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

Theorem

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

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

let:

and let:

Then:

Also see

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