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

Theorem

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

Proof
Let: