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

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

Then:
 * $\displaystyle \int \left({a x^2 + b x + c}\right)^{n + \frac 1 2} \ \mathrm d x = \frac {\left({2 a x + b}\right) \left({a x^2 + b x + c}\right)^{n + \frac 1 2} } {4 a \left({n + 1}\right)} + \frac {\left({2 n + 1}\right) \left({4 a c - b^2}\right)} {8 a \left({n + 1}\right)} \int \left({a x^2 + b x + c}\right)^{n - \frac 1 2} \ \mathrm d x$