Integrand size = 29, antiderivative size = 310 \[ \int \frac {x^2}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=\frac {x^2 \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {x^2 \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}+\frac {2 x \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}-\frac {2 x \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}-\frac {2 \operatorname {PolyLog}\left (3,-\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^3 \log ^3(f)}+\frac {2 \operatorname {PolyLog}\left (3,-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^3 \log ^3(f)} \]
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Time = 0.39 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2299, 2296, 2221, 2611, 2320, 6724} \[ \int \frac {x^2}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=-\frac {2 \operatorname {PolyLog}\left (3,-\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{d^3 \log ^3(f) \sqrt {a^2-4 b c}}+\frac {2 \operatorname {PolyLog}\left (3,-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{d^3 \log ^3(f) \sqrt {a^2-4 b c}}+\frac {2 x \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{d^2 \log ^2(f) \sqrt {a^2-4 b c}}-\frac {2 x \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{d^2 \log ^2(f) \sqrt {a^2-4 b c}}+\frac {x^2 \log \left (\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}+1\right )}{d \log (f) \sqrt {a^2-4 b c}}-\frac {x^2 \log \left (\frac {2 c f^{c+d x}}{\sqrt {a^2-4 b c}+a}+1\right )}{d \log (f) \sqrt {a^2-4 b c}} \]
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Rule 2221
Rule 2296
Rule 2299
Rule 2320
Rule 2611
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \frac {f^{c+d x} x^2}{b+a f^{c+d x}+c f^{2 (c+d x)}} \, dx \\ & = \frac {(2 c) \int \frac {f^{c+d x} x^2}{a-\sqrt {a^2-4 b c}+2 c f^{c+d x}} \, dx}{\sqrt {a^2-4 b c}}-\frac {(2 c) \int \frac {f^{c+d x} x^2}{a+\sqrt {a^2-4 b c}+2 c f^{c+d x}} \, dx}{\sqrt {a^2-4 b c}} \\ & = \frac {x^2 \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {x^2 \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {2 \int x \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right ) \, dx}{\sqrt {a^2-4 b c} d \log (f)}+\frac {2 \int x \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right ) \, dx}{\sqrt {a^2-4 b c} d \log (f)} \\ & = \frac {x^2 \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {x^2 \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}+\frac {2 x \text {Li}_2\left (-\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}-\frac {2 x \text {Li}_2\left (-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}-\frac {2 \int \text {Li}_2\left (-\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right ) \, dx}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}+\frac {2 \int \text {Li}_2\left (-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right ) \, dx}{\sqrt {a^2-4 b c} d^2 \log ^2(f)} \\ & = \frac {x^2 \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {x^2 \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}+\frac {2 x \text {Li}_2\left (-\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}-\frac {2 x \text {Li}_2\left (-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}-\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,f^{c+d x}\right )}{\sqrt {a^2-4 b c} d^3 \log ^3(f)}+\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,f^{c+d x}\right )}{\sqrt {a^2-4 b c} d^3 \log ^3(f)} \\ & = \frac {x^2 \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}-\frac {x^2 \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d \log (f)}+\frac {2 x \text {Li}_2\left (-\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}-\frac {2 x \text {Li}_2\left (-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^2 \log ^2(f)}-\frac {2 \text {Li}_3\left (-\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^3 \log ^3(f)}+\frac {2 \text {Li}_3\left (-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^3 \log ^3(f)} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.76 \[ \int \frac {x^2}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=\frac {d^2 x^2 \log ^2(f) \log \left (1+\frac {2 c f^{c+d x}}{a-\sqrt {a^2-4 b c}}\right )-d^2 x^2 \log ^2(f) \log \left (1+\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )+2 d x \log (f) \operatorname {PolyLog}\left (2,\frac {2 c f^{c+d x}}{-a+\sqrt {a^2-4 b c}}\right )-2 d x \log (f) \operatorname {PolyLog}\left (2,-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )-2 \operatorname {PolyLog}\left (3,\frac {2 c f^{c+d x}}{-a+\sqrt {a^2-4 b c}}\right )+2 \operatorname {PolyLog}\left (3,-\frac {2 c f^{c+d x}}{a+\sqrt {a^2-4 b c}}\right )}{\sqrt {a^2-4 b c} d^3 \log ^3(f)} \]
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\[\int \frac {x^{2}}{a +b \,f^{-d x -c}+c \,f^{d x +c}}d x\]
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Time = 0.33 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.58 \[ \int \frac {x^2}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=-\frac {b c^{2} \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} \log \left (2 \, c f^{d x + c} + b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} + a\right ) \log \left (f\right )^{2} - b c^{2} \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} \log \left (2 \, c f^{d x + c} - b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} + a\right ) \log \left (f\right )^{2} - 2 \, b d x \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} {\rm Li}_2\left (-\frac {{\left (b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} + a\right )} f^{d x + c} + 2 \, b}{2 \, b} + 1\right ) \log \left (f\right ) + 2 \, b d x \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} {\rm Li}_2\left (\frac {{\left (b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} - a\right )} f^{d x + c} - 2 \, b}{2 \, b} + 1\right ) \log \left (f\right ) - {\left (b d^{2} x^{2} - b c^{2}\right )} \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} \log \left (f\right )^{2} \log \left (\frac {{\left (b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} + a\right )} f^{d x + c} + 2 \, b}{2 \, b}\right ) + {\left (b d^{2} x^{2} - b c^{2}\right )} \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} \log \left (f\right )^{2} \log \left (-\frac {{\left (b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} - a\right )} f^{d x + c} - 2 \, b}{2 \, b}\right ) + 2 \, b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} {\rm polylog}\left (3, -\frac {{\left (b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} + a\right )} f^{d x + c}}{2 \, b}\right ) - 2 \, b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} {\rm polylog}\left (3, \frac {{\left (b \sqrt {\frac {a^{2} - 4 \, b c}{b^{2}}} - a\right )} f^{d x + c}}{2 \, b}\right )}{{\left (a^{2} - 4 \, b c\right )} d^{3} \log \left (f\right )^{3}} \]
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\[ \int \frac {x^2}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=\int \frac {x^{2}}{a + b f^{- c - d x} + c f^{c + d x}}\, dx \]
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Exception generated. \[ \int \frac {x^2}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {x^2}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=\int { \frac {x^{2}}{c f^{d x + c} + b f^{-d x - c} + a} \,d x } \]
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Timed out. \[ \int \frac {x^2}{a+b f^{-c-d x}+c f^{c+d x}} \, dx=\int \frac {x^2}{a+c\,f^{c+d\,x}+\frac {b}{f^{c+d\,x}}} \,d x \]
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