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

Source Work

1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions

$3$: Elementary Analytic Methods

$3.3$ Rules for Differentiation and Integration:

Integrals of Irrational Algebraic Functions: $3.3.28$

Mistake

$\ds \int \frac {\d x} {\sqbrk {\paren {a + b x} \paren {c + d x} }^{1/2} } = \dfrac 2 {\paren {b d}^{1/2} } \ln \size {\sqbrk {b d \paren {a + b x} }^{1/2} + \sqbrk {b \paren {c + d x} }^{1/2} } + C$

Correction

This should read:

$\ds \int \frac {\d x} {\sqbrk {\paren {a + b x} \paren {c + d x} }^{1/2} } = \dfrac 2 {\paren {b d}^{1/2} } \ln \size {\sqbrk {d \paren {a + b x} }^{1/2} + \sqbrk {b \paren {c + d x} }^{1/2} } + C$

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: $3.3.28$