Primitive of x squared by Root of a x + b

Theorem

 * $\displaystyle \int x^2 \sqrt{a x + b} \ \mathrm d x = \frac {2 \left({15 a^2 x^2 - 12 a b x + 8 b^2}\right) } {105 a^3} \sqrt{\left({a x + b}\right)^3}$

Proof
Let:

Then: