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

Theorem

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

for $\size x \ge a$.

Proof
Let:

Also see

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