Primitive of Root of a x + b over Power of x

Theorem

Formulation 1

$\displaystyle \int \frac {\sqrt{a x + b} } {x^m} \ \mathrm d x = -\frac {\sqrt{a x + b} } {\left({m - 1}\right) x^{m-1} } + \frac a {2 \left({m - 1}\right)} \int \frac {\mathrm d x} {x^{m - 1} \sqrt{a x + b} }$

Formulation 2

$\displaystyle \int \frac {\sqrt{a x + b} } {x^m} \ \mathrm d x = -\frac {\left({\sqrt{a x + b} }\right)^3} {\left({m - 1}\right) b x^{m-1} } - \frac {\left({2 m - 5}\right) a} {\left({2 m - 2}\right) b} \int \frac {\sqrt{a x + b} } {x^{m - 1} } \ \mathrm d x$