Optimal. Leaf size=43 \[ \frac{1}{b^2 (a x+b)}-\frac{\log (a x+b)}{b^3}+\frac{1}{2 b (a x+b)^2}+\frac{\log (x)}{b^3} \]
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Rubi [A] time = 0.0231611, antiderivative size = 43, 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} \[ \frac{1}{b^2 (a x+b)}-\frac{\log (a x+b)}{b^3}+\frac{1}{2 b (a x+b)^2}+\frac{\log (x)}{b^3} \]
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 )^3 x^4} \, dx &=\int \frac{1}{x (b+a x)^3} \, dx\\ &=\int \left (\frac{1}{b^3 x}-\frac{a}{b (b+a x)^3}-\frac{a}{b^2 (b+a x)^2}-\frac{a}{b^3 (b+a x)}\right ) \, dx\\ &=\frac{1}{2 b (b+a x)^2}+\frac{1}{b^2 (b+a x)}+\frac{\log (x)}{b^3}-\frac{\log (b+a x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0262438, size = 37, normalized size = 0.86 \[ \frac{\frac{b (2 a x+3 b)}{(a x+b)^2}-2 \log (a x+b)+2 \log (x)}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 42, normalized size = 1. \begin{align*}{\frac{1}{2\,b \left ( ax+b \right ) ^{2}}}+{\frac{1}{{b}^{2} \left ( ax+b \right ) }}+{\frac{\ln \left ( x \right ) }{{b}^{3}}}-{\frac{\ln \left ( ax+b \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982694, size = 69, normalized size = 1.6 \begin{align*} \frac{2 \, a x + 3 \, b}{2 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )}} - \frac{\log \left (a x + b\right )}{b^{3}} + \frac{\log \left (x\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67602, size = 182, normalized size = 4.23 \begin{align*} \frac{2 \, a b x + 3 \, b^{2} - 2 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (a x + b\right ) + 2 \,{\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.422403, size = 46, normalized size = 1.07 \begin{align*} \frac{2 a x + 3 b}{2 a^{2} b^{2} x^{2} + 4 a b^{3} x + 2 b^{4}} + \frac{\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11302, size = 58, normalized size = 1.35 \begin{align*} -\frac{\log \left ({\left | a x + b \right |}\right )}{b^{3}} + \frac{\log \left ({\left | x \right |}\right )}{b^{3}} + \frac{2 \, a b x + 3 \, b^{2}}{2 \,{\left (a x + b\right )}^{2} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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