Primitive of Reciprocal of p x + q by Root of a x + b/Lemma

Lemma for Primitive of $\frac 1 {\paren {p x + q} \sqrt {a x + b} }$
Let $a, b, p, q \in \R$ such that $a p \ne b q$ and such that $p \ne 0$.

Then:
 * $\ds \int \frac {\d x} {\paren {p x + q} \sqrt {a x + b} } = \dfrac 2 p \int \frac {\d u} {u^2 - \paren {\dfrac {b p - a q} p} }$

where:
 * $u = \sqrt {a x + b}$

Proof
Let:

and:

Then: