Integrand size = 18, antiderivative size = 333 \[ \int \frac {f^x x^2}{\left (a+b f^{2 x}\right )^2} \, dx=-\frac {x \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^2 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {i x \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {i x \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {i \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)} \]
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Time = 0.24 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2281, 205, 211, 2277, 14, 12, 2320, 4940, 2438, 5251, 2611, 6724} \[ \int \frac {f^x x^2}{\left (a+b f^{2 x}\right )^2} \, dx=\frac {x^2 \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {x \arctan \left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {i \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {i \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {i x \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i x \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {x^2 f^x}{2 a \log (f) \left (a+b f^{2 x}\right )} \]
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Rule 12
Rule 14
Rule 205
Rule 211
Rule 2277
Rule 2281
Rule 2320
Rule 2438
Rule 2611
Rule 4940
Rule 5251
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {f^x x^2}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-2 \int x \left (\frac {f^x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}\right ) \, dx \\ & = \frac {f^x x^2}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-2 \int \left (\frac {f^x x}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}\right ) \, dx \\ & = \frac {f^x x^2}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {\int \frac {f^x x}{a+b f^{2 x}} \, dx}{a \log (f)}-\frac {\int x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{a^{3/2} \sqrt {b} \log (f)} \\ & = -\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {\int \frac {\tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (f)} \, dx}{a \log (f)}-\frac {i \int x \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {i \int x \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log (f)} \\ & = -\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i \int \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {i \int \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {\int \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{a^{3/2} \sqrt {b} \log ^2(f)} \\ & = -\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {i \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {\text {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{a^{3/2} \sqrt {b} \log ^3(f)} \\ & = -\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {i \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)} \\ & = -\frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^2}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {i \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {i \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {i \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {i \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.43 \[ \int \frac {f^x x^2}{\left (a+b f^{2 x}\right )^2} \, dx=\frac {f^x x^2}{2 a \left (a+b f^{2 x}\right ) \log (f)}-\frac {\frac {\frac {i x^2}{2 \sqrt {a}}-\frac {i x \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log (f)}-\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log ^2(f)}}{2 \sqrt {b}}+\frac {-\frac {i x^2}{2 \sqrt {a}}+\frac {i x \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log (f)}+\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log ^2(f)}}{2 \sqrt {b}}}{a \log (f)}+\frac {\frac {\frac {i x^3}{3 \sqrt {a}}-\frac {i x^2 \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log (f)}-\frac {2 i x \operatorname {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log ^2(f)}+\frac {2 i \operatorname {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log ^3(f)}}{2 \sqrt {b}}+\frac {-\frac {i x^3}{3 \sqrt {a}}+\frac {i x^2 \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log (f)}+\frac {2 i x \operatorname {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log ^2(f)}-\frac {2 i \operatorname {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \log ^3(f)}}{2 \sqrt {b}}}{2 a} \]
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\[\int \frac {f^{x} x^{2}}{\left (a +b \,f^{2 x}\right )^{2}}d x\]
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Time = 0.27 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.17 \[ \int \frac {f^x x^2}{\left (a+b f^{2 x}\right )^2} \, dx=\frac {2 \, b f^{x} x^{2} \log \left (f\right )^{2} + 2 \, {\left ({\left (b x \log \left (f\right ) - b\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x \log \left (f\right ) - a\right )} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (f^{x} \sqrt {-\frac {b}{a}}\right ) - 2 \, {\left ({\left (b x \log \left (f\right ) - b\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x \log \left (f\right ) - a\right )} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {b}{a}}\right ) - {\left ({\left (b x^{2} \log \left (f\right )^{2} - 2 \, b x \log \left (f\right )\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x^{2} \log \left (f\right )^{2} - 2 \, a x \log \left (f\right )\right )} \sqrt {-\frac {b}{a}}\right )} \log \left (f^{x} \sqrt {-\frac {b}{a}} + 1\right ) + {\left ({\left (b x^{2} \log \left (f\right )^{2} - 2 \, b x \log \left (f\right )\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x^{2} \log \left (f\right )^{2} - 2 \, a x \log \left (f\right )\right )} \sqrt {-\frac {b}{a}}\right )} \log \left (-f^{x} \sqrt {-\frac {b}{a}} + 1\right ) - 2 \, {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} {\rm polylog}\left (3, f^{x} \sqrt {-\frac {b}{a}}\right ) + 2 \, {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} {\rm polylog}\left (3, -f^{x} \sqrt {-\frac {b}{a}}\right )}{4 \, {\left (a b^{2} f^{2 \, x} \log \left (f\right )^{3} + a^{2} b \log \left (f\right )^{3}\right )}} \]
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\[ \int \frac {f^x x^2}{\left (a+b f^{2 x}\right )^2} \, dx=\frac {f^{x} x^{2}}{2 a^{2} \log {\left (f \right )} + 2 a b f^{2 x} \log {\left (f \right )}} + \frac {\int \left (- \frac {2 f^{x} x}{a + b f^{2 x}}\right )\, dx + \int \frac {f^{x} x^{2} \log {\left (f \right )}}{a + b f^{2 x}}\, dx}{2 a \log {\left (f \right )}} \]
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\[ \int \frac {f^x x^2}{\left (a+b f^{2 x}\right )^2} \, dx=\int { \frac {f^{x} x^{2}}{{\left (b f^{2 \, x} + a\right )}^{2}} \,d x } \]
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\[ \int \frac {f^x x^2}{\left (a+b f^{2 x}\right )^2} \, dx=\int { \frac {f^{x} x^{2}}{{\left (b f^{2 \, x} + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {f^x x^2}{\left (a+b f^{2 x}\right )^2} \, dx=\int \frac {f^x\,x^2}{{\left (a+b\,f^{2\,x}\right )}^2} \,d x \]
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