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

Theorem

 * $\ds \int \frac {\d x} {\paren {p x + q} \sqrt {a x + b} } = \begin {cases}

\dfrac 1 {\sqrt {b p - a q} \sqrt p} \ln \size {\dfrac {\sqrt {p \paren {a x + b} } - \sqrt {b p - a q} } {\sqrt {p \paren {a x + b} } + \sqrt {b p - a q} } } & : p \paren {b p - a q} > 0 \\ \dfrac 2 {\sqrt {a q - b p} \sqrt p} \arctan \sqrt {\dfrac {p \paren {a x + b} } {a q - b p} } & : p \paren {b p - a q} < 0 \\ \end {cases}$

Proof
Let:

Then:

Now let:

Thus:

Let $b p - a q > 0$.

Let $d = \sqrt {b p - a q}$.

Then:

Let $b p - a q < 0$.

Let $d = \sqrt {-\paren {b p - a q} } = \sqrt {a q - b p}$.

Then: