Primitive of Power of x over Power of a x squared plus b x plus c

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

Then:

Proof
With a view to expressing the problem in the form:
 * $\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \mathrm d x$

let:

and let:

Then:

Meanwhile: