Primitive of p x + q over Root of a x + b

Theorem

 * $\ds \int \frac {p x + q} {\sqrt {a x + b} } \rd x = \frac {2 \paren {a p x + 3 a q - 2 b p} } {3 a^2} \sqrt{a x + b}$

Proof
Let:

Then: