Optimal. Leaf size=82 \[ \frac{a^3}{2 b^4 x^2}-\frac{a^2}{3 b^3 x^3}-\frac{a^4}{b^5 x}-\frac{a^5 \log (x)}{b^6}+\frac{a^5 \log (a x+b)}{b^6}+\frac{a}{4 b^2 x^4}-\frac{1}{5 b x^5} \]
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Rubi [A] time = 0.0342849, antiderivative size = 82, 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} \[ \frac{a^3}{2 b^4 x^2}-\frac{a^2}{3 b^3 x^3}-\frac{a^4}{b^5 x}-\frac{a^5 \log (x)}{b^6}+\frac{a^5 \log (a x+b)}{b^6}+\frac{a}{4 b^2 x^4}-\frac{1}{5 b x^5} \]
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^7} \, dx &=\int \frac{1}{x^6 (b+a x)} \, dx\\ &=\int \left (\frac{1}{b x^6}-\frac{a}{b^2 x^5}+\frac{a^2}{b^3 x^4}-\frac{a^3}{b^4 x^3}+\frac{a^4}{b^5 x^2}-\frac{a^5}{b^6 x}+\frac{a^6}{b^6 (b+a x)}\right ) \, dx\\ &=-\frac{1}{5 b x^5}+\frac{a}{4 b^2 x^4}-\frac{a^2}{3 b^3 x^3}+\frac{a^3}{2 b^4 x^2}-\frac{a^4}{b^5 x}-\frac{a^5 \log (x)}{b^6}+\frac{a^5 \log (b+a x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0053832, size = 82, normalized size = 1. \[ \frac{a^3}{2 b^4 x^2}-\frac{a^2}{3 b^3 x^3}-\frac{a^4}{b^5 x}-\frac{a^5 \log (x)}{b^6}+\frac{a^5 \log (a x+b)}{b^6}+\frac{a}{4 b^2 x^4}-\frac{1}{5 b x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 75, normalized size = 0.9 \begin{align*} -{\frac{1}{5\,b{x}^{5}}}+{\frac{a}{4\,{b}^{2}{x}^{4}}}-{\frac{{a}^{2}}{3\,{b}^{3}{x}^{3}}}+{\frac{{a}^{3}}{2\,{b}^{4}{x}^{2}}}-{\frac{{a}^{4}}{{b}^{5}x}}-{\frac{{a}^{5}\ln \left ( x \right ) }{{b}^{6}}}+{\frac{{a}^{5}\ln \left ( ax+b \right ) }{{b}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.64672, size = 99, normalized size = 1.21 \begin{align*} \frac{a^{5} \log \left (a x + b\right )}{b^{6}} - \frac{a^{5} \log \left (x\right )}{b^{6}} - \frac{60 \, a^{4} x^{4} - 30 \, a^{3} b x^{3} + 20 \, a^{2} b^{2} x^{2} - 15 \, a b^{3} x + 12 \, b^{4}}{60 \, b^{5} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44079, size = 180, normalized size = 2.2 \begin{align*} \frac{60 \, a^{5} x^{5} \log \left (a x + b\right ) - 60 \, a^{5} x^{5} \log \left (x\right ) - 60 \, a^{4} b x^{4} + 30 \, a^{3} b^{2} x^{3} - 20 \, a^{2} b^{3} x^{2} + 15 \, a b^{4} x - 12 \, b^{5}}{60 \, b^{6} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.438308, size = 68, normalized size = 0.83 \begin{align*} \frac{a^{5} \left (- \log{\left (x \right )} + \log{\left (x + \frac{b}{a} \right )}\right )}{b^{6}} - \frac{60 a^{4} x^{4} - 30 a^{3} b x^{3} + 20 a^{2} b^{2} x^{2} - 15 a b^{3} x + 12 b^{4}}{60 b^{5} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10621, size = 105, normalized size = 1.28 \begin{align*} \frac{a^{5} \log \left ({\left | a x + b \right |}\right )}{b^{6}} - \frac{a^{5} \log \left ({\left | x \right |}\right )}{b^{6}} - \frac{60 \, a^{4} b x^{4} - 30 \, a^{3} b^{2} x^{3} + 20 \, a^{2} b^{3} x^{2} - 15 \, a b^{4} x + 12 \, b^{5}}{60 \, b^{6} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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