<s>[INST] Substitution for $\int \frac {dx} {ax^2 + bx + c}$ I'm looking for the substitution that makes easier to solve integral containing quadratic polynomial in denominator (!) when such polynomial cannot be broken into parts (if it can, then it's possible to use partial fraction decomposition). Example:
$$\int \frac {dx} {5x^2 + x - 2}$$
Formulas suggested by wikipedia are hard to remember. So I hope there is some kind of substitution.
link: http://en.wikipedia.org/wiki/List_of_integrals_of_rational_functions#Integrands_of_the_form_xm_.2F_.28a_x2_.2B_b_x_.2B_c.29n
 [/INST] HINT:
$$\frac1{ax^2+bx+c}=\frac{4a}{(2ax+b)^2+4ca-b^2}$$
Case $1:$ If $4ca-b^2=0,$ $$\int \frac{dx}{ax^2+bx+c}=4a\int\frac{dx}{(2ax+b)^2} $$ Put $y=2ax+b$
Case $2:$ If $4ca-b^2>0, 4ca-b^2=d^2$(say)
$$\int \frac{dx}{ax^2+bx+c}=4a\int\frac{dx}{(2ax+b)^2+d^2} $$ 
Put $2ax+b=d\tan\theta$ as $\tan^2\theta+1=\sec^2\theta$
Case $3:$  If $4ca-b^2<0, 4ca-b^2=-d^2$(say)
$$\int \frac{dx}{ax^2+bx+c}=4a\int\frac{dx}{(2ax+b)^2-d^2} $$ 
Put $2ax+b=d\sec\theta$  as $\sec^2\theta-1=\tan^2\theta$
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