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

Theorem

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

for $\size x \ge a$.

Proof
Let:

Also see

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