Primitive of Root of x squared plus a squared cubed

Theorem

 * $\displaystyle \int \left({\sqrt {x^2 + a^2} }\right)^3 \ \mathrm d x = \frac {x \left({\sqrt {x^2 + a^2} }\right)^3} 4 + \frac {3 a^2 x \sqrt {x^2 + a^2} } 8 + \frac {3 a^4} 8 \ln \left({x + \sqrt {x^2 + a^2} }\right) + C$

Proof
Let: