Primitive of Reciprocal of Root of a x squared plus b x plus c/Examples/2 x + x^2

Example of Use of Primitive of $\dfrac 1 {a x^2 + b x + c}$

 * $\ds \int \dfrac {\d x} {2 x + x^2} = $

Proof
We aim to use Primitive of $\dfrac 1 {a x^2 + b x + c}$ with:

We note that:

Hence from Primitive of $\dfrac 1 {a x^2 + b x + c}$:
 * $\ds \int \frac {\d x} {a x^2 + b x + c} = \dfrac 1 {\sqrt a} \ln \size {2 \sqrt a \sqrt {a x^2 + b x + c} + 2 a x + b} + C$

Substituting for $a$, $b$ and $c$ and simplifying: