Optimal. Leaf size=68 \[ -\frac{a^2}{2 b^3 x^2}+\frac{a^3}{b^4 x}+\frac{a^4 \log (x)}{b^5}-\frac{a^4 \log (a x+b)}{b^5}+\frac{a}{3 b^2 x^3}-\frac{1}{4 b x^4} \]
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Rubi [A] time = 0.0303397, antiderivative size = 68, 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} \[ -\frac{a^2}{2 b^3 x^2}+\frac{a^3}{b^4 x}+\frac{a^4 \log (x)}{b^5}-\frac{a^4 \log (a x+b)}{b^5}+\frac{a}{3 b^2 x^3}-\frac{1}{4 b x^4} \]
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^6} \, dx &=\int \frac{1}{x^5 (b+a x)} \, dx\\ &=\int \left (\frac{1}{b x^5}-\frac{a}{b^2 x^4}+\frac{a^2}{b^3 x^3}-\frac{a^3}{b^4 x^2}+\frac{a^4}{b^5 x}-\frac{a^5}{b^5 (b+a x)}\right ) \, dx\\ &=-\frac{1}{4 b x^4}+\frac{a}{3 b^2 x^3}-\frac{a^2}{2 b^3 x^2}+\frac{a^3}{b^4 x}+\frac{a^4 \log (x)}{b^5}-\frac{a^4 \log (b+a x)}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0046514, size = 68, normalized size = 1. \[ -\frac{a^2}{2 b^3 x^2}+\frac{a^3}{b^4 x}+\frac{a^4 \log (x)}{b^5}-\frac{a^4 \log (a x+b)}{b^5}+\frac{a}{3 b^2 x^3}-\frac{1}{4 b x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 63, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,b{x}^{4}}}+{\frac{a}{3\,{b}^{2}{x}^{3}}}-{\frac{{a}^{2}}{2\,{b}^{3}{x}^{2}}}+{\frac{{a}^{3}}{{b}^{4}x}}+{\frac{{a}^{4}\ln \left ( x \right ) }{{b}^{5}}}-{\frac{{a}^{4}\ln \left ( ax+b \right ) }{{b}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11261, size = 84, normalized size = 1.24 \begin{align*} -\frac{a^{4} \log \left (a x + b\right )}{b^{5}} + \frac{a^{4} \log \left (x\right )}{b^{5}} + \frac{12 \, a^{3} x^{3} - 6 \, a^{2} b x^{2} + 4 \, a b^{2} x - 3 \, b^{3}}{12 \, b^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48762, size = 154, normalized size = 2.26 \begin{align*} -\frac{12 \, a^{4} x^{4} \log \left (a x + b\right ) - 12 \, a^{4} x^{4} \log \left (x\right ) - 12 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 4 \, a b^{3} x + 3 \, b^{4}}{12 \, b^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.402119, size = 56, normalized size = 0.82 \begin{align*} \frac{a^{4} \left (\log{\left (x \right )} - \log{\left (x + \frac{b}{a} \right )}\right )}{b^{5}} + \frac{12 a^{3} x^{3} - 6 a^{2} b x^{2} + 4 a b^{2} x - 3 b^{3}}{12 b^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10883, size = 90, normalized size = 1.32 \begin{align*} -\frac{a^{4} \log \left ({\left | a x + b \right |}\right )}{b^{5}} + \frac{a^{4} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac{12 \, a^{3} b x^{3} - 6 \, a^{2} b^{2} x^{2} + 4 \, a b^{3} x - 3 \, b^{4}}{12 \, b^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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