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

Theorem

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

Proof
Let:

Also see

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