<s>[INST] Analytic solution of $ \int_t^{t+r} \frac{x^2}{(t+r-\sqrt{2rx-x^2})^4}dx $ Could anyone help me to solve this defined integral analytically? 
$$
\int_t^{t+r} \dfrac{x^2}{(t+r-\sqrt{2rx-x^2})^4}dx
$$
 [/INST] The antiderivative of the integrand of the integral under consideration can be found by the Euler changes $\sqrt{2rx-x^2}=tx$ or/and $\sqrt{2rx-x^2}=t(x-2r)$. The output done with Maple by the command $int(x^2/(r+t-2*sqrt(2*r*x-x^2))^4, x)$ is huge. It can be seen in the worksheet exported as a pdf file (20MB). The definite integral under consideration depends on the two parameters $t$ and $r$ so parametric analysis is necessary.
I give only its results:$ t \le r, t \ge r/3$ under the assumption $t \ge 0$. In other cases the integral may diverge because of the singularity caused by the zeros of the denominator of the integrand $r+t-2\sqrt{2rx-x^2}=0$:$$x_1=r+\frac 1 2\,\sqrt {3\,{r}^{2}-2\,rt-{t}^{2}},x_2=r-\frac 1 2\,\sqrt {3\,{r}^{2}-2\,rt-{t}^{2}}.
$$
For concrete values of the parameters the integral under consideration can be found with Maple by the command
$$J:=(t,r)->int(x^2/(r+t-2*sqrt(2*r*x-x^2))^4,x=t..t+r).$$ For example,
$$J(1,2)= {\frac {1440}{2401}}\,{\it arctanh} \left( 2/7\,\sqrt {7} \right)
\sqrt {7}+{\frac {360}{2401}}\,\sqrt {7}\ln  \left( \sqrt {7}\sqrt {3}
+8\,\sqrt {3}-9 \right) -$$ $${\frac {360}{2401}}\,\sqrt {7}\ln  \left(
\sqrt {7}\sqrt {3}+8\,\sqrt {3}+9 \right) +$$ $${\frac {360}{2401}}\,\sqrt
{7}\ln  \left( -\sqrt {7}\sqrt {3}+8\,\sqrt {3}+9 \right) -$$ $${\frac {360
}{2401}}\,\sqrt {7}\ln  \left( -\sqrt {7}\sqrt {3}+8\,\sqrt {3}-9
 \right) +{\frac {27404}{3087}}\,\sqrt {3}+{\frac {15494}{1029}}
$$
and
$$J\left(\frac 1 {10},2\right) = \infty.$$
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