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

Theorem

 * $\displaystyle \int x^3 \paren {\sqrt {x^2 + a^2} }^3 \rd x = \frac {\paren {\sqrt {x^2 + a^2} }^7} 7 - \frac {a^2 \paren {\sqrt {x^2 + a^2} }^5} 5 + C$

Proof
Let:

Also see

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