Optimal. Leaf size=59 \[ -x \text {Li}_2\left (-\frac {b e^x}{a}\right )+\text {Li}_3\left (-\frac {b e^x}{a}\right )+\frac {1}{2} x^2 \log \left (a+b e^x\right )-\frac {1}{2} x^2 \log \left (\frac {b e^x}{a}+1\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2532, 2531, 2282, 6589} \[ -x \text {PolyLog}\left (2,-\frac {b e^x}{a}\right )+\text {PolyLog}\left (3,-\frac {b e^x}{a}\right )+\frac {1}{2} x^2 \log \left (a+b e^x\right )-\frac {1}{2} x^2 \log \left (\frac {b e^x}{a}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 2532
Rule 6589
Rubi steps
\begin {align*} \int x \log \left (a+b e^x\right ) \, dx &=\frac {1}{2} x^2 \log \left (a+b e^x\right )-\frac {1}{2} x^2 \log \left (1+\frac {b e^x}{a}\right )+\int x \log \left (1+\frac {b e^x}{a}\right ) \, dx\\ &=\frac {1}{2} x^2 \log \left (a+b e^x\right )-\frac {1}{2} x^2 \log \left (1+\frac {b e^x}{a}\right )-x \text {Li}_2\left (-\frac {b e^x}{a}\right )+\int \text {Li}_2\left (-\frac {b e^x}{a}\right ) \, dx\\ &=\frac {1}{2} x^2 \log \left (a+b e^x\right )-\frac {1}{2} x^2 \log \left (1+\frac {b e^x}{a}\right )-x \text {Li}_2\left (-\frac {b e^x}{a}\right )+\operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {b x}{a}\right )}{x} \, dx,x,e^x\right )\\ &=\frac {1}{2} x^2 \log \left (a+b e^x\right )-\frac {1}{2} x^2 \log \left (1+\frac {b e^x}{a}\right )-x \text {Li}_2\left (-\frac {b e^x}{a}\right )+\text {Li}_3\left (-\frac {b e^x}{a}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 59, normalized size = 1.00 \[ -x \text {Li}_2\left (-\frac {b e^x}{a}\right )+\text {Li}_3\left (-\frac {b e^x}{a}\right )+\frac {1}{2} x^2 \log \left (a+b e^x\right )-\frac {1}{2} x^2 \log \left (\frac {b e^x}{a}+1\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.60, size = 56, normalized size = 0.95 \[ \frac {1}{2} \, x^{2} \log \left (b e^{x} + a\right ) - \frac {1}{2} \, x^{2} \log \left (\frac {b e^{x} + a}{a}\right ) - x {\rm Li}_2\left (-\frac {b e^{x} + a}{a} + 1\right ) + {\rm polylog}\left (3, -\frac {b e^{x}}{a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \log \left (b e^{x} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 52, normalized size = 0.88 \[ -\frac {x^{2} \ln \left (\frac {b \,{\mathrm e}^{x}}{a}+1\right )}{2}+\frac {x^{2} \ln \left (b \,{\mathrm e}^{x}+a \right )}{2}-x \polylog \left (2, -\frac {b \,{\mathrm e}^{x}}{a}\right )+\polylog \left (3, -\frac {b \,{\mathrm e}^{x}}{a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 50, normalized size = 0.85 \[ \frac {1}{2} \, x^{2} \log \left (b e^{x} + a\right ) - \frac {1}{2} \, x^{2} \log \left (\frac {b e^{x}}{a} + 1\right ) - x {\rm Li}_2\left (-\frac {b e^{x}}{a}\right ) + {\rm Li}_{3}(-\frac {b e^{x}}{a}) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\ln \left (a+b\,{\mathrm {e}}^x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {b \int \frac {x^{2} e^{x}}{a + b e^{x}}\, dx}{2} + \frac {x^{2} \log {\left (a + b e^{x} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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