Optimal. Leaf size=81 \[ -\frac{2 a^3 x^2}{b^5}+\frac{a^2 x^3}{b^4}-\frac{a^6}{b^7 (a+b x)}+\frac{5 a^4 x}{b^6}-\frac{6 a^5 \log (a+b x)}{b^7}-\frac{a x^4}{2 b^3}+\frac{x^5}{5 b^2} \]
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Rubi [A] time = 0.0592848, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ -\frac{2 a^3 x^2}{b^5}+\frac{a^2 x^3}{b^4}-\frac{a^6}{b^7 (a+b x)}+\frac{5 a^4 x}{b^6}-\frac{6 a^5 \log (a+b x)}{b^7}-\frac{a x^4}{2 b^3}+\frac{x^5}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^6}{(a+b x)^2} \, dx &=\int \left (\frac{5 a^4}{b^6}-\frac{4 a^3 x}{b^5}+\frac{3 a^2 x^2}{b^4}-\frac{2 a x^3}{b^3}+\frac{x^4}{b^2}+\frac{a^6}{b^6 (a+b x)^2}-\frac{6 a^5}{b^6 (a+b x)}\right ) \, dx\\ &=\frac{5 a^4 x}{b^6}-\frac{2 a^3 x^2}{b^5}+\frac{a^2 x^3}{b^4}-\frac{a x^4}{2 b^3}+\frac{x^5}{5 b^2}-\frac{a^6}{b^7 (a+b x)}-\frac{6 a^5 \log (a+b x)}{b^7}\\ \end{align*}
Mathematica [A] time = 0.0356202, size = 77, normalized size = 0.95 \[ \frac{-20 a^3 b^2 x^2+10 a^2 b^3 x^3-\frac{10 a^6}{a+b x}+50 a^4 b x-60 a^5 \log (a+b x)-5 a b^4 x^4+2 b^5 x^5}{10 b^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 78, normalized size = 1. \begin{align*} 5\,{\frac{{a}^{4}x}{{b}^{6}}}-2\,{\frac{{a}^{3}{x}^{2}}{{b}^{5}}}+{\frac{{a}^{2}{x}^{3}}{{b}^{4}}}-{\frac{a{x}^{4}}{2\,{b}^{3}}}+{\frac{{x}^{5}}{5\,{b}^{2}}}-{\frac{{a}^{6}}{{b}^{7} \left ( bx+a \right ) }}-6\,{\frac{{a}^{5}\ln \left ( bx+a \right ) }{{b}^{7}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05872, size = 111, normalized size = 1.37 \begin{align*} -\frac{a^{6}}{b^{8} x + a b^{7}} - \frac{6 \, a^{5} \log \left (b x + a\right )}{b^{7}} + \frac{2 \, b^{4} x^{5} - 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} - 20 \, a^{3} b x^{2} + 50 \, a^{4} x}{10 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48875, size = 208, normalized size = 2.57 \begin{align*} \frac{2 \, b^{6} x^{6} - 3 \, a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{4} - 10 \, a^{3} b^{3} x^{3} + 30 \, a^{4} b^{2} x^{2} + 50 \, a^{5} b x - 10 \, a^{6} - 60 \,{\left (a^{5} b x + a^{6}\right )} \log \left (b x + a\right )}{10 \,{\left (b^{8} x + a b^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.497526, size = 78, normalized size = 0.96 \begin{align*} - \frac{a^{6}}{a b^{7} + b^{8} x} - \frac{6 a^{5} \log{\left (a + b x \right )}}{b^{7}} + \frac{5 a^{4} x}{b^{6}} - \frac{2 a^{3} x^{2}}{b^{5}} + \frac{a^{2} x^{3}}{b^{4}} - \frac{a x^{4}}{2 b^{3}} + \frac{x^{5}}{5 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22041, size = 139, normalized size = 1.72 \begin{align*} -\frac{{\left (b x + a\right )}^{5}{\left (\frac{15 \, a}{b x + a} - \frac{50 \, a^{2}}{{\left (b x + a\right )}^{2}} + \frac{100 \, a^{3}}{{\left (b x + a\right )}^{3}} - \frac{150 \, a^{4}}{{\left (b x + a\right )}^{4}} - 2\right )}}{10 \, b^{7}} + \frac{6 \, a^{5} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{7}} - \frac{a^{6}}{{\left (b x + a\right )} b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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