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

Theorem

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

Proof
With a view to expressing the problem in the form:
 * $\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \mathrm d x$

let:

and let:

Then:

Also see

 * Primitive of $\dfrac {x^3} {\sqrt {x^2 - a^2} }$
 * Primitive of $\dfrac {x^3} {\sqrt {a^2 - x^2} }$