Optimal. Leaf size=77 \[ \frac{a^5}{2 b^6 (a+b x)^2}-\frac{5 a^4}{b^6 (a+b x)}+\frac{6 a^2 x}{b^5}-\frac{10 a^3 \log (a+b x)}{b^6}-\frac{3 a x^2}{2 b^4}+\frac{x^3}{3 b^3} \]
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Rubi [A] time = 0.0460517, antiderivative size = 77, 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} \[ \frac{a^5}{2 b^6 (a+b x)^2}-\frac{5 a^4}{b^6 (a+b x)}+\frac{6 a^2 x}{b^5}-\frac{10 a^3 \log (a+b x)}{b^6}-\frac{3 a x^2}{2 b^4}+\frac{x^3}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{(a+b x)^3} \, dx &=\int \left (\frac{6 a^2}{b^5}-\frac{3 a x}{b^4}+\frac{x^2}{b^3}-\frac{a^5}{b^5 (a+b x)^3}+\frac{5 a^4}{b^5 (a+b x)^2}-\frac{10 a^3}{b^5 (a+b x)}\right ) \, dx\\ &=\frac{6 a^2 x}{b^5}-\frac{3 a x^2}{2 b^4}+\frac{x^3}{3 b^3}+\frac{a^5}{2 b^6 (a+b x)^2}-\frac{5 a^4}{b^6 (a+b x)}-\frac{10 a^3 \log (a+b x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0352137, size = 67, normalized size = 0.87 \[ \frac{\frac{3 a^5}{(a+b x)^2}-\frac{30 a^4}{a+b x}+36 a^2 b x-60 a^3 \log (a+b x)-9 a b^2 x^2+2 b^3 x^3}{6 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 72, normalized size = 0.9 \begin{align*} 6\,{\frac{{a}^{2}x}{{b}^{5}}}-{\frac{3\,a{x}^{2}}{2\,{b}^{4}}}+{\frac{{x}^{3}}{3\,{b}^{3}}}+{\frac{{a}^{5}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}-5\,{\frac{{a}^{4}}{{b}^{6} \left ( bx+a \right ) }}-10\,{\frac{{a}^{3}\ln \left ( bx+a \right ) }{{b}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08787, size = 109, normalized size = 1.42 \begin{align*} -\frac{10 \, a^{4} b x + 9 \, a^{5}}{2 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} - \frac{10 \, a^{3} \log \left (b x + a\right )}{b^{6}} + \frac{2 \, b^{2} x^{3} - 9 \, a b x^{2} + 36 \, a^{2} x}{6 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39161, size = 227, normalized size = 2.95 \begin{align*} \frac{2 \, b^{5} x^{5} - 5 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 63 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x - 27 \, a^{5} - 60 \,{\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{6 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.574858, size = 83, normalized size = 1.08 \begin{align*} - \frac{10 a^{3} \log{\left (a + b x \right )}}{b^{6}} + \frac{6 a^{2} x}{b^{5}} - \frac{3 a x^{2}}{2 b^{4}} - \frac{9 a^{5} + 10 a^{4} b x}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} + \frac{x^{3}}{3 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25648, size = 99, normalized size = 1.29 \begin{align*} -\frac{10 \, a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{6}} - \frac{10 \, a^{4} b x + 9 \, a^{5}}{2 \,{\left (b x + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} x^{3} - 9 \, a b^{5} x^{2} + 36 \, a^{2} b^{4} x}{6 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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