Optimal. Leaf size=70 \[ -\frac {b^5 \log (a x+b)}{a^6}+\frac {b^4 x}{a^5}-\frac {b^3 x^2}{2 a^4}+\frac {b^2 x^3}{3 a^3}-\frac {b x^4}{4 a^2}+\frac {x^5}{5 a} \]
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Rubi [A] time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, 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 43
Rule 263
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*}
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Mathematica [A] time = 0.01, size = 70, normalized size = 1.00 \[ -\frac {b^5 \log (a x+b)}{a^6}+\frac {b^4 x}{a^5}-\frac {b^3 x^2}{2 a^4}+\frac {b^2 x^3}{3 a^3}-\frac {b x^4}{4 a^2}+\frac {x^5}{5 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 63, normalized size = 0.90 \[ \frac {12 \, a^{5} x^{5} - 15 \, a^{4} b x^{4} + 20 \, a^{3} b^{2} x^{3} - 30 \, a^{2} b^{3} x^{2} + 60 \, a b^{4} x - 60 \, b^{5} \log \left (a x + b\right )}{60 \, a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 65, normalized size = 0.93 \[ -\frac {b^{5} \log \left ({\left | a x + b \right |}\right )}{a^{6}} + \frac {12 \, a^{4} x^{5} - 15 \, a^{3} b x^{4} + 20 \, a^{2} b^{2} x^{3} - 30 \, a b^{3} x^{2} + 60 \, b^{4} x}{60 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 63, normalized size = 0.90 \[ \frac {x^{5}}{5 a}-\frac {b \,x^{4}}{4 a^{2}}+\frac {b^{2} x^{3}}{3 a^{3}}-\frac {b^{3} x^{2}}{2 a^{4}}+\frac {b^{4} x}{a^{5}}-\frac {b^{5} \ln \left (a x +b \right )}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 64, normalized size = 0.91 \[ -\frac {b^{5} \log \left (a x + b\right )}{a^{6}} + \frac {12 \, a^{4} x^{5} - 15 \, a^{3} b x^{4} + 20 \, a^{2} b^{2} x^{3} - 30 \, a b^{3} x^{2} + 60 \, b^{4} x}{60 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 62, normalized size = 0.89 \[ \frac {x^5}{5\,a}-\frac {b^5\,\ln \left (b+a\,x\right )}{a^6}-\frac {b\,x^4}{4\,a^2}+\frac {b^4\,x}{a^5}+\frac {b^2\,x^3}{3\,a^3}-\frac {b^3\,x^2}{2\,a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 61, normalized size = 0.87 \[ \frac {x^{5}}{5 a} - \frac {b x^{4}}{4 a^{2}} + \frac {b^{2} x^{3}}{3 a^{3}} - \frac {b^{3} x^{2}}{2 a^{4}} + \frac {b^{4} x}{a^{5}} - \frac {b^{5} \log {\left (a x + b \right )}}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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