Primitive of Reciprocal of Cube of Root of a x squared plus b x plus c

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

Then:
 * $\displaystyle \int \frac {\mathrm d x} {\left({\sqrt {a x^2 + b x + c} }\right)^3} = \frac {2 \left({2 a x + b}\right)} {\left({4 a c - b^2}\right) \sqrt {a x^2 + b x + c} } + C$

Proof
Then:

Let $4 a c - b^2 > 0$.

Then:

Let $4 a c - b^2 < 0$.

Then:

Thus in both cases the same result applies, and so: