How find the following integral?

Question

I want find a closed Form for below integral$$\int \frac{1}{-1-aX+\frac{1}{2}bX^2} dX$$.

thanks for help

Answer 1

Hint

Complete the square first $$\frac{b}{2}X^2-a X-1=\frac{b}{2}\Big(X^2-\frac{2a}{b}X-\frac{2}{b}\Big)=\frac{b}{2}\Big((X-\frac ab)^2-(\frac {a^2}{b^2}+\frac 2b)\Big)$$ In the most general case, the change of variable becomes clear.

I am sure that you can take from here.

Answer 2

HINT: If $b=0$, then (moved and scaled) logaritm, if $a^2+2b<0$, then (moved and scaled) arctan, if $a^2+2b=0$, then (moved and scaled) $1/x$, if $a^2+2b>0$, then sum of two (moved and scaled) logarithms.