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

Theorem

 * $\displaystyle \int x^2 \left({\sqrt {x^2 + a^2} }\right)^3 \ \mathrm d x = \frac {x \left({\sqrt {x^2 + a^2} }\right)^5} 6 - \frac {a^2 x \left({\sqrt {x^2 + a^2} }\right)^3} {24} - \frac {a^4 x \sqrt {x^2 + a^2} } {16} - \frac {a^6} {16} \ln \left({x + \sqrt {x^2 + a^2} }\right) + C$

Proof
Let: