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

Theorem

 * $\displaystyle \int x^3 \left({\sqrt {x^2 - a^2} }\right)^3 \ \mathrm d x = \frac {\left({\sqrt {x^2 - a^2} }\right)^7} 7 + \frac {a^2 \left({\sqrt {x^2 - a^2} }\right)^5} 5 + C$

Proof
Let:

Also see

 * Primitive of $x^3 \left({\sqrt {x^2 + a^2} }\right)^3$
 * Primitive of $x^3 \left({\sqrt {a^2 - z^2} }\right)^3$