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

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

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^3 \paren {a x + b}^2}
      | r = \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
      | c = Partial Fraction Expansion
}}
{{eqn | r = \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}
      | c = Linear Combination of Primitives
}}
{{eqn | r = \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
      | c = Primitive of Power
}}
{{eqn | r = \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
      | c = Primitive of Reciprocal
}}
{{eqn | r = \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
      | c = Primitive of Reciprocal of a x + b
}}
{{eqn | r = \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
      | c = Primitive of Reciprocal of a x + b squared
}}
{{eqn | r = \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
      | c = Difference of Logarithms
}}
{{eqn | r = \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 = C is an arbitrary constant
}}
{{eqn | r = \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
      | c = Represent with common denominators
}}
{{eqn | r = \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
      | c = Combine fractions
}}
{{eqn | r = \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
      | c = Expansion
}}
{{eqn | r = \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
      | c = Factorisation
}}
{{eqn | r = \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
      | c = Split into separate fractions
}}
{{eqn | r = \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
      | c = Simplify
}}
{{end-eqn}}
{{qed}}