Optimal. Leaf size=93 \[ -x^3 \text{PolyLog}\left (2,-\frac{b e^x}{a}\right )+3 x^2 \text{PolyLog}\left (3,-\frac{b e^x}{a}\right )-6 x \text{PolyLog}\left (4,-\frac{b e^x}{a}\right )+6 \text{PolyLog}\left (5,-\frac{b e^x}{a}\right )+\frac{1}{4} x^4 \log \left (a+b e^x\right )-\frac{1}{4} x^4 \log \left (\frac{b e^x}{a}+1\right ) \]
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Rubi [A] time = 0.0730538, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {2532, 2531, 6609, 2282, 6589} \[ -x^3 \text{PolyLog}\left (2,-\frac{b e^x}{a}\right )+3 x^2 \text{PolyLog}\left (3,-\frac{b e^x}{a}\right )-6 x \text{PolyLog}\left (4,-\frac{b e^x}{a}\right )+6 \text{PolyLog}\left (5,-\frac{b e^x}{a}\right )+\frac{1}{4} x^4 \log \left (a+b e^x\right )-\frac{1}{4} x^4 \log \left (\frac{b e^x}{a}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2532
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^3 \log \left (a+b e^x\right ) \, dx &=\frac{1}{4} x^4 \log \left (a+b e^x\right )-\frac{1}{4} x^4 \log \left (1+\frac{b e^x}{a}\right )+\int x^3 \log \left (1+\frac{b e^x}{a}\right ) \, dx\\ &=\frac{1}{4} x^4 \log \left (a+b e^x\right )-\frac{1}{4} x^4 \log \left (1+\frac{b e^x}{a}\right )-x^3 \text{Li}_2\left (-\frac{b e^x}{a}\right )+3 \int x^2 \text{Li}_2\left (-\frac{b e^x}{a}\right ) \, dx\\ &=\frac{1}{4} x^4 \log \left (a+b e^x\right )-\frac{1}{4} x^4 \log \left (1+\frac{b e^x}{a}\right )-x^3 \text{Li}_2\left (-\frac{b e^x}{a}\right )+3 x^2 \text{Li}_3\left (-\frac{b e^x}{a}\right )-6 \int x \text{Li}_3\left (-\frac{b e^x}{a}\right ) \, dx\\ &=\frac{1}{4} x^4 \log \left (a+b e^x\right )-\frac{1}{4} x^4 \log \left (1+\frac{b e^x}{a}\right )-x^3 \text{Li}_2\left (-\frac{b e^x}{a}\right )+3 x^2 \text{Li}_3\left (-\frac{b e^x}{a}\right )-6 x \text{Li}_4\left (-\frac{b e^x}{a}\right )+6 \int \text{Li}_4\left (-\frac{b e^x}{a}\right ) \, dx\\ &=\frac{1}{4} x^4 \log \left (a+b e^x\right )-\frac{1}{4} x^4 \log \left (1+\frac{b e^x}{a}\right )-x^3 \text{Li}_2\left (-\frac{b e^x}{a}\right )+3 x^2 \text{Li}_3\left (-\frac{b e^x}{a}\right )-6 x \text{Li}_4\left (-\frac{b e^x}{a}\right )+6 \operatorname{Subst}\left (\int \frac{\text{Li}_4\left (-\frac{b x}{a}\right )}{x} \, dx,x,e^x\right )\\ &=\frac{1}{4} x^4 \log \left (a+b e^x\right )-\frac{1}{4} x^4 \log \left (1+\frac{b e^x}{a}\right )-x^3 \text{Li}_2\left (-\frac{b e^x}{a}\right )+3 x^2 \text{Li}_3\left (-\frac{b e^x}{a}\right )-6 x \text{Li}_4\left (-\frac{b e^x}{a}\right )+6 \text{Li}_5\left (-\frac{b e^x}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.0071216, size = 93, normalized size = 1. \[ -x^3 \text{PolyLog}\left (2,-\frac{b e^x}{a}\right )+3 x^2 \text{PolyLog}\left (3,-\frac{b e^x}{a}\right )-6 x \text{PolyLog}\left (4,-\frac{b e^x}{a}\right )+6 \text{PolyLog}\left (5,-\frac{b e^x}{a}\right )+\frac{1}{4} x^4 \log \left (a+b e^x\right )-\frac{1}{4} x^4 \log \left (\frac{b e^x}{a}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 84, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}\ln \left ( a+b{{\rm e}^{x}} \right ) }{4}}-{\frac{{x}^{4}}{4}\ln \left ( 1+{\frac{b{{\rm e}^{x}}}{a}} \right ) }-{x}^{3}{\it polylog} \left ( 2,-{\frac{b{{\rm e}^{x}}}{a}} \right ) +3\,{x}^{2}{\it polylog} \left ( 3,-{\frac{b{{\rm e}^{x}}}{a}} \right ) -6\,x{\it polylog} \left ( 4,-{\frac{b{{\rm e}^{x}}}{a}} \right ) +6\,{\it polylog} \left ( 5,-{\frac{b{{\rm e}^{x}}}{a}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20719, size = 111, normalized size = 1.19 \begin{align*} \frac{1}{4} \, x^{4} \log \left (b e^{x} + a\right ) - \frac{1}{4} \, x^{4} \log \left (\frac{b e^{x}}{a} + 1\right ) - x^{3}{\rm Li}_2\left (-\frac{b e^{x}}{a}\right ) + 3 \, x^{2}{\rm Li}_{3}(-\frac{b e^{x}}{a}) - 6 \, x{\rm Li}_{4}(-\frac{b e^{x}}{a}) + 6 \,{\rm Li}_{5}(-\frac{b e^{x}}{a}) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.04759, size = 224, normalized size = 2.41 \begin{align*} \frac{1}{4} \, x^{4} \log \left (b e^{x} + a\right ) - \frac{1}{4} \, x^{4} \log \left (\frac{b e^{x} + a}{a}\right ) - x^{3}{\rm Li}_2\left (-\frac{b e^{x} + a}{a} + 1\right ) + 3 \, x^{2}{\rm polylog}\left (3, -\frac{b e^{x}}{a}\right ) - 6 \, x{\rm polylog}\left (4, -\frac{b e^{x}}{a}\right ) + 6 \,{\rm polylog}\left (5, -\frac{b e^{x}}{a}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{b \int \frac{x^{4} e^{x}}{a + b e^{x}}\, dx}{4} + \frac{x^{4} \log{\left (a + b e^{x} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \log \left (b e^{x} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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