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

Theorem

 * $\ds \int x^2 \paren {\sqrt {a^2 - x^2} }^3 \rd x = \frac {-x \paren {\sqrt {a^2 - x^2} }^5} 6 + \frac {a^2 x \paren {\sqrt {a^2 - x^2} }^3} {24} + \frac {a^4 x \sqrt {a^2 - x^2} } {16} + \frac {a^6} {16} \arcsin \frac x a + C$

Proof
Let:

Also see

 * Primitive of $x^2 \paren {\sqrt {x^2 + a^2} }^3$
 * Primitive of $x^2 \paren {\sqrt {x^2 - a^2} }^3$