Primitive of x squared by Root of a squared minus x squared

Theorem

 * $\displaystyle \int x^2 \sqrt {a^2 - x^2} \ \mathrm d x = \frac {-x \left({\sqrt {a^2 - x^2} }\right)^3} 4 + \frac {a^2 x \sqrt {a^2 - x^2} } 8 + \frac {a^4} 8 \sinh^{-1} \frac x a + C$

Proof
Let:

Also see

 * Primitive of $x^2 \sqrt{x^2 + a^2}$
 * Primitive of $x^2 \sqrt{x^2 - a^2}$