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

:$\ds\int \frac {\d x} {x \paren {a x + b}^3} = \frac {a^2 x^2} {2 b^3 \paren {a x + b}^2} - \frac {2 a x} {b^3 \paren {a x + b} } + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C$

Proof:
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x \paren {a x + b}^3}
      | r = \int \paren {\frac 1 {b^3 x} - \frac a {b^3 \paren {a x + b} } - \frac a {b^2 \paren {a x + b}^2} - \frac a {b \paren {a x + b}^3} } \rd x
      | c = Partial Fraction Expansion
}}
{{eqn | r = \frac 1 {b^3} \int \frac {\d x} x - \frac a {b^3} \int \frac {\d x} {a x + b} - \frac a {b^2} \int \frac {\d x} {\paren {a x + b}^2} - \frac a b \int \frac {\d x} {\paren {a x + b}^3}
      | c = Linear Combination of Primitives
}}
{{eqn | r = \frac 1 {b^3} \ln \size x - \frac a {b^3} \int \frac {\d x} {a x + b} - \frac a {b^2} \int \frac {\d x} {\paren {a x + b}^2} - \frac a b \int \frac {\d x} {\paren {a x + b}^3} + C
      | c = Primitive of Reciprocal
}}
{{eqn | r = \frac 1 {b^3} \ln \size x - \frac 1 {b^3} \ln \size {a x + b} - \frac a {b^2} \int \frac {\d x} {\paren {a x + b}^2} - \frac a b \int \frac {\d x} {\paren {a x + b}^3} + C
      | c = Primitive of $\dfrac 1 {a x + b}$
}}
{{eqn | r = \frac 1 {b^3} \ln \size x - \frac 1 {b^3} \ln \size {a x + b} - \frac a {b^2} \frac {-1} {a \paren {a x + b} } - \frac a b \int \frac {\d x} {\paren {a x + b}^3} + C
      | c = Primitive of $\dfrac 1 {\paren {a x + b}^2}$
}}
{{eqn | r = \frac 1 {b^3} \ln \size x - \frac 1 {b^3} \ln \size {a x + b} - \frac a {b^2} \frac {-1} {a \paren {a x + b} } - \frac a b \frac {-1} {2 a \paren {a x + b}^2} + C
      | c = Primitive of $\dfrac 1 {\paren {a x + b}^3}$
}}
{{eqn | r = \frac 1 {b^2 \paren {a x + b} } + \frac 1 {2 b \paren {a x + b}^2} + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C
      | c = Difference of Logarithms and rearranging
}}
{{eqn | r = \frac {b} {b^3 \paren {a x + b} } + \frac {b^2} {2 b^3 \paren {a x + b}^2} + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C
      | c = 
}}
{{eqn | r = \paren {\frac {a x + b} {b^3 \paren {a x + b} } - \frac {a x} {b^3 \paren {a x + b} } } + \paren {\frac {a^2 x^2 + 2 a b x + b^2} {2 b^3 \paren {a x + b}^2} - \frac {a^2 x^2 + 2 a b x} {2 b^3 \paren {a x + b}^2} } + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C
      | c = 
}}
{{eqn | r = \frac {a x + b} {b^3 \paren {a x + b} } - \frac {a x} {b^3 \paren {a x + b} } + \frac {\paren {a x + b}^2} {2 b^3 \paren {a x + b}^2} - \frac {a^2 x^2 + 2 a b x} {2 b^3 \paren {a x + b}^2} + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C
      | c = 
}}
{{eqn | r = \frac 1 {b^3} - \frac {a x} {b^3 \paren {a x + b} } + \frac 1 {2 b^3} - \frac {a^2 x^2 + 2 a b x} {2 b^3 \paren {a x + b}^2} + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C
      | c = 
}}
{{eqn | r = -\frac {a x} {b^3 \paren {a x + b} } - \frac {a^2 x^2 + 2 a b x} {2 b^3 \paren {a x + b}^2} + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C + \frac 3 {2 b^3}
      | c = 
}}
{{eqn | r = -\frac {a x} {b^3 \paren {a x + b} } - \frac {a^2 x^2} {2 b^3 \paren {a x + b}^2} - \frac {2 a b x} {2 b^3 \paren {a x + b}^2} + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C + \frac 3 {2 b^3}
      | c = 
}}
{{eqn | r = -\frac {a x} {b^3 \paren {a x + b} } - \frac {a^2 x^2} {2 b^3 \paren {a x + b}^2} - \frac {2 a x \paren {a x + b} } {2 b^3 \paren {a x + b}^2} + \frac {2 a^2 x^2} {2 b^3 \paren {a x + b}^2} + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C + \frac 3 {2 b^3}
      | c = 
}}
{{eqn | r = -\frac {a x} {b^3 \paren {a x + b} } - \frac {a^2 x^2} {2 b^3 \paren {a x + b}^2} - \frac {2 a x} {2 b^3 \paren {a x + b} } + \frac {2 a^2 x^2} {2 b^3 \paren {a x + b}2} + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C + \frac 3 {2 b^3}
      | c = 
}}
{{eqn | r = -\frac {a x} {b^3 \paren {a x + b} } + \frac {a^2 x^2} {2 b^3 \paren {a x + b}^2} - \frac {a x} {b^3 \paren {a x + b} } + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C + \frac 3 {2 b^3}
      | c = 
}}
{{eqn | r = \frac {a^2 x^2} {2 b^3 \paren {a x + b}^2} - \frac {2 a x} {b^3 \paren {a x + b} } + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C + \frac 3 {2 b^3}
      | c = 
}}
{{eqn | r = \frac {a^2 x^2} {2 b^3 \paren {a x + b}^2} - \frac {2 a x} {b^3 \paren {a x + b} } + \frac 1 {b^3} \ln \size {\frac x {a x + b} } + C
      | c = $C + \dfrac 3 {2 b^3}$ is constant
}}
{{end-eqn}}
{{qed}}