Optimal. Leaf size=29 \[ -\frac{\log (a x+b)}{b^2}+\frac{1}{b (a x+b)}+\frac{\log (x)}{b^2} \]
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Rubi [A] time = 0.0168673, antiderivative size = 29, 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} \[ -\frac{\log (a x+b)}{b^2}+\frac{1}{b (a x+b)}+\frac{\log (x)}{b^2} \]
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 )^2 x^3} \, dx &=\int \frac{1}{x (b+a x)^2} \, dx\\ &=\int \left (\frac{1}{b^2 x}-\frac{a}{b (b+a x)^2}-\frac{a}{b^2 (b+a x)}\right ) \, dx\\ &=\frac{1}{b (b+a x)}+\frac{\log (x)}{b^2}-\frac{\log (b+a x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.0095023, size = 24, normalized size = 0.83 \[ \frac{\frac{b}{a x+b}-\log (a x+b)+\log (x)}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 30, normalized size = 1. \begin{align*}{\frac{1}{b \left ( ax+b \right ) }}+{\frac{\ln \left ( x \right ) }{{b}^{2}}}-{\frac{\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 = 1.10529, size = 38, normalized size = 1.31 \begin{align*} \frac{1}{a b x + b^{2}} - \frac{\log \left (a x + b\right )}{b^{2}} + \frac{\log \left (x\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4592, size = 89, normalized size = 3.07 \begin{align*} -\frac{{\left (a x + b\right )} \log \left (a x + b\right ) -{\left (a x + b\right )} \log \left (x\right ) - b}{a b^{2} x + b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.338869, size = 22, normalized size = 0.76 \begin{align*} \frac{1}{a b x + b^{2}} + \frac{\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12537, size = 42, normalized size = 1.45 \begin{align*} -\frac{\log \left ({\left | a x + b \right |}\right )}{b^{2}} + \frac{\log \left ({\left | x \right |}\right )}{b^{2}} + \frac{1}{{\left (a x + b\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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