{{begin-eqn}} {{eqn | l = \int \frac {\mathrm d x} {x^3 \left({a x + b}\right)^2}       | r = \int \left({\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 \left({a x + b}\right)} + \frac {-a^3} {b^3 \left({a x + b}\right)^2} }\right) \ \mathrm d x       | c = Partial Fraction Expansion }} {{eqn | r = \frac {3 a^2} {b^4} \int \frac {\mathrm d x} x + \frac {-2 a} {b^3} \int \frac {\mathrm d x} {x^2} + \frac 1 {b^2} \int \frac {\mathrm d x} {x^3} + \frac {-3 a^3} {b^4} \int \frac {\mathrm d x} {a x + b} + \frac {-a^3} {b^3} \int \frac {\mathrm d x} {\left({a x + b}\right)^2}       | c = Linear Combination of Integrals }} {{eqn | r = \frac {3 a^2} {b^4} \int \frac {\mathrm 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 {\mathrm d x} {a x + b} + \frac {-a^3} {b^3} \int \frac {\mathrm d x} {\left({a x + b}\right)^2} + C       | c = Primitive of Power }} {{eqn | r = \frac {3 a^2} {b^4} \ln \left\vert{x}\right\vert + \frac {2 a} {b^3 x} - \frac 1 {2 b^2 x^2} + \frac {-3 a^3} {b^4} \int \frac {\mathrm d x} {a x + b} + \frac {-a^3} {b^3} \int \frac {\mathrm d x} {\left({a x + b}\right)^2} + C       | c = Primitive of Reciprocal }} {{eqn | r = \frac {3 a^2} {b^4} \ln \left\vert{x}\right\vert + \frac {2 a} {b^3 x} - \frac 1 {2 b^2 x^2} + \frac {-3 a^3} {b^4} \frac 1 a \ln \left\vert{a x + b}\right\vert + \frac {-a^3} {b^3} \int \frac {\mathrm d x} {\left({a x + b}\right)^2} + C       | c = Primitive of Reciprocal of a x + b }} {{eqn | r = \frac {3 a^2} {b^4} \ln \left\vert{x}\right\vert + \frac {2 a} {b^3 x} - \frac 1 {2 b^2 x^2} + \frac {-3 a^2} {b^4} \ln \left\vert{a x + b}\right\vert + \frac {-a^3} {b^3} \frac {-1} {a \left({a x + b}\right)} + 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 \left({a x + b}\right)} + \frac {3 a^2} {b^4} \ln \left\vert{\frac x {a x + b} }\right\vert + 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 \left({a x + b}\right)} + \frac {3 a^2} {b^4} \ln \left\vert{\frac x {a x + b} }\right\vert + \frac {3 a^2} {2 b^4} + C       | c = C is an arbitrary constant }} {{eqn | r = \frac {4 a b \left({a x + b}\right) x} {2 b^4 \left({a x + b}\right) x^2} - \frac {b^2 \left({a x + b}\right)} {2 b^4 \left({a x + b}\right) x^2} + \frac {2 a^2 b x^2} {2 b^4 \left({a x + b}\right) x^2} + \frac {3 a^2} {b^4} \ln \left\vert{\frac x {a x + b} }\right\vert + \frac {3 a^2 \left({a x + b}\right) x^2} {2 b^4 \left({a x + b}\right) x^2} + C       | c = Represent with common denominators }} {{eqn | r = \frac {4 a b \left({a x + b}\right) x - b^2 \left({a x + b}\right) + 2 a^2 b x^2 + 3 a^2 \left({a x + b}\right) x^2} {2 b^4 \left({a x + b}\right) x^2} + \frac {3 a^2} {b^4} \ln \left\vert{\frac x {a x + b} }\right\vert + 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 \left({a x + b}\right) x^2} + \frac {3 a^2} {b^4} \ln \left\vert{\frac x {a x + b} }\right\vert + C       | c = Expansion }} {{eqn | r = \frac {6 a \left({a x + b}\right)^2 x - \left({a x + b}\right)^3 - 2 a^3 x^3} {2 b^4 \left({a x + b}\right) x^2} + \frac {3 a^2} {b^4} \ln \left\vert{\frac x {a x + b} }\right\vert + C       | c = Factorisation }} {{eqn | r = \frac {6 a \left({a x + b}\right)^2 x} {2 b^4 \left({a x + b}\right) x^2} - \frac {\left({a x + b}\right)^3} {2 b^4 \left({a x + b}\right) x^2} - \frac {2 a^3 x^3} {2 b^4 \left({a x + b}\right) x^2} + \frac {3 a^2} {b^4} \ln \left\vert{\frac x {a x + b} }\right\vert + C       | c = Split into separate fractions }} {{eqn | r = \frac {3 a \left({a x + b}\right)} {b^4 x} - \frac {\left({a x + b}\right)^2} {2 b^4 x^2} - \frac {a^3 x} {b^4 \left({a x + b}\right)} + \frac {3 a^2} {b^4} \ln \left\vert{\frac x {a x + b} }\right\vert + C       | c = Simplify }} {{end-eqn}} {{qed}}