apply_utils Antiderivative((~c + ~(d') * (~x) ^ ~(n')) ^ ~q * (~a + ~(b') * (~x) ^ ~(nn')) ^ ~p, ~x) => Antiderivative(ExpandIntegrand((~a + ~b * (~x) ^ ~nn) ^ ~p, (~c / ((~c) ^ 2 - (~d) ^ 2 * (~x) ^ (2 * ~n)) - (~d * (~x) ^ ~n) / ((~c) ^ 2 - (~d) ^ 2 * (~x) ^ (2 * ~n))) ^ -(~q), ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~n, ~nn, ~p], ~x) && (Not(IntegerQ(~p)) && (ILtQ(~q, 0) && IGtQ(log(2, ~nn / ~n), 0)))
    @apply_utils Antiderivative((~(e') * ~x) ^ ~(m') * (~c + ~(d') * (~x) ^ ~(n')) ^ ~q * (~a + ~(b') * (~x) ^ ~(nn')) ^ ~p, ~x) => ((~e * ~x) ^ ~m / (~x) ^ ~m) * Antiderivative(ExpandIntegrand((~x) ^ ~m * (~a + ~b * (~x) ^ ~nn) ^ ~p, (~c / ((~c) ^ 2 - (~d) ^ 2 * (~x) ^ (2 * ~n)) - (~d * (~x) ^ ~n) / ((~c) ^ 2 - (~d) ^ 2 * (~x) ^ (2 * ~n))) ^ -(~q), ~x), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~m, ~n, ~nn, ~p], ~x) && (Not(IntegerQ(~p)) && (ILtQ(~q, 0) && IGtQ(log(2, ~nn / ~n), 0)))
    @apply_utils Antiderivative((~x) ^ ~(m') / (~c + ~(d') * (~x) ^ ~n + ~(e') * sqrt(~a + ~(b') * (~x) ^ ~n)), ~x) => (1 / ~n) * Subst(Antiderivative((~x) ^ ((~m + 1) / ~n - 1) / (~c + ~d * ~x + ~e * sqrt(~a + ~b * ~x)), ~x), ~x, (~x) ^ ~n) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~m, ~n], ~x) && (EqQ(~b * ~c - ~a * ~d, 0) && IntegerQ((~m + 1) / ~n))
    @apply_utils Antiderivative(~(u') / (~c + ~(d') * (~x) ^ ~n + ~(e') * sqrt(~a + ~(b') * (~x) ^ ~n)), ~x) => ~c * Antiderivative(~u / (((~c) ^ 2 - ~a * (~e) ^ 2) + ~c * ~d * (~x) ^ ~n), ~x) - ~a * ~e * Antiderivative(~u / ((((~c) ^ 2 - ~a * (~e) ^ 2) + ~c * ~d * (~x) ^ ~n) * sqrt(~a + ~b * (~x) ^ ~n)), ~x) <-- FreeQ([~a, ~b, ~c, ~d, ~e, ~n], ~x) && EqQ(~b * ~c - ~a * ~d, 0)
end
