Optimal. Leaf size=84 \[ -\frac{3 a^2}{2 b^4 x^2}+\frac{a^4}{b^5 (a x+b)}+\frac{4 a^3}{b^5 x}+\frac{5 a^4 \log (x)}{b^6}-\frac{5 a^4 \log (a x+b)}{b^6}+\frac{2 a}{3 b^3 x^3}-\frac{1}{4 b^2 x^4} \]
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Rubi [A] time = 0.0471178, antiderivative size = 84, 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} \[ -\frac{3 a^2}{2 b^4 x^2}+\frac{a^4}{b^5 (a x+b)}+\frac{4 a^3}{b^5 x}+\frac{5 a^4 \log (x)}{b^6}-\frac{5 a^4 \log (a x+b)}{b^6}+\frac{2 a}{3 b^3 x^3}-\frac{1}{4 b^2 x^4} \]
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 )^2 x^7} \, dx &=\int \frac{1}{x^5 (b+a x)^2} \, dx\\ &=\int \left (\frac{1}{b^2 x^5}-\frac{2 a}{b^3 x^4}+\frac{3 a^2}{b^4 x^3}-\frac{4 a^3}{b^5 x^2}+\frac{5 a^4}{b^6 x}-\frac{a^5}{b^5 (b+a x)^2}-\frac{5 a^5}{b^6 (b+a x)}\right ) \, dx\\ &=-\frac{1}{4 b^2 x^4}+\frac{2 a}{3 b^3 x^3}-\frac{3 a^2}{2 b^4 x^2}+\frac{4 a^3}{b^5 x}+\frac{a^4}{b^5 (b+a x)}+\frac{5 a^4 \log (x)}{b^6}-\frac{5 a^4 \log (b+a x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0406603, size = 79, normalized size = 0.94 \[ \frac{\frac{b \left (-10 a^2 b^2 x^2+30 a^3 b x^3+60 a^4 x^4+5 a b^3 x-3 b^4\right )}{x^4 (a x+b)}-60 a^4 \log (a x+b)+60 a^4 \log (x)}{12 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 79, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{b}^{2}{x}^{4}}}+{\frac{2\,a}{3\,{b}^{3}{x}^{3}}}-{\frac{3\,{a}^{2}}{2\,{b}^{4}{x}^{2}}}+4\,{\frac{{a}^{3}}{{b}^{5}x}}+{\frac{{a}^{4}}{{b}^{5} \left ( ax+b \right ) }}+5\,{\frac{{a}^{4}\ln \left ( x \right ) }{{b}^{6}}}-5\,{\frac{{a}^{4}\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 = 1.05828, size = 116, normalized size = 1.38 \begin{align*} \frac{60 \, a^{4} x^{4} + 30 \, a^{3} b x^{3} - 10 \, a^{2} b^{2} x^{2} + 5 \, a b^{3} x - 3 \, b^{4}}{12 \,{\left (a b^{5} x^{5} + b^{6} x^{4}\right )}} - \frac{5 \, a^{4} \log \left (a x + b\right )}{b^{6}} + \frac{5 \, a^{4} \log \left (x\right )}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49315, size = 231, normalized size = 2.75 \begin{align*} \frac{60 \, a^{4} b x^{4} + 30 \, a^{3} b^{2} x^{3} - 10 \, a^{2} b^{3} x^{2} + 5 \, a b^{4} x - 3 \, b^{5} - 60 \,{\left (a^{5} x^{5} + a^{4} b x^{4}\right )} \log \left (a x + b\right ) + 60 \,{\left (a^{5} x^{5} + a^{4} b x^{4}\right )} \log \left (x\right )}{12 \,{\left (a b^{6} x^{5} + b^{7} x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.500878, size = 80, normalized size = 0.95 \begin{align*} \frac{5 a^{4} \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} - 10 a^{2} b^{2} x^{2} + 5 a b^{3} x - 3 b^{4}}{12 a b^{5} x^{5} + 12 b^{6} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13629, size = 116, normalized size = 1.38 \begin{align*} -\frac{5 \, a^{4} \log \left ({\left | a x + b \right |}\right )}{b^{6}} + \frac{5 \, a^{4} \log \left ({\left | x \right |}\right )}{b^{6}} + \frac{60 \, a^{4} b x^{4} + 30 \, a^{3} b^{2} x^{3} - 10 \, a^{2} b^{3} x^{2} + 5 \, a b^{4} x - 3 \, b^{5}}{12 \,{\left (a x + b\right )} b^{6} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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