Primitive of Root of x squared minus a squared cubed

Theorem

 * $\ds \int \paren {\sqrt {x^2 - a^2} }^3 \rd x = \frac {x \paren {\sqrt {x^2 - a^2} }^3} 4 - \frac {3 a^2 x \sqrt {x^2 - a^2} } 8 + \frac {3 a^4} 8 \ln \size {x + \sqrt {x^2 - a^2} } + C$

for $\size x \ge a$.

Proof
Let:

Also see

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