Primitives of Functions involving Power of Root of a x + b

Theorem

This page gathers together the primitives of some rational functions involving $\sqrt {a x + b}$.

Primitive of $\left({\sqrt {a x + b} }\right)^m$

$\ds \int \paren {\sqrt {a x + b} }^m \rd x = \frac {2 \paren {\sqrt {a x + b} }^{m + 2} } {a \paren {m + 2} } + C$

for $m \ne -2$.

Primitive of $x \left({\sqrt {a x + b} }\right)^m$

$\ds \int x \paren {\sqrt {a x + b} }^m \rd x = \frac {2 \paren {\sqrt {a x + b} }^{m + 4} } {a^2 \paren {m + 4} } - \frac {2 b \paren {\sqrt {a x + b} }^{m + 2} } {a^2 \paren {m + 2} } + C$

Primitive of $x^2 \left({\sqrt {a x + b} }\right)^m$

$\ds \int x^2 \paren {\sqrt {a x + b} }^m \rd x = \frac {2 \paren {\sqrt {a x + b} }^{m + 6} } {a^3 \paren {m + 6} } - \frac {4 b \paren {\sqrt {a x + b} }^{m + 4} } {a^3 \paren {m + 4} } + \frac {2 b^2 \paren {\sqrt {a x + b} }^{m + 2} } {a^3 \paren {m + 2} } + C$

Primitive of $\dfrac {\left({\sqrt {a x + b} }\right)^m} x$

$\ds \int \frac {\paren {\sqrt{a x + b} }^m} x \rd x = \frac {2 \paren {\sqrt{a x + b} }^m } m + b \int \frac {\paren {\sqrt{a x + b} }^{m - 2} } x \rd x$

Primitive of $\dfrac {\left({\sqrt {a x + b} }\right)^m} {x^2}$

$\ds \int \frac {\paren {\sqrt {a x + b} }^m} {x^2} \rd x = -\frac {\paren {\sqrt {a x + b} }^{m + 2} } {b x} + \frac {m a} {2 b} \int \frac {\paren {\sqrt {a x + b} }^m} x \rd x$

Primitive of $\dfrac {\left({\sqrt {a x + b} }\right)^m} {x^2}$

$\ds \int \frac {\paren {\sqrt {a x + b} }^m} {x^2} \rd x = -\frac {\paren {\sqrt {a x + b} }^{m + 2} } {b x} + \frac {m a} {2 b} \int \frac {\paren {\sqrt {a x + b} }^m} x \rd x$

Primitive of $\dfrac 1 {x \left({\sqrt {a x + b} }\right)^m}$

$\ds \int \frac {\d x} {x \paren {\sqrt {a x + b} }^m} = \frac 2 {\paren {m - 2} b \paren {\sqrt {a x + b} }^{m - 2} } + \frac 1 b \int \frac {\d x} {x \paren {\sqrt {a x + b} }^{m - 2} }$

Also see

• Primitives of Functions involving $\sqrt {a x + b}$