Integral with $ax^2+bx+c$

29

$$\int \frac{1}{ax^2+bx+c}\ \mathrm{d}x= \left\{\begin{aligned} &\frac{2}{\sqrt{4ac-b^2}}\arctan{\frac{2ax+b}{\sqrt{4ac-b^2}}}+C \left(4ac>{b^2}\right)\\ &\frac{1}{\sqrt{b^2-4ac}}\ln {\left|\frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right|}+C\left(4ac<{b^2}\right) \end{aligned} \right.$$

Derivation

30

$$\int \frac{x}{ax^2+bx+c}\ \mathrm{d}x=\frac{1}{2a}\ln{\left|ax^2+bx+c\right|}-\frac{b}{2a}\int \frac{1}{ax^2+bx+c}\ \mathrm{d}x$$

Derivation