Optimal. Leaf size=77 \[ -x^2 \text{PolyLog}\left (2,-\frac{b e^x}{a}\right )+2 x \text{PolyLog}\left (3,-\frac{b e^x}{a}\right )-2 \text{PolyLog}\left (4,-\frac{b e^x}{a}\right )+\frac{1}{3} x^3 \log \left (a+b e^x\right )-\frac{1}{3} x^3 \log \left (\frac{b e^x}{a}+1\right ) \]
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Rubi [A] time = 0.0584256, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ -x^2 \text{PolyLog}\left (2,-\frac{b e^x}{a}\right )+2 x \text{PolyLog}\left (3,-\frac{b e^x}{a}\right )-2 \text{PolyLog}\left (4,-\frac{b e^x}{a}\right )+\frac{1}{3} x^3 \log \left (a+b e^x\right )-\frac{1}{3} x^3 \log \left (\frac{b e^x}{a}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2532
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \log \left (a+b e^x\right ) \, dx &=\frac{1}{3} x^3 \log \left (a+b e^x\right )-\frac{1}{3} x^3 \log \left (1+\frac{b e^x}{a}\right )+\int x^2 \log \left (1+\frac{b e^x}{a}\right ) \, dx\\ &=\frac{1}{3} x^3 \log \left (a+b e^x\right )-\frac{1}{3} x^3 \log \left (1+\frac{b e^x}{a}\right )-x^2 \text{Li}_2\left (-\frac{b e^x}{a}\right )+2 \int x \text{Li}_2\left (-\frac{b e^x}{a}\right ) \, dx\\ &=\frac{1}{3} x^3 \log \left (a+b e^x\right )-\frac{1}{3} x^3 \log \left (1+\frac{b e^x}{a}\right )-x^2 \text{Li}_2\left (-\frac{b e^x}{a}\right )+2 x \text{Li}_3\left (-\frac{b e^x}{a}\right )-2 \int \text{Li}_3\left (-\frac{b e^x}{a}\right ) \, dx\\ &=\frac{1}{3} x^3 \log \left (a+b e^x\right )-\frac{1}{3} x^3 \log \left (1+\frac{b e^x}{a}\right )-x^2 \text{Li}_2\left (-\frac{b e^x}{a}\right )+2 x \text{Li}_3\left (-\frac{b e^x}{a}\right )-2 \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{b x}{a}\right )}{x} \, dx,x,e^x\right )\\ &=\frac{1}{3} x^3 \log \left (a+b e^x\right )-\frac{1}{3} x^3 \log \left (1+\frac{b e^x}{a}\right )-x^2 \text{Li}_2\left (-\frac{b e^x}{a}\right )+2 x \text{Li}_3\left (-\frac{b e^x}{a}\right )-2 \text{Li}_4\left (-\frac{b e^x}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.0052255, size = 77, normalized size = 1. \[ -x^2 \text{PolyLog}\left (2,-\frac{b e^x}{a}\right )+2 x \text{PolyLog}\left (3,-\frac{b e^x}{a}\right )-2 \text{PolyLog}\left (4,-\frac{b e^x}{a}\right )+\frac{1}{3} x^3 \log \left (a+b e^x\right )-\frac{1}{3} x^3 \log \left (\frac{b e^x}{a}+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 69, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}\ln \left ( a+b{{\rm e}^{x}} \right ) }{3}}-{\frac{{x}^{3}}{3}\ln \left ( 1+{\frac{b{{\rm e}^{x}}}{a}} \right ) }-{x}^{2}{\it polylog} \left ( 2,-{\frac{b{{\rm e}^{x}}}{a}} \right ) +2\,x{\it polylog} \left ( 3,-{\frac{b{{\rm e}^{x}}}{a}} \right ) -2\,{\it polylog} \left ( 4,-{\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 = 1.12855, size = 90, normalized size = 1.17 \begin{align*} \frac{1}{3} \, x^{3} \log \left (b e^{x} + a\right ) - \frac{1}{3} \, x^{3} \log \left (\frac{b e^{x}}{a} + 1\right ) - x^{2}{\rm Li}_2\left (-\frac{b e^{x}}{a}\right ) + 2 \, x{\rm Li}_{3}(-\frac{b e^{x}}{a}) - 2 \,{\rm Li}_{4}(-\frac{b e^{x}}{a}) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.05659, size = 185, normalized size = 2.4 \begin{align*} \frac{1}{3} \, x^{3} \log \left (b e^{x} + a\right ) - \frac{1}{3} \, x^{3} \log \left (\frac{b e^{x} + a}{a}\right ) - x^{2}{\rm Li}_2\left (-\frac{b e^{x} + a}{a} + 1\right ) + 2 \, x{\rm polylog}\left (3, -\frac{b e^{x}}{a}\right ) - 2 \,{\rm polylog}\left (4, -\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^{3} e^{x}}{a + b e^{x}}\, dx}{3} + \frac{x^{3} \log{\left (a + b e^{x} \right )}}{3} \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^{2} \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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