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

Theorem

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

for $\size x \ge a$.

Proof
Let:

Also see

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