Optimal. Leaf size=84 \[ -\frac{2 x \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a d^2}+\frac{2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a}\right )}{a d^3}-\frac{x^2 \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a d}+\frac{x^3}{3 a} \]
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Rubi [A] time = 0.167754, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {2184, 2190, 2531, 2282, 6589} \[ -\frac{2 x \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a d^2}+\frac{2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a}\right )}{a d^3}-\frac{x^2 \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a d}+\frac{x^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 2184
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{x^2}{a+b e^{c+d x}} \, dx &=\frac{x^3}{3 a}-\frac{b \int \frac{e^{c+d x} x^2}{a+b e^{c+d x}} \, dx}{a}\\ &=\frac{x^3}{3 a}-\frac{x^2 \log \left (1+\frac{b e^{c+d x}}{a}\right )}{a d}+\frac{2 \int x \log \left (1+\frac{b e^{c+d x}}{a}\right ) \, dx}{a d}\\ &=\frac{x^3}{3 a}-\frac{x^2 \log \left (1+\frac{b e^{c+d x}}{a}\right )}{a d}-\frac{2 x \text{Li}_2\left (-\frac{b e^{c+d x}}{a}\right )}{a d^2}+\frac{2 \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a}\right ) \, dx}{a d^2}\\ &=\frac{x^3}{3 a}-\frac{x^2 \log \left (1+\frac{b e^{c+d x}}{a}\right )}{a d}-\frac{2 x \text{Li}_2\left (-\frac{b e^{c+d x}}{a}\right )}{a d^2}+\frac{2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{a}\right )}{x} \, dx,x,e^{c+d x}\right )}{a d^3}\\ &=\frac{x^3}{3 a}-\frac{x^2 \log \left (1+\frac{b e^{c+d x}}{a}\right )}{a d}-\frac{2 x \text{Li}_2\left (-\frac{b e^{c+d x}}{a}\right )}{a d^2}+\frac{2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a}\right )}{a d^3}\\ \end{align*}
Mathematica [A] time = 0.0060089, size = 83, normalized size = 0.99 \[ \frac{2 x \text{PolyLog}\left (2,-\frac{a e^{-c-d x}}{b}\right )}{a d^2}+\frac{2 \text{PolyLog}\left (3,-\frac{a e^{-c-d x}}{b}\right )}{a d^3}-\frac{x^2 \log \left (\frac{a e^{-c-d x}}{b}+1\right )}{a d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 166, normalized size = 2. \begin{align*}{\frac{{x}^{3}}{3\,a}}-{\frac{{c}^{2}x}{a{d}^{2}}}-{\frac{2\,{c}^{3}}{3\,{d}^{3}a}}-{\frac{{x}^{2}}{ad}\ln \left ( 1+{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }+{\frac{{c}^{2}}{{d}^{3}a}\ln \left ( 1+{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }-2\,{\frac{x}{a{d}^{2}}{\it polylog} \left ( 2,-{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }+2\,{\frac{1}{{d}^{3}a}{\it polylog} \left ( 3,-{\frac{b{{\rm e}^{dx+c}}}{a}} \right ) }+{\frac{{c}^{2}\ln \left ({{\rm e}^{dx+c}} \right ) }{{d}^{3}a}}-{\frac{{c}^{2}\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{{d}^{3}a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07884, size = 97, normalized size = 1.15 \begin{align*} \frac{x^{3}}{3 \, a} - \frac{d^{2} x^{2} \log \left (\frac{b e^{\left (d x + c\right )}}{a} + 1\right ) + 2 \, d x{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )}}{a}\right ) - 2 \,{\rm Li}_{3}(-\frac{b e^{\left (d x + c\right )}}{a})}{a d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.52682, size = 238, normalized size = 2.83 \begin{align*} \frac{d^{3} x^{3} - 6 \, d x{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )} + a}{a} + 1\right ) - 3 \, c^{2} \log \left (b e^{\left (d x + c\right )} + a\right ) - 3 \,{\left (d^{2} x^{2} - c^{2}\right )} \log \left (\frac{b e^{\left (d x + c\right )} + a}{a}\right ) + 6 \,{\rm polylog}\left (3, -\frac{b e^{\left (d x + c\right )}}{a}\right )}{3 \, a d^{3}} \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^{2}}{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^{2}}{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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