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

Theorem

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

Proof
Let:

Also see

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