Optimal. Leaf size=59 \[ -x \text{PolyLog}\left (2,-\frac{b e^x}{a}\right )+\text{PolyLog}\left (3,-\frac{b e^x}{a}\right )+\frac{1}{2} x^2 \log \left (a+b e^x\right )-\frac{1}{2} x^2 \log \left (\frac{b e^x}{a}+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.036611, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, 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.
[In]
[Out]
Rule 2532
Rule 2531
Rule 2282
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*}
Mathematica [A] time = 0.0045427, size = 59, normalized size = 1. \[ -x \text{PolyLog}\left (2,-\frac{b e^x}{a}\right )+\text{PolyLog}\left (3,-\frac{b e^x}{a}\right )+\frac{1}{2} x^2 \log \left (a+b e^x\right )-\frac{1}{2} x^2 \log \left (\frac{b e^x}{a}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 52, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}\ln \left ( a+b{{\rm e}^{x}} \right ) }{2}}-{\frac{{x}^{2}}{2}\ln \left ( 1+{\frac{b{{\rm e}^{x}}}{a}} \right ) }-x{\it polylog} \left ( 2,-{\frac{b{{\rm e}^{x}}}{a}} \right ) +{\it polylog} \left ( 3,-{\frac{b{{\rm e}^{x}}}{a}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.12416, size = 68, normalized size = 1.15 \begin{align*} \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}) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 2.13465, size = 143, normalized size = 2.42 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \log \left (b e^{x} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]