Primitive of x by Half Integer Power of a x squared plus b x plus c

Theorem
Let $a \in \R_{\ne 0}$.

Then:
 * $\displaystyle \int x \left({a x^2 + b x + c}\right)^{n + \frac 1 2} \ \mathrm d x = \frac {\left({a x^2 + b x + c}\right)^{n + \frac 3 2} } {a \left({2 n + 3}\right)} - \frac b {2 a} \int \left({a x^2 + b x + c}\right)^{n + \frac 1 2} \ \mathrm d x$

Proof
Let:

So: