Primitive of x squared by Root of a x squared plus b x plus c

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

Then:
 * $\displaystyle \int x^2 \sqrt {a x^2 + b x + c} \rd x = \frac {6 a x - 5 b} {24 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {5 b^2 - 4 a c} {16 a^2} \int \sqrt {a x^2 + b x + c} \rd x$

Proof
Let:

With a view to expressing the primitive $\displaystyle \int x \paren {2 a x + b} \sqrt {a x^2 + b x + c} \rd x$ in the form:
 * $\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

and let:

Then:

Now consider:

Thus:

Hence: