Primitive of Reciprocal of x cubed by a x + b squared

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Theorem

$\ds \int \frac {\d x} {x^3 \paren {a x + b}^2} = - \frac {\paren {a x + b}^2} {2 b^4 x^2} + \frac {3 a \paren {a x + b} } {b^4 x} - \frac {a^3 x} {b^4 \paren {a x + b} } + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C$


Proof

\(\ds \int \frac {\d x} {x^3 \paren {a x + b}^2}\) \(=\) \(\ds \int \paren {\frac {3 a^2} {b^4 x} + \frac {-2 a} {b^3 x^2} + \frac 1 {b^2 x^3} + \frac {-3 a^3} {b^4 \paren {a x + b} } + \frac {-a^3} {b^3 \paren {a x + b}^2} } \rd x\) Partial Fraction Expansion
\(\ds \) \(=\) \(\ds \frac {3 a^2} {b^4} \int \frac {\d x} x + \frac {-2 a} {b^3} \int \frac {\d x} {x^2} + \frac 1 {b^2} \int \frac {\d x} {x^3} + \frac {-3 a^3} {b^4} \int \frac {\d x} {a x + b} + \frac {-a^3} {b^3} \int \frac {\d x} {\paren {a x + b}^2}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac {3 a^2} {b^4} \int \frac {\d x} x + \frac {-2 a} {b^3} \frac {-1} x + \frac 1 {b^2} \frac {-1} {2 x^2} + \frac {-3 a^3} {b^4} \int \frac {\d x} {a x + b} + \frac {-a^3} {b^3} \int \frac {\d x} {\paren {a x + b}^2} + C\) Primitive of Power
\(\ds \) \(=\) \(\ds \frac {3 a^2} {b^4} \ln \size x + \frac {2 a} {b^3 x} - \frac 1 {2 b^2 x^2} + \frac {-3 a^3} {b^4} \int \frac {\d x} {a x + b} + \frac {-a^3} {b^3} \int \frac {\d x} {\paren {a x + b}^2} + C\) Primitive of Reciprocal
\(\ds \) \(=\) \(\ds \frac {3 a^2} {b^4} \ln \size x + \frac {2 a} {b^3 x} - \frac 1 {2 b^2 x^2} + \frac {-3 a^3} {b^4} \frac 1 a \ln \size {a x + b} + \frac {-a^3} {b^3} \int \frac {\d x} {\paren {a x + b}^2} + C\) Primitive of Reciprocal of a x + b
\(\ds \) \(=\) \(\ds \frac {3 a^2} {b^4} \ln \size x + \frac {2 a} {b^3 x} - \frac 1 {2 b^2 x^2} + \frac {-3 a^2} {b^4} \ln \size {a x + b} + \frac {-a^3} {b^3} \frac {-1} {a \paren {a x + b} } + C\) Primitive of Reciprocal of a x + b squared
\(\ds \) \(=\) \(\ds \frac {2 a} {b^3 x} - \frac 1 {2 b^2 x^2} + \frac {a^2} {b^3 \paren {a x + b} } + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C\) Difference of Logarithms
\(\ds \) \(=\) \(\ds \frac {2 a} {b^3 x} - \frac 1 {2 b^2 x^2} + \frac {a^2} {b^3 \paren {a x + b} } + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + \frac {3 a^2} {2 b^4} + C\) C is an arbitrary constant
\(\ds \) \(=\) \(\ds \frac {4 a b \paren {a x + b} x} {2 b^4 \paren {a x + b} x^2} - \frac {b^2 \paren {a x + b} } {2 b^4 \paren {a x + b} x^2} + \frac {2 a^2 b x^2} {2 b^4 \paren {a x + b} x^2} + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + \frac {3 a^2 \paren {a x + b} x^2} {2 b^4 \paren {a x + b} x^2} + C\) Represent with common denominators
\(\ds \) \(=\) \(\ds \frac {4 a b \paren {a x + b} x - b^2 \paren {a x + b} + 2 a^2 b x^2 + 3 a^2 \paren {a x + b} x^2} {2 b^4 \paren {a x + b} x^2} + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C\) Combine fractions
\(\ds \) \(=\) \(\ds \frac {4 a^2 b x^2 + 4 a b^2 x - a b^2 x - b^3 + 2 a^2 b x^2 + 3 a^3 x^3 + 3 a^2 b x^2} {2 b^4 \paren {a x + b} x^2} + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C\) Expansion
\(\ds \) \(=\) \(\ds \frac {6 a \paren {a x + b}^2 x - \paren {a x + b}^3 - 2 a^3 x^3} {2 b^4 \paren {a x + b} x^2} + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C\) Factorisation
\(\ds \) \(=\) \(\ds \frac {6 a \paren {a x + b}^2 x} {2 b^4 \paren {a x + b} x^2} - \frac {\paren {a x + b}^3} {2 b^4 \paren {a x + b} x^2} - \frac {2 a^3 x^3} {2 b^4 \paren {a x + b} x^2} + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C\) Split into separate fractions
\(\ds \) \(=\) \(\ds \frac {3 a \paren {a x + b} } {b^4 x} - \frac {\paren {a x + b}^2} {2 b^4 x^2} - \frac {a^3 x} {b^4 \paren {a x + b} } + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C\) Simplify

$\blacksquare$


Sources