Optimal. Leaf size=41 \[ -\frac{b^2}{2 a^3 (a x+b)^2}+\frac{2 b}{a^3 (a x+b)}+\frac{\log (a x+b)}{a^3} \]
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Rubi [A] time = 0.0222989, antiderivative size = 41, 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, 43} \[ -\frac{b^2}{2 a^3 (a x+b)^2}+\frac{2 b}{a^3 (a x+b)}+\frac{\log (a x+b)}{a^3} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^3 x} \, dx &=\int \frac{x^2}{(b+a x)^3} \, dx\\ &=\int \left (\frac{b^2}{a^2 (b+a x)^3}-\frac{2 b}{a^2 (b+a x)^2}+\frac{1}{a^2 (b+a x)}\right ) \, dx\\ &=-\frac{b^2}{2 a^3 (b+a x)^2}+\frac{2 b}{a^3 (b+a x)}+\frac{\log (b+a x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0124631, size = 33, normalized size = 0.8 \[ \frac{\frac{b (4 a x+3 b)}{(a x+b)^2}+2 \log (a x+b)}{2 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 40, normalized size = 1. \begin{align*} -{\frac{{b}^{2}}{2\,{a}^{3} \left ( ax+b \right ) ^{2}}}+2\,{\frac{b}{{a}^{3} \left ( ax+b \right ) }}+{\frac{\ln \left ( ax+b \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07854, size = 65, normalized size = 1.59 \begin{align*} \frac{4 \, a b x + 3 \, b^{2}}{2 \,{\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}} + \frac{\log \left (a x + b\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68191, size = 132, normalized size = 3.22 \begin{align*} \frac{4 \, 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^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.34727, size = 46, normalized size = 1.12 \begin{align*} \frac{4 a b x + 3 b^{2}}{2 a^{5} x^{2} + 4 a^{4} b x + 2 a^{3} b^{2}} + \frac{\log{\left (a x + b \right )}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08731, size = 50, normalized size = 1.22 \begin{align*} \frac{\log \left ({\left | a x + b \right |}\right )}{a^{3}} + \frac{4 \, b x + \frac{3 \, b^{2}}{a}}{2 \,{\left (a x + b\right )}^{2} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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