Primitive of Reciprocal of Power of x by Power of a x squared plus b x plus c

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

Then:

Proof
First:

Next, with a view to obtaining an expression 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:

Thus: