Optimal. Leaf size=28 \[ -\frac{a \log (x)}{b^2}+\frac{a \log (a x+b)}{b^2}-\frac{1}{b x} \]
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Rubi [A] time = 0.0162563, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 44} \[ -\frac{a \log (x)}{b^2}+\frac{a \log (a x+b)}{b^2}-\frac{1}{b x} \]
Antiderivative was successfully verified.
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Rule 263
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right ) x^3} \, dx &=\int \frac{1}{x^2 (b+a x)} \, dx\\ &=\int \left (\frac{1}{b x^2}-\frac{a}{b^2 x}+\frac{a^2}{b^2 (b+a x)}\right ) \, dx\\ &=-\frac{1}{b x}-\frac{a \log (x)}{b^2}+\frac{a \log (b+a x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0044804, size = 28, normalized size = 1. \[ -\frac{a \log (x)}{b^2}+\frac{a \log (a x+b)}{b^2}-\frac{1}{b x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 29, normalized size = 1. \begin{align*} -{\frac{1}{bx}}-{\frac{a\ln \left ( x \right ) }{{b}^{2}}}+{\frac{a\ln \left ( ax+b \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.977844, size = 38, normalized size = 1.36 \begin{align*} \frac{a \log \left (a x + b\right )}{b^{2}} - \frac{a \log \left (x\right )}{b^{2}} - \frac{1}{b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4317, size = 61, normalized size = 2.18 \begin{align*} \frac{a x \log \left (a x + b\right ) - a x \log \left (x\right ) - b}{b^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.312985, size = 19, normalized size = 0.68 \begin{align*} \frac{a \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{2}} - \frac{1}{b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12027, size = 41, normalized size = 1.46 \begin{align*} \frac{a \log \left ({\left | a x + b \right |}\right )}{b^{2}} - \frac{a \log \left ({\left | x \right |}\right )}{b^{2}} - \frac{1}{b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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