Primitive of Reciprocal of Root of a x + b by Root of p x + q/a greater than 0, p less than 0

Theorem
Let $a, b, p, q \in \R$ such that $a p \ne b q$. Let $a > 0$ and $p < 0$.

Then:


 * $\ds \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } } = \dfrac {-1} {\sqrt {-a p} } \map \arcsin {\dfrac {2 a p x + b p + a q} {a q - b p} } + C$

for all $x \in \R$ such that $\paren {a x + b} \paren {p x + q} > 0$.

Completing the Square
Hence: