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

Theorem

 * $\ds \int x^2 \sqrt {x^2 + a^2} \rd x = \frac {x \paren {\sqrt {x^2 + a^2} }^3} 4 - \frac {a^2 x \sqrt {x^2 + a^2} } 8 - \frac {a^4} 8 \map \ln {x + \sqrt {x^2 + a^2} } + C$

Proof
Let:

Also see

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