Optimal. Leaf size=56 \[ -\frac{a^2}{b^3 x}-\frac{a^3 \log (x)}{b^4}+\frac{a^3 \log (a x+b)}{b^4}+\frac{a}{2 b^2 x^2}-\frac{1}{3 b x^3} \]
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Rubi [A] time = 0.0231498, antiderivative size = 56, 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} \[ -\frac{a^2}{b^3 x}-\frac{a^3 \log (x)}{b^4}+\frac{a^3 \log (a x+b)}{b^4}+\frac{a}{2 b^2 x^2}-\frac{1}{3 b x^3} \]
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 ) x^5} \, dx &=\int \frac{1}{x^4 (b+a x)} \, dx\\ &=\int \left (\frac{1}{b x^4}-\frac{a}{b^2 x^3}+\frac{a^2}{b^3 x^2}-\frac{a^3}{b^4 x}+\frac{a^4}{b^4 (b+a x)}\right ) \, dx\\ &=-\frac{1}{3 b x^3}+\frac{a}{2 b^2 x^2}-\frac{a^2}{b^3 x}-\frac{a^3 \log (x)}{b^4}+\frac{a^3 \log (b+a x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0049534, size = 56, normalized size = 1. \[ -\frac{a^2}{b^3 x}-\frac{a^3 \log (x)}{b^4}+\frac{a^3 \log (a x+b)}{b^4}+\frac{a}{2 b^2 x^2}-\frac{1}{3 b x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 53, normalized size = 1. \begin{align*} -{\frac{1}{3\,b{x}^{3}}}+{\frac{a}{2\,{b}^{2}{x}^{2}}}-{\frac{{a}^{2}}{{b}^{3}x}}-{\frac{{a}^{3}\ln \left ( x \right ) }{{b}^{4}}}+{\frac{{a}^{3}\ln \left ( ax+b \right ) }{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02314, size = 69, normalized size = 1.23 \begin{align*} \frac{a^{3} \log \left (a x + b\right )}{b^{4}} - \frac{a^{3} \log \left (x\right )}{b^{4}} - \frac{6 \, a^{2} x^{2} - 3 \, a b x + 2 \, b^{2}}{6 \, b^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47402, size = 126, normalized size = 2.25 \begin{align*} \frac{6 \, a^{3} x^{3} \log \left (a x + b\right ) - 6 \, a^{3} x^{3} \log \left (x\right ) - 6 \, a^{2} b x^{2} + 3 \, a b^{2} x - 2 \, b^{3}}{6 \, b^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.372969, size = 44, normalized size = 0.79 \begin{align*} \frac{a^{3} \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{4}} - \frac{6 a^{2} x^{2} - 3 a b x + 2 b^{2}}{6 b^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10682, size = 76, normalized size = 1.36 \begin{align*} \frac{a^{3} \log \left ({\left | a x + b \right |}\right )}{b^{4}} - \frac{a^{3} \log \left ({\left | x \right |}\right )}{b^{4}} - \frac{6 \, a^{2} b x^{2} - 3 \, a b^{2} x + 2 \, b^{3}}{6 \, b^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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