Optimal. Leaf size=391 \[ \frac{4 c x \text{PolyLog}\left (2,-\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{4 c x \text{PolyLog}\left (2,-\frac{2 c e^x}{\sqrt{b^2-4 a c}+b}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{4 c \text{PolyLog}\left (3,-\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{4 c \text{PolyLog}\left (3,-\frac{2 c e^x}{\sqrt{b^2-4 a c}+b}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c x^3}{3 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{2 c x^3}{3 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{2 c x^2 \log \left (\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}+1\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{2 c x^2 \log \left (\frac{2 c e^x}{\sqrt{b^2-4 a c}+b}+1\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2} \]
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Rubi [A] time = 0.665303, antiderivative size = 391, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2263, 2184, 2190, 2531, 2282, 6589} \[ \frac{4 c x \text{PolyLog}\left (2,-\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{4 c x \text{PolyLog}\left (2,-\frac{2 c e^x}{\sqrt{b^2-4 a c}+b}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{4 c \text{PolyLog}\left (3,-\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{4 c \text{PolyLog}\left (3,-\frac{2 c e^x}{\sqrt{b^2-4 a c}+b}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{2 c x^3}{3 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{2 c x^3}{3 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{2 c x^2 \log \left (\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}+1\right )}{-b \sqrt{b^2-4 a c}-4 a c+b^2}+\frac{2 c x^2 \log \left (\frac{2 c e^x}{\sqrt{b^2-4 a c}+b}+1\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2} \]
Antiderivative was successfully verified.
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Rule 2263
Rule 2184
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{x^2}{a+b e^x+c e^{2 x}} \, dx &=\frac{(2 c) \int \frac{x^2}{b-\sqrt{b^2-4 a c}+2 c e^x} \, dx}{\sqrt{b^2-4 a c}}-\frac{(2 c) \int \frac{x^2}{b+\sqrt{b^2-4 a c}+2 c e^x} \, dx}{\sqrt{b^2-4 a c}}\\ &=-\frac{2 c x^3}{3 \left (b^2-4 a c-b \sqrt{b^2-4 a c}\right )}-\frac{2 c x^3}{3 \left (b^2-4 a c+b \sqrt{b^2-4 a c}\right )}+\frac{\left (4 c^2\right ) \int \frac{e^x x^2}{b-\sqrt{b^2-4 a c}+2 c e^x} \, dx}{b^2-4 a c-b \sqrt{b^2-4 a c}}+\frac{\left (4 c^2\right ) \int \frac{e^x x^2}{b+\sqrt{b^2-4 a c}+2 c e^x} \, dx}{b^2-4 a c+b \sqrt{b^2-4 a c}}\\ &=-\frac{2 c x^3}{3 \left (b^2-4 a c-b \sqrt{b^2-4 a c}\right )}-\frac{2 c x^3}{3 \left (b^2-4 a c+b \sqrt{b^2-4 a c}\right )}+\frac{2 c x^2 \log \left (1+\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt{b^2-4 a c}}+\frac{2 c x^2 \log \left (1+\frac{2 c e^x}{b+\sqrt{b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt{b^2-4 a c}}-\frac{(4 c) \int x \log \left (1+\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}\right ) \, dx}{b^2-4 a c-b \sqrt{b^2-4 a c}}-\frac{(4 c) \int x \log \left (1+\frac{2 c e^x}{b+\sqrt{b^2-4 a c}}\right ) \, dx}{b^2-4 a c+b \sqrt{b^2-4 a c}}\\ &=-\frac{2 c x^3}{3 \left (b^2-4 a c-b \sqrt{b^2-4 a c}\right )}-\frac{2 c x^3}{3 \left (b^2-4 a c+b \sqrt{b^2-4 a c}\right )}+\frac{2 c x^2 \log \left (1+\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt{b^2-4 a c}}+\frac{2 c x^2 \log \left (1+\frac{2 c e^x}{b+\sqrt{b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt{b^2-4 a c}}+\frac{4 c x \text{Li}_2\left (-\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt{b^2-4 a c}}+\frac{4 c x \text{Li}_2\left (-\frac{2 c e^x}{b+\sqrt{b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt{b^2-4 a c}}-\frac{(4 c) \int \text{Li}_2\left (-\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}\right ) \, dx}{b^2-4 a c-b \sqrt{b^2-4 a c}}-\frac{(4 c) \int \text{Li}_2\left (-\frac{2 c e^x}{b+\sqrt{b^2-4 a c}}\right ) \, dx}{b^2-4 a c+b \sqrt{b^2-4 a c}}\\ &=-\frac{2 c x^3}{3 \left (b^2-4 a c-b \sqrt{b^2-4 a c}\right )}-\frac{2 c x^3}{3 \left (b^2-4 a c+b \sqrt{b^2-4 a c}\right )}+\frac{2 c x^2 \log \left (1+\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt{b^2-4 a c}}+\frac{2 c x^2 \log \left (1+\frac{2 c e^x}{b+\sqrt{b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt{b^2-4 a c}}+\frac{4 c x \text{Li}_2\left (-\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt{b^2-4 a c}}+\frac{4 c x \text{Li}_2\left (-\frac{2 c e^x}{b+\sqrt{b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt{b^2-4 a c}}-\frac{(4 c) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{2 c x}{-b+\sqrt{b^2-4 a c}}\right )}{x} \, dx,x,e^x\right )}{b^2-4 a c-b \sqrt{b^2-4 a c}}-\frac{(4 c) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{x} \, dx,x,e^x\right )}{b^2-4 a c+b \sqrt{b^2-4 a c}}\\ &=-\frac{2 c x^3}{3 \left (b^2-4 a c-b \sqrt{b^2-4 a c}\right )}-\frac{2 c x^3}{3 \left (b^2-4 a c+b \sqrt{b^2-4 a c}\right )}+\frac{2 c x^2 \log \left (1+\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt{b^2-4 a c}}+\frac{2 c x^2 \log \left (1+\frac{2 c e^x}{b+\sqrt{b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt{b^2-4 a c}}+\frac{4 c x \text{Li}_2\left (-\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt{b^2-4 a c}}+\frac{4 c x \text{Li}_2\left (-\frac{2 c e^x}{b+\sqrt{b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt{b^2-4 a c}}-\frac{4 c \text{Li}_3\left (-\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt{b^2-4 a c}}-\frac{4 c \text{Li}_3\left (-\frac{2 c e^x}{b+\sqrt{b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt{b^2-4 a c}}\\ \end{align*}
Mathematica [A] time = 0.184272, size = 407, normalized size = 1.04 \[ \frac{-6 x \left (\sqrt{b^2-4 a c}+b\right ) \text{PolyLog}\left (2,\frac{2 c e^x}{\sqrt{b^2-4 a c}-b}\right )+6 x \left (b-\sqrt{b^2-4 a c}\right ) \text{PolyLog}\left (2,-\frac{2 c e^x}{\sqrt{b^2-4 a c}+b}\right )+6 b \text{PolyLog}\left (3,\frac{2 c e^x}{\sqrt{b^2-4 a c}-b}\right )+6 \sqrt{b^2-4 a c} \text{PolyLog}\left (3,\frac{2 c e^x}{\sqrt{b^2-4 a c}-b}\right )-6 b \text{PolyLog}\left (3,-\frac{2 c e^x}{\sqrt{b^2-4 a c}+b}\right )+6 \sqrt{b^2-4 a c} \text{PolyLog}\left (3,-\frac{2 c e^x}{\sqrt{b^2-4 a c}+b}\right )+2 x^3 \sqrt{b^2-4 a c}-3 b x^2 \log \left (\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}+1\right )-3 x^2 \sqrt{b^2-4 a c} \log \left (\frac{2 c e^x}{b-\sqrt{b^2-4 a c}}+1\right )+3 b x^2 \log \left (\frac{2 c e^x}{\sqrt{b^2-4 a c}+b}+1\right )-3 x^2 \sqrt{b^2-4 a c} \log \left (\frac{2 c e^x}{\sqrt{b^2-4 a c}+b}+1\right )}{6 a \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.029, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{a+b{{\rm e}^{x}}+c{{\rm e}^{2\,x}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.55385, size = 977, normalized size = 2.5 \begin{align*} \frac{2 \,{\left (b^{2} - 4 \, a c\right )} x^{3} - 6 \,{\left (a b x \sqrt{\frac{b^{2} - 4 \, a c}{a^{2}}} +{\left (b^{2} - 4 \, a c\right )} x\right )}{\rm Li}_2\left (-\frac{a \sqrt{\frac{b^{2} - 4 \, a c}{a^{2}}} e^{x} + b e^{x} + 2 \, a}{2 \, a} + 1\right ) + 6 \,{\left (a b x \sqrt{\frac{b^{2} - 4 \, a c}{a^{2}}} -{\left (b^{2} - 4 \, a c\right )} x\right )}{\rm Li}_2\left (\frac{a \sqrt{\frac{b^{2} - 4 \, a c}{a^{2}}} e^{x} - b e^{x} - 2 \, a}{2 \, a} + 1\right ) - 3 \,{\left (a b x^{2} \sqrt{\frac{b^{2} - 4 \, a c}{a^{2}}} +{\left (b^{2} - 4 \, a c\right )} x^{2}\right )} \log \left (\frac{a \sqrt{\frac{b^{2} - 4 \, a c}{a^{2}}} e^{x} + b e^{x} + 2 \, a}{2 \, a}\right ) + 3 \,{\left (a b x^{2} \sqrt{\frac{b^{2} - 4 \, a c}{a^{2}}} -{\left (b^{2} - 4 \, a c\right )} x^{2}\right )} \log \left (-\frac{a \sqrt{\frac{b^{2} - 4 \, a c}{a^{2}}} e^{x} - b e^{x} - 2 \, a}{2 \, a}\right ) + 6 \,{\left (a b \sqrt{\frac{b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 4 \, a c\right )}{\rm polylog}\left (3, -\frac{a \sqrt{\frac{b^{2} - 4 \, a c}{a^{2}}} e^{x} + b e^{x}}{2 \, a}\right ) - 6 \,{\left (a b \sqrt{\frac{b^{2} - 4 \, a c}{a^{2}}} - b^{2} + 4 \, a c\right )}{\rm polylog}\left (3, \frac{a \sqrt{\frac{b^{2} - 4 \, a c}{a^{2}}} e^{x} - b e^{x}}{2 \, a}\right )}{6 \,{\left (a b^{2} - 4 \, a^{2} c\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*} \int \frac{x^{2}}{a + b e^{x} + c e^{2 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}}{c e^{\left (2 \, x\right )} + b e^{x} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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