Primitive of Reciprocal of p x + q by Root of a x + b/p (b p - a q) less than 0/Also presented as

Primitive of $\frac 1 {\paren {p x + q} \sqrt {a x + b} }$ where $p \paren {b p - a q} < 0$: Also presented as

This result can also be seen presented as:

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

but this presupposes both that $p > 0$ and $b p - a q < 0$.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {a x + b}$ and $p x + q$: $14.114$