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

Theorem

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

Proof
Let: