Optimal. Leaf size=70 \[ -\frac{a^3 x^2}{2 b^4}+\frac{a^2 x^3}{3 b^3}+\frac{a^4 x}{b^5}-\frac{a^5 \log (a+b x)}{b^6}-\frac{a x^4}{4 b^2}+\frac{x^5}{5 b} \]
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Rubi [A] time = 0.0344801, antiderivative size = 70, 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} \[ -\frac{a^3 x^2}{2 b^4}+\frac{a^2 x^3}{3 b^3}+\frac{a^4 x}{b^5}-\frac{a^5 \log (a+b x)}{b^6}-\frac{a x^4}{4 b^2}+\frac{x^5}{5 b} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{a+b x} \, dx &=\int \left (\frac{a^4}{b^5}-\frac{a^3 x}{b^4}+\frac{a^2 x^2}{b^3}-\frac{a x^3}{b^2}+\frac{x^4}{b}-\frac{a^5}{b^5 (a+b x)}\right ) \, dx\\ &=\frac{a^4 x}{b^5}-\frac{a^3 x^2}{2 b^4}+\frac{a^2 x^3}{3 b^3}-\frac{a x^4}{4 b^2}+\frac{x^5}{5 b}-\frac{a^5 \log (a+b x)}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0053479, size = 70, normalized size = 1. \[ -\frac{a^3 x^2}{2 b^4}+\frac{a^2 x^3}{3 b^3}+\frac{a^4 x}{b^5}-\frac{a^5 \log (a+b x)}{b^6}-\frac{a x^4}{4 b^2}+\frac{x^5}{5 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 63, normalized size = 0.9 \begin{align*}{\frac{{a}^{4}x}{{b}^{5}}}-{\frac{{a}^{3}{x}^{2}}{2\,{b}^{4}}}+{\frac{{a}^{2}{x}^{3}}{3\,{b}^{3}}}-{\frac{a{x}^{4}}{4\,{b}^{2}}}+{\frac{{x}^{5}}{5\,b}}-{\frac{{a}^{5}\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 = 1.06738, size = 86, normalized size = 1.23 \begin{align*} -\frac{a^{5} \log \left (b x + a\right )}{b^{6}} + \frac{12 \, b^{4} x^{5} - 15 \, a b^{3} x^{4} + 20 \, a^{2} b^{2} x^{3} - 30 \, a^{3} b x^{2} + 60 \, a^{4} x}{60 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49602, size = 144, normalized size = 2.06 \begin{align*} \frac{12 \, b^{5} x^{5} - 15 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} - 30 \, a^{3} b^{2} x^{2} + 60 \, a^{4} b x - 60 \, a^{5} \log \left (b x + a\right )}{60 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.373252, size = 61, normalized size = 0.87 \begin{align*} - \frac{a^{5} \log{\left (a + b x \right )}}{b^{6}} + \frac{a^{4} x}{b^{5}} - \frac{a^{3} x^{2}}{2 b^{4}} + \frac{a^{2} x^{3}}{3 b^{3}} - \frac{a x^{4}}{4 b^{2}} + \frac{x^{5}}{5 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15778, size = 88, normalized size = 1.26 \begin{align*} -\frac{a^{5} \log \left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{12 \, b^{4} x^{5} - 15 \, a b^{3} x^{4} + 20 \, a^{2} b^{2} x^{3} - 30 \, a^{3} b x^{2} + 60 \, a^{4} x}{60 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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