Optimal. Leaf size=244 \[ \frac {x^2 \log \left (1+\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}-\frac {x^2 \log \left (1+\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}+\frac {2 x \text {Li}_2\left (-\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}-\frac {2 x \text {Li}_2\left (-\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}-\frac {2 \text {Li}_3\left (-\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}+\frac {2 \text {Li}_3\left (-\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}} \]
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Rubi [A]
time = 0.34, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2299, 2296,
2221, 2611, 2320, 6724} \begin {gather*} \frac {2 x \text {PolyLog}\left (2,-\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}-\frac {2 x \text {PolyLog}\left (2,-\frac {2 c e^x}{\sqrt {a^2-4 b c}+a}\right )}{\sqrt {a^2-4 b c}}-\frac {2 \text {PolyLog}\left (3,-\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}+\frac {2 \text {PolyLog}\left (3,-\frac {2 c e^x}{\sqrt {a^2-4 b c}+a}\right )}{\sqrt {a^2-4 b c}}+\frac {x^2 \log \left (\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}+1\right )}{\sqrt {a^2-4 b c}}-\frac {x^2 \log \left (\frac {2 c e^x}{\sqrt {a^2-4 b c}+a}+1\right )}{\sqrt {a^2-4 b c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2296
Rule 2299
Rule 2320
Rule 2611
Rule 6724
Rubi steps
\begin {align*} \int \frac {x^2}{a+b e^{-x}+c e^x} \, dx &=\int \frac {e^x x^2}{b+a e^x+c e^{2 x}} \, dx\\ &=\frac {(2 c) \int \frac {e^x x^2}{a-\sqrt {a^2-4 b c}+2 c e^x} \, dx}{\sqrt {a^2-4 b c}}-\frac {(2 c) \int \frac {e^x x^2}{a+\sqrt {a^2-4 b c}+2 c e^x} \, dx}{\sqrt {a^2-4 b c}}\\ &=\frac {x^2 \log \left (1+\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}-\frac {x^2 \log \left (1+\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}-\frac {2 \int x \log \left (1+\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right ) \, dx}{\sqrt {a^2-4 b c}}+\frac {2 \int x \log \left (1+\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right ) \, dx}{\sqrt {a^2-4 b c}}\\ &=\frac {x^2 \log \left (1+\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}-\frac {x^2 \log \left (1+\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}+\frac {2 x \text {Li}_2\left (-\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}-\frac {2 x \text {Li}_2\left (-\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}-\frac {2 \int \text {Li}_2\left (-\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right ) \, dx}{\sqrt {a^2-4 b c}}+\frac {2 \int \text {Li}_2\left (-\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right ) \, dx}{\sqrt {a^2-4 b c}}\\ &=\frac {x^2 \log \left (1+\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}-\frac {x^2 \log \left (1+\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}+\frac {2 x \text {Li}_2\left (-\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}-\frac {2 x \text {Li}_2\left (-\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}-\frac {2 \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {2 c x}{-a+\sqrt {a^2-4 b c}}\right )}{x} \, dx,x,e^x\right )}{\sqrt {a^2-4 b c}}+\frac {2 \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {2 c x}{a+\sqrt {a^2-4 b c}}\right )}{x} \, dx,x,e^x\right )}{\sqrt {a^2-4 b c}}\\ &=\frac {x^2 \log \left (1+\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}-\frac {x^2 \log \left (1+\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}+\frac {2 x \text {Li}_2\left (-\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}-\frac {2 x \text {Li}_2\left (-\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}-\frac {2 \text {Li}_3\left (-\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}+\frac {2 \text {Li}_3\left (-\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 185, normalized size = 0.76 \begin {gather*} \frac {x^2 \log \left (1+\frac {2 c e^x}{a-\sqrt {a^2-4 b c}}\right )-x^2 \log \left (1+\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right )+2 x \text {Li}_2\left (\frac {2 c e^x}{-a+\sqrt {a^2-4 b c}}\right )-2 x \text {Li}_2\left (-\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right )-2 \text {Li}_3\left (\frac {2 c e^x}{-a+\sqrt {a^2-4 b c}}\right )+2 \text {Li}_3\left (-\frac {2 c e^x}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{a +b \,{\mathrm e}^{-x}+c \,{\mathrm e}^{x}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 316, normalized size = 1.30 \begin {gather*} \frac {b x^{2} \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} \log \left (\frac {b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} e^{x} + a e^{x} + 2 \, b}{2 \, b}\right ) - b x^{2} \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} \log \left (-\frac {b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} e^{x} - a e^{x} - 2 \, b}{2 \, b}\right ) + 2 \, b x \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} {\rm Li}_2\left (-\frac {b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} e^{x} + a e^{x} + 2 \, b}{2 \, b} + 1\right ) - 2 \, b x \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} {\rm Li}_2\left (\frac {b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} e^{x} - a e^{x} - 2 \, b}{2 \, b} + 1\right ) - 2 \, b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} {\rm polylog}\left (3, -\frac {b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} e^{x} + a e^{x}}{2 \, b}\right ) + 2 \, b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} {\rm polylog}\left (3, \frac {b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} e^{x} - a e^{x}}{2 \, b}\right )}{a^{2} - 4 \, b c} \end {gather*}
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 \begin {gather*} \int \frac {x^{2} e^{x}}{a e^{x} + b + c e^{2 x}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{a+c\,{\mathrm {e}}^x+b\,{\mathrm {e}}^{-x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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