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

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

Then:
 * $\displaystyle \int x^2 \sqrt {a x^2 + b x + c} \ \mathrm d x = \frac {6 a x - 5 b} {24 a^2} \left({\sqrt {a x^2 + b x + c} }\right)^3 + \frac {5 b^2 - 4 a c} {16 a^2} \int \sqrt {a x^2 + b x + c} \ \mathrm d x$

Proof
Let:

With a view to expressing the primitive $\displaystyle \int x \left({2 a x + b}\right) \sqrt {a x^2 + b x + c} \ \mathrm d x$ 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:

Now consider:

Thus:

Hence: