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

Theorem

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

Proof
Let:

Also see

 * Primitive of $\dfrac{\left({\sqrt {x^2 - a^2} }\right)^3} {x^3}$
 * Primitive of $\dfrac{\left({\sqrt {a^2 - x^2} }\right)^3} {x^3}$