Primitives of Functions involving a x + b and p x + q

Theorem

This page gathers together the primitives of some rational functions involving $a x + b$ and $p x + q$.

Primitive of Reciprocal of $\left({a x + b}\right) \left({p x + q}\right)$

$\ds \int \frac {\d x} {\paren {a x + b} \paren {p x + q} } = \frac 1 {b p - a q} \ln \size {\frac {p x + q} {a x + b} } + C$

where $b p \ne a q$.

Primitive of $x$ over $\left({a x + b}\right) \left({p x + q}\right)$

$\ds \int \frac {x \rd x} {\paren {a x + b} \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac b a \ln \size {a x + b} - \frac q p \ln \size {p x + q} } + C$

Primitive of Reciprocal of $\left({a x + b}\right)^2 \left({p x + q}\right)$

$\ds \int \frac {\d x} {\paren {a x + b}^2 \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac 1 {a x + b} + \frac p {b p - a q} \ln \size {\frac {p x + q} {a x + b} } } + C$

Primitive of $x$ over $\left({a x + b}\right)^2 \left({p x + q}\right)$

$\ds \int \frac {x \rd x} {\paren {a x + b}^2 \paren {p x + q} } = \frac 1 {b p - a q} \paren {\frac q {b p - a q} \ln \size {\frac {a x + b} {p x + q} } - \frac b {a \paren {a x + b} } } + C$

Primitive of $x^2$ over $\left({a x + b}\right)^2 \left({p x + q}\right)$

$\ds \int \frac {x^2 \rd x} {\paren {a x + b}^2 \paren {p x + q} } = \frac {b^2} {\paren {b p - a q} a^2 \paren {a x + b} } + \frac 1 {\paren {b p - a q}^2} \paren {\frac {q^2} p \ln \size {p x + q} + \frac {b \paren {b p - 2 a q} } {a^2} \ln \size {a x + b} } + C$

Primitive of Reciprocal of $\left({a x + b}\right)^m \left({p x + q}\right)^n$

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

Primitive of $a x + b$ over $p x + q$

$\ds \int \frac {a x + b} {p x + q} \rd x = \frac {a x} p + \frac {b p - a q} {p^2} \ln \size {p x + q} + C$

Primitive of $\left({a x + b}\right)^m$ over $\left({p x + q}\right)^n$

Formulation 1

$\ds \int \frac {\paren {a x + b}^m} {\paren {p x + q}^n} \rd x = \frac {-1} {\paren {n - 1} \paren {b p - a q} } \paren {\frac {\paren {a x + b}^{m + 1} } {\paren {p x + q}^{n - 1} } + \paren {n - m - 2} a \int \frac {\paren {a x + b}^m} {\paren {p x + q}^{n - 1} } \rd x}$

Formulation 2

$\ds \int \frac {\paren {a x + b}^m} {\paren {p x + q}^n} \rd x = \frac {-1} {\paren {n - m - 1} p} \paren {\frac {\paren {a x + b}^m} {\paren {p x + q}^{n - 1} } + m \paren {b p - a q} \int \frac {\paren {a x + b}^{m - 1} } {\paren {p x + q}^n} \rd x}$

Formulation 3

$\ds \int \frac {\paren {a x + b}^m} {\paren {p x + q}^n} \rd x = \frac {-1} {\paren {n - 1} p} \paren {\frac {\paren {a x + b}^m} {\paren {p x + q}^{n - 1} } - m a \int \frac {\paren {a x + b}^{m - 1} } {\paren {p x + q}^{n - 1}} \rd x}$