Optimal. Leaf size=70 \[ -\frac{b^3 x^2}{2 a^4}+\frac{b^2 x^3}{3 a^3}+\frac{b^4 x}{a^5}-\frac{b^5 \log (a x+b)}{a^6}-\frac{b x^4}{4 a^2}+\frac{x^5}{5 a} \]
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Rubi [A] time = 0.0372933, antiderivative size = 70, 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} \[ -\frac{b^3 x^2}{2 a^4}+\frac{b^2 x^3}{3 a^3}+\frac{b^4 x}{a^5}-\frac{b^5 \log (a x+b)}{a^6}-\frac{b x^4}{4 a^2}+\frac{x^5}{5 a} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{x^4}{a+\frac{b}{x}} \, dx &=\int \frac{x^5}{b+a x} \, dx\\ &=\int \left (\frac{b^4}{a^5}-\frac{b^3 x}{a^4}+\frac{b^2 x^2}{a^3}-\frac{b x^3}{a^2}+\frac{x^4}{a}-\frac{b^5}{a^5 (b+a x)}\right ) \, dx\\ &=\frac{b^4 x}{a^5}-\frac{b^3 x^2}{2 a^4}+\frac{b^2 x^3}{3 a^3}-\frac{b x^4}{4 a^2}+\frac{x^5}{5 a}-\frac{b^5 \log (b+a x)}{a^6}\\ \end{align*}
Mathematica [A] time = 0.004117, size = 70, normalized size = 1. \[ -\frac{b^3 x^2}{2 a^4}+\frac{b^2 x^3}{3 a^3}+\frac{b^4 x}{a^5}-\frac{b^5 \log (a x+b)}{a^6}-\frac{b x^4}{4 a^2}+\frac{x^5}{5 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 63, normalized size = 0.9 \begin{align*}{\frac{{b}^{4}x}{{a}^{5}}}-{\frac{{b}^{3}{x}^{2}}{2\,{a}^{4}}}+{\frac{{b}^{2}{x}^{3}}{3\,{a}^{3}}}-{\frac{b{x}^{4}}{4\,{a}^{2}}}+{\frac{{x}^{5}}{5\,a}}-{\frac{{b}^{5}\ln \left ( ax+b \right ) }{{a}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01661, size = 86, normalized size = 1.23 \begin{align*} -\frac{b^{5} \log \left (a x + b\right )}{a^{6}} + \frac{12 \, a^{4} x^{5} - 15 \, a^{3} b x^{4} + 20 \, a^{2} b^{2} x^{3} - 30 \, a b^{3} x^{2} + 60 \, b^{4} x}{60 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3889, size = 144, normalized size = 2.06 \begin{align*} \frac{12 \, a^{5} x^{5} - 15 \, a^{4} b x^{4} + 20 \, a^{3} b^{2} x^{3} - 30 \, a^{2} b^{3} x^{2} + 60 \, a b^{4} x - 60 \, b^{5} \log \left (a x + b\right )}{60 \, a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.283082, size = 61, normalized size = 0.87 \begin{align*} \frac{x^{5}}{5 a} - \frac{b x^{4}}{4 a^{2}} + \frac{b^{2} x^{3}}{3 a^{3}} - \frac{b^{3} x^{2}}{2 a^{4}} + \frac{b^{4} x}{a^{5}} - \frac{b^{5} \log{\left (a x + b \right )}}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10506, size = 88, normalized size = 1.26 \begin{align*} -\frac{b^{5} \log \left ({\left | a x + b \right |}\right )}{a^{6}} + \frac{12 \, a^{4} x^{5} - 15 \, a^{3} b x^{4} + 20 \, a^{2} b^{2} x^{3} - 30 \, a b^{3} x^{2} + 60 \, b^{4} x}{60 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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