Integrand size = 20, antiderivative size = 391 \[ \int \frac {x^2}{a+b e^x+c e^{2 x}} \, dx=-\frac {2 c x^3}{3 \left (b^2-4 a c-b \sqrt {b^2-4 a c}\right )}-\frac {2 c x^3}{3 \left (b^2-4 a c+b \sqrt {b^2-4 a c}\right )}+\frac {2 c x^2 \log \left (1+\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {2 c x^2 \log \left (1+\frac {2 c e^x}{b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}+\frac {4 c x \operatorname {PolyLog}\left (2,-\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}+\frac {4 c x \operatorname {PolyLog}\left (2,-\frac {2 c e^x}{b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}-\frac {4 c \operatorname {PolyLog}\left (3,-\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {4 c \operatorname {PolyLog}\left (3,-\frac {2 c e^x}{b+\sqrt {b^2-4 a c}}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}} \]
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Time = 0.37 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2295, 2215, 2221, 2611, 2320, 6724} \[ \int \frac {x^2}{a+b e^x+c e^{2 x}} \, dx=\frac {4 c x \operatorname {PolyLog}\left (2,-\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}\right )}{-b \sqrt {b^2-4 a c}-4 a c+b^2}+\frac {4 c x \operatorname {PolyLog}\left (2,-\frac {2 c e^x}{b+\sqrt {b^2-4 a c}}\right )}{b \sqrt {b^2-4 a c}-4 a c+b^2}-\frac {4 c \operatorname {PolyLog}\left (3,-\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}\right )}{-b \sqrt {b^2-4 a c}-4 a c+b^2}-\frac {4 c \operatorname {PolyLog}\left (3,-\frac {2 c e^x}{b+\sqrt {b^2-4 a c}}\right )}{b \sqrt {b^2-4 a c}-4 a c+b^2}-\frac {2 c x^3}{3 \left (-b \sqrt {b^2-4 a c}-4 a c+b^2\right )}-\frac {2 c x^3}{3 \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )}+\frac {2 c x^2 \log \left (\frac {2 c e^x}{b-\sqrt {b^2-4 a c}}+1\right )}{-b \sqrt {b^2-4 a c}-4 a c+b^2}+\frac {2 c x^2 \log \left (\frac {2 c e^x}{\sqrt {b^2-4 a c}+b}+1\right )}{b \sqrt {b^2-4 a c}-4 a c+b^2} \]
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Rule 2215
Rule 2221
Rule 2295
Rule 2320
Rule 2611
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \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) \text {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) \text {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*}
Time = 0.19 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.71 \[ \int \frac {x^2}{a+b e^x+c e^{2 x}} \, dx=\frac {2 c \left (\frac {x^2 \log \left (1+\frac {\left (b-\sqrt {b^2-4 a c}\right ) e^{-x}}{2 c}\right )}{-b+\sqrt {b^2-4 a c}}+\frac {x^2 \log \left (1+\frac {\left (b+\sqrt {b^2-4 a c}\right ) e^{-x}}{2 c}\right )}{b+\sqrt {b^2-4 a c}}+\frac {2 \left (x \operatorname {PolyLog}\left (2,\frac {\left (-b+\sqrt {b^2-4 a c}\right ) e^{-x}}{2 c}\right )+\operatorname {PolyLog}\left (3,\frac {\left (-b+\sqrt {b^2-4 a c}\right ) e^{-x}}{2 c}\right )\right )}{b-\sqrt {b^2-4 a c}}-\frac {2 \left (x \operatorname {PolyLog}\left (2,-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e^{-x}}{2 c}\right )+\operatorname {PolyLog}\left (3,-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e^{-x}}{2 c}\right )\right )}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c}} \]
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\[\int \frac {x^{2}}{a +b \,{\mathrm e}^{x}+c \,{\mathrm e}^{2 x}}d x\]
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Time = 0.30 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.06 \[ \int \frac {x^2}{a+b e^x+c e^{2 x}} \, dx=\frac {2 \, {\left (b^{2} - 4 \, a c\right )} x^{3} - 6 \, {\left (a b x \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + {\left (b^{2} - 4 \, a c\right )} x\right )} {\rm Li}_2\left (-\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{x} + b e^{x} + 2 \, a}{2 \, a} + 1\right ) + 6 \, {\left (a b x \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - {\left (b^{2} - 4 \, a c\right )} x\right )} {\rm Li}_2\left (\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{x} - b e^{x} - 2 \, a}{2 \, a} + 1\right ) - 3 \, {\left (a b x^{2} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + {\left (b^{2} - 4 \, a c\right )} x^{2}\right )} \log \left (\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{x} + b e^{x} + 2 \, a}{2 \, a}\right ) + 3 \, {\left (a b x^{2} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - {\left (b^{2} - 4 \, a c\right )} x^{2}\right )} \log \left (-\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{x} - b e^{x} - 2 \, a}{2 \, a}\right ) + 6 \, {\left (a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 4 \, a c\right )} {\rm polylog}\left (3, -\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{x} + b e^{x}}{2 \, a}\right ) - 6 \, {\left (a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b^{2} + 4 \, a c\right )} {\rm polylog}\left (3, \frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} e^{x} - b e^{x}}{2 \, a}\right )}{6 \, {\left (a b^{2} - 4 \, a^{2} c\right )}} \]
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\[ \int \frac {x^2}{a+b e^x+c e^{2 x}} \, dx=\int \frac {x^{2}}{a + b e^{x} + c e^{2 x}}\, dx \]
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Exception generated. \[ \int \frac {x^2}{a+b e^x+c e^{2 x}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^2}{a+b e^x+c e^{2 x}} \, dx=\int { \frac {x^{2}}{c e^{\left (2 \, x\right )} + b e^{x} + a} \,d x } \]
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Timed out. \[ \int \frac {x^2}{a+b e^x+c e^{2 x}} \, dx=\int \frac {x^2}{a+b\,{\mathrm {e}}^x+c\,{\mathrm {e}}^{2\,x}} \,d x \]
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