Optimal. Leaf size=98 \[ \frac{5 b^4 x^2}{2 a^6}-\frac{4 b^3 x^3}{3 a^5}+\frac{3 b^2 x^4}{4 a^4}+\frac{b^7}{a^8 (a x+b)}-\frac{6 b^5 x}{a^7}+\frac{7 b^6 \log (a x+b)}{a^8}-\frac{2 b x^5}{5 a^3}+\frac{x^6}{6 a^2} \]
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Rubi [A] time = 0.0655405, antiderivative size = 98, 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} \[ \frac{5 b^4 x^2}{2 a^6}-\frac{4 b^3 x^3}{3 a^5}+\frac{3 b^2 x^4}{4 a^4}+\frac{b^7}{a^8 (a x+b)}-\frac{6 b^5 x}{a^7}+\frac{7 b^6 \log (a x+b)}{a^8}-\frac{2 b x^5}{5 a^3}+\frac{x^6}{6 a^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+\frac{b}{x}\right )^2} \, dx &=\int \frac{x^7}{(b+a x)^2} \, dx\\ &=\int \left (-\frac{6 b^5}{a^7}+\frac{5 b^4 x}{a^6}-\frac{4 b^3 x^2}{a^5}+\frac{3 b^2 x^3}{a^4}-\frac{2 b x^4}{a^3}+\frac{x^5}{a^2}-\frac{b^7}{a^7 (b+a x)^2}+\frac{7 b^6}{a^7 (b+a x)}\right ) \, dx\\ &=-\frac{6 b^5 x}{a^7}+\frac{5 b^4 x^2}{2 a^6}-\frac{4 b^3 x^3}{3 a^5}+\frac{3 b^2 x^4}{4 a^4}-\frac{2 b x^5}{5 a^3}+\frac{x^6}{6 a^2}+\frac{b^7}{a^8 (b+a x)}+\frac{7 b^6 \log (b+a x)}{a^8}\\ \end{align*}
Mathematica [A] time = 0.0206441, size = 88, normalized size = 0.9 \[ \frac{150 a^2 b^4 x^2-80 a^3 b^3 x^3+45 a^4 b^2 x^4-24 a^5 b x^5+10 a^6 x^6+\frac{60 b^7}{a x+b}-360 a b^5 x+420 b^6 \log (a x+b)}{60 a^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 89, normalized size = 0.9 \begin{align*} -6\,{\frac{{b}^{5}x}{{a}^{7}}}+{\frac{5\,{b}^{4}{x}^{2}}{2\,{a}^{6}}}-{\frac{4\,{b}^{3}{x}^{3}}{3\,{a}^{5}}}+{\frac{3\,{b}^{2}{x}^{4}}{4\,{a}^{4}}}-{\frac{2\,b{x}^{5}}{5\,{a}^{3}}}+{\frac{{x}^{6}}{6\,{a}^{2}}}+{\frac{{b}^{7}}{{a}^{8} \left ( ax+b \right ) }}+7\,{\frac{{b}^{6}\ln \left ( ax+b \right ) }{{a}^{8}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02057, size = 124, normalized size = 1.27 \begin{align*} \frac{b^{7}}{a^{9} x + a^{8} b} + \frac{7 \, b^{6} \log \left (a x + b\right )}{a^{8}} + \frac{10 \, a^{5} x^{6} - 24 \, a^{4} b x^{5} + 45 \, a^{3} b^{2} x^{4} - 80 \, a^{2} b^{3} x^{3} + 150 \, a b^{4} x^{2} - 360 \, b^{5} x}{60 \, a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44011, size = 239, normalized size = 2.44 \begin{align*} \frac{10 \, a^{7} x^{7} - 14 \, a^{6} b x^{6} + 21 \, a^{5} b^{2} x^{5} - 35 \, a^{4} b^{3} x^{4} + 70 \, a^{3} b^{4} x^{3} - 210 \, a^{2} b^{5} x^{2} - 360 \, a b^{6} x + 60 \, b^{7} + 420 \,{\left (a b^{6} x + b^{7}\right )} \log \left (a x + b\right )}{60 \,{\left (a^{9} x + a^{8} b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.384867, size = 99, normalized size = 1.01 \begin{align*} \frac{b^{7}}{a^{9} x + a^{8} b} + \frac{x^{6}}{6 a^{2}} - \frac{2 b x^{5}}{5 a^{3}} + \frac{3 b^{2} x^{4}}{4 a^{4}} - \frac{4 b^{3} x^{3}}{3 a^{5}} + \frac{5 b^{4} x^{2}}{2 a^{6}} - \frac{6 b^{5} x}{a^{7}} + \frac{7 b^{6} \log{\left (a x + b \right )}}{a^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08777, size = 128, normalized size = 1.31 \begin{align*} \frac{7 \, b^{6} \log \left ({\left | a x + b \right |}\right )}{a^{8}} + \frac{b^{7}}{{\left (a x + b\right )} a^{8}} + \frac{10 \, a^{10} x^{6} - 24 \, a^{9} b x^{5} + 45 \, a^{8} b^{2} x^{4} - 80 \, a^{7} b^{3} x^{3} + 150 \, a^{6} b^{4} x^{2} - 360 \, a^{5} b^{5} x}{60 \, a^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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