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

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Theorem

$\displaystyle \int \frac {\mathrm d x} {x^2 \left({a x + b}\right)^2} = \frac {-a} {b^2 \left({a x + b}\right)} - \frac 1 {b^2 x} + \frac {2 a} {b^3} \ln \left\vert{\frac {a x + b} x}\right\vert + C$


Proof

\(\displaystyle \int \frac {\mathrm d x} {x^2 \left({a x + b}\right)^2}\) \(=\) \(\displaystyle \int \left({-\frac {2 a} {b^3 x} + \frac 1 {b^2 x^2} + \frac {2 a^2} {b^3 \left({a x + b}\right)} + \frac {a^2} {b^2 \left({a x + b}\right)^2} }\right) \ \mathrm d x\) Partial Fraction Expansion
\(\displaystyle \) \(=\) \(\displaystyle -\frac {2 a} {b^3} \int \frac {\mathrm d x} x + \frac 1 {b^2} \int \frac {\mathrm d x} {x^2} + \frac {2 a^2} {b^3} \int \frac {\mathrm d x} {\left({a x + b}\right)} + \frac {a^2} {b^2} \int \frac {\mathrm d x} {\left({a x + b}\right)^2}\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle -\frac {2 a} {b^3} \int \frac {\mathrm d x} x + \frac 1 {b^2} \frac {-1} x + \frac {2 a^2} {b^3} \int \frac {\mathrm d x} {\left({a x + b}\right)} + \frac {a^2} {b^2} \int \frac {\mathrm d x} {\left({a x + b}\right)^2} + C\) Primitive of Power
\(\displaystyle \) \(=\) \(\displaystyle -\frac {2 a} {b^3} \ln \left\vert{x}\right\vert + \frac 1 {b^2} \frac {-1} x + \frac {2 a^2} {b^3} \int \frac {\mathrm d x} {\left({a x + b}\right)} + \frac {a^2} {b^2} \int \frac {\mathrm d x} {\left({a x + b}\right)^2} + C\) Primitive of $\dfrac 1 x$
\(\displaystyle \) \(=\) \(\displaystyle -\frac {2 a} {b^3} \ln \left\vert{x}\right\vert - \frac 1 {b^2 x} + \frac {2 a^2} {a b^3} \ln \left\vert{a x + b}\right\vert + \frac {a^2} {b^2} \int \frac {\mathrm d x} {\left({a x + b}\right)^2} + C\) Primitive of $\dfrac 1 {a x + b}$
\(\displaystyle \) \(=\) \(\displaystyle -\frac {2 a} {b^3} \ln \left\vert{x}\right\vert - \frac 1 {b^2 x} + \frac {2 a} {b^3} \ln \left\vert{a x + b}\right\vert + \frac {a^2} {b^2} \frac {-1} {a \left({a x + b}\right)} + C\) Primitive of $\dfrac 1 {\left({a x + b}\right)^2}$
\(\displaystyle \) \(=\) \(\displaystyle \frac {-a} {b^2 \left({a x + b}\right)} - \frac 1 {b^2 x} + \frac {2 a} {b^3} \ln \left\vert{\frac {a x + b} x}\right\vert + C\) Difference of Logarithms and rearranging

$\blacksquare$


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