Primitive of x by Root of a x squared plus b x plus c/Lemma

Lemma for Primitive of $x \sqrt {a x^2 + b x + c}$
Let $a \in \R_{\ne 0}$.

Then:
 * $\displaystyle \int x \sqrt {a x^2 + b x + c} \ \mathrm d x = \frac {\left({\sqrt {a x^2 + b x + c} }\right)^3} {3 a} - \frac b {2 a} \int \sqrt {a x^2 + b x + c} \ \mathrm d x$

Proof
Let:

So: