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:
 * $\ds \int x \sqrt {a x^2 + b x + c} \rd x = \frac {\paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {2 a} \int \sqrt {a x^2 + b x + c} \rd x$

Proof
Let:

So: