Optimal. Leaf size=81 \[ -\frac{2 b^3 x^2}{a^5}+\frac{b^2 x^3}{a^4}-\frac{b^6}{a^7 (a x+b)}+\frac{5 b^4 x}{a^6}-\frac{6 b^5 \log (a x+b)}{a^7}-\frac{b x^4}{2 a^3}+\frac{x^5}{5 a^2} \]
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Rubi [A] time = 0.0527342, antiderivative size = 81, 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} \[ -\frac{2 b^3 x^2}{a^5}+\frac{b^2 x^3}{a^4}-\frac{b^6}{a^7 (a x+b)}+\frac{5 b^4 x}{a^6}-\frac{6 b^5 \log (a x+b)}{a^7}-\frac{b x^4}{2 a^3}+\frac{x^5}{5 a^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a+\frac{b}{x}\right )^2} \, dx &=\int \frac{x^6}{(b+a x)^2} \, dx\\ &=\int \left (\frac{5 b^4}{a^6}-\frac{4 b^3 x}{a^5}+\frac{3 b^2 x^2}{a^4}-\frac{2 b x^3}{a^3}+\frac{x^4}{a^2}+\frac{b^6}{a^6 (b+a x)^2}-\frac{6 b^5}{a^6 (b+a x)}\right ) \, dx\\ &=\frac{5 b^4 x}{a^6}-\frac{2 b^3 x^2}{a^5}+\frac{b^2 x^3}{a^4}-\frac{b x^4}{2 a^3}+\frac{x^5}{5 a^2}-\frac{b^6}{a^7 (b+a x)}-\frac{6 b^5 \log (b+a x)}{a^7}\\ \end{align*}
Mathematica [A] time = 0.0229799, size = 77, normalized size = 0.95 \[ \frac{-20 a^2 b^3 x^2+10 a^3 b^2 x^3-5 a^4 b x^4+2 a^5 x^5-\frac{10 b^6}{a x+b}+50 a b^4 x-60 b^5 \log (a x+b)}{10 a^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 78, normalized size = 1. \begin{align*} 5\,{\frac{{b}^{4}x}{{a}^{6}}}-2\,{\frac{{b}^{3}{x}^{2}}{{a}^{5}}}+{\frac{{b}^{2}{x}^{3}}{{a}^{4}}}-{\frac{b{x}^{4}}{2\,{a}^{3}}}+{\frac{{x}^{5}}{5\,{a}^{2}}}-{\frac{{b}^{6}}{{a}^{7} \left ( ax+b \right ) }}-6\,{\frac{{b}^{5}\ln \left ( ax+b \right ) }{{a}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11633, size = 111, normalized size = 1.37 \begin{align*} -\frac{b^{6}}{a^{8} x + a^{7} b} - \frac{6 \, b^{5} \log \left (a x + b\right )}{a^{7}} + \frac{2 \, a^{4} x^{5} - 5 \, a^{3} b x^{4} + 10 \, a^{2} b^{2} x^{3} - 20 \, a b^{3} x^{2} + 50 \, b^{4} x}{10 \, a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39507, size = 208, normalized size = 2.57 \begin{align*} \frac{2 \, a^{6} x^{6} - 3 \, a^{5} b x^{5} + 5 \, a^{4} b^{2} x^{4} - 10 \, a^{3} b^{3} x^{3} + 30 \, a^{2} b^{4} x^{2} + 50 \, a b^{5} x - 10 \, b^{6} - 60 \,{\left (a b^{5} x + b^{6}\right )} \log \left (a x + b\right )}{10 \,{\left (a^{8} x + a^{7} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.369558, size = 78, normalized size = 0.96 \begin{align*} - \frac{b^{6}}{a^{8} x + a^{7} b} + \frac{x^{5}}{5 a^{2}} - \frac{b x^{4}}{2 a^{3}} + \frac{b^{2} x^{3}}{a^{4}} - \frac{2 b^{3} x^{2}}{a^{5}} + \frac{5 b^{4} x}{a^{6}} - \frac{6 b^{5} \log{\left (a x + b \right )}}{a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11966, size = 115, normalized size = 1.42 \begin{align*} -\frac{6 \, b^{5} \log \left ({\left | a x + b \right |}\right )}{a^{7}} - \frac{b^{6}}{{\left (a x + b\right )} a^{7}} + \frac{2 \, a^{8} x^{5} - 5 \, a^{7} b x^{4} + 10 \, a^{6} b^{2} x^{3} - 20 \, a^{5} b^{3} x^{2} + 50 \, a^{4} b^{4} x}{10 \, a^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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