Primitive of Root of a x + b by Root of p x + q/Also presented as

Primitive of $\sqrt {\paren {a x + b} \paren {p x + q} }$: Also presented as

This result can also be seen presented in this form:

$\ds \int \sqrt {\paren {a x + b} \paren {p x + q} } \rd x = \frac {\paren {b p - a q} + 2 a {p x + q} } {4 a p} \sqrt {\paren {a x + b} \paren {p x + q} } - \frac {\paren {b p - a q}^2} {8 a p} \int \frac {\d x} {\sqrt {\paren {a x + b} \paren {p x + q} } }$

Sources

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.3$ Rules for Differentiation and Integration: Integrals of Irrational Algebraic Functions: $3.3.31$