Optimal. Leaf size=58 \[ -\frac{\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a d^2}-\frac{x \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a d}+\frac{x^2}{2 a} \]
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Rubi [A] time = 0.0991882, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {2184, 2190, 2279, 2391} \[ -\frac{\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a d^2}-\frac{x \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a d}+\frac{x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{x}{a+b e^{c+d x}} \, dx &=\frac{x^2}{2 a}-\frac{b \int \frac{e^{c+d x} x}{a+b e^{c+d x}} \, dx}{a}\\ &=\frac{x^2}{2 a}-\frac{x \log \left (1+\frac{b e^{c+d x}}{a}\right )}{a d}+\frac{\int \log \left (1+\frac{b e^{c+d x}}{a}\right ) \, dx}{a d}\\ &=\frac{x^2}{2 a}-\frac{x \log \left (1+\frac{b e^{c+d x}}{a}\right )}{a d}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a}\right )}{x} \, dx,x,e^{c+d x}\right )}{a d^2}\\ &=\frac{x^2}{2 a}-\frac{x \log \left (1+\frac{b e^{c+d x}}{a}\right )}{a d}-\frac{\text{Li}_2\left (-\frac{b e^{c+d x}}{a}\right )}{a d^2}\\ \end{align*}
Mathematica [A] time = 0.0030927, size = 53, normalized size = 0.91 \[ \frac{\text{PolyLog}\left (2,-\frac{a e^{-c-d x}}{b}\right )}{a d^2}-\frac{x \log \left (\frac{a e^{-c-d x}}{b}+1\right )}{a d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 133, normalized size = 2.3 \begin{align*}{\frac{{x}^{2}}{2\,a}}+{\frac{cx}{ad}}+{\frac{{c}^{2}}{2\,a{d}^{2}}}-{\frac{x}{ad}\ln \left ( 1+{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }-{\frac{c}{a{d}^{2}}\ln \left ( 1+{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }-{\frac{1}{a{d}^{2}}{\it polylog} \left ( 2,-{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }-{\frac{c\ln \left ({{\rm e}^{dx+c}} \right ) }{a{d}^{2}}}+{\frac{c\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{a{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08588, size = 65, normalized size = 1.12 \begin{align*} \frac{x^{2}}{2 \, a} - \frac{d x \log \left (\frac{b e^{\left (d x + c\right )}}{a} + 1\right ) +{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )}}{a}\right )}{a d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49365, size = 177, normalized size = 3.05 \begin{align*} \frac{d^{2} x^{2} + 2 \, c \log \left (b e^{\left (d x + c\right )} + a\right ) - 2 \,{\left (d x + c\right )} \log \left (\frac{b e^{\left (d x + c\right )} + a}{a}\right ) - 2 \,{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )} + a}{a} + 1\right )}{2 \, a d^{2}} \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*} \int \frac{x}{a + b e^{c} e^{d x}}\, dx \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 \frac{x}{b e^{\left (d x + c\right )} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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