Optimal. Leaf size=501 \[ -\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {3 i \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}-\frac {3 i \text {Li}_4\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i \text {Li}_4\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i x \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {x^3 f^x}{2 a \log (f) \left (a+b f^{2 x}\right )} \]
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Rubi [A] time = 0.51, antiderivative size = 501, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {2249, 199, 205, 2245, 14, 12, 5143, 2531, 2282, 6589, 6609} \[ -\frac {3 i x^2 \text {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 i x^2 \text {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i x \text {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i \text {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}-\frac {3 i x \text {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i \text {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}-\frac {3 i \text {PolyLog}\left (4,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i \text {PolyLog}\left (4,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}-\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {x^3 f^x}{2 a \log (f) \left (a+b f^{2 x}\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 199
Rule 205
Rule 2245
Rule 2249
Rule 2282
Rule 2531
Rule 5143
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {f^x x^3}{\left (a+b f^{2 x}\right )^2} \, dx &=\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-3 \int x^2 \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^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-3 \int \left (\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)}\right ) \, dx\\ &=\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {3 \int \frac {f^x x^2}{a+b f^{2 x}} \, dx}{2 a \log (f)}-\frac {3 \int x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log (f)}\\ &=-\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {3 \int \frac {x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (f)} \, dx}{a \log (f)}-\frac {(3 i) \int x^2 \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{4 a^{3/2} \sqrt {b} \log (f)}+\frac {(3 i) \int x^2 \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{4 a^{3/2} \sqrt {b} \log (f)}\\ &=-\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {(3 i) \int x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {(3 i) \int x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 \int x \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{a^{3/2} \sqrt {b} \log ^2(f)}\\ &=-\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}-\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {(3 i) \int \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {(3 i) \int \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {(3 i) \int x \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {(3 i) \int x \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^2(f)}\\ &=-\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {(3 i) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {(3 i) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {i \sqrt {b} x}{\sqrt {a}}\right )}{x} \, dx,x,f^x\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}-\frac {(3 i) \int \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {(3 i) \int \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 a^{3/2} \sqrt {b} \log ^3(f)}\\ &=-\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i \text {Li}_4\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i \text {Li}_4\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}-\frac {(3 i) \operatorname {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 ^4(f)}+\frac {(3 i) \operatorname {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 ^4(f)}\\ &=-\frac {3 x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^2(f)}+\frac {f^x x^3}{2 a \left (a+b f^{2 x}\right ) \log (f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log (f)}+\frac {3 i x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {3 i x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b} \log ^2(f)}-\frac {3 i \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i x \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}+\frac {3 i \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}-\frac {3 i x \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^3(f)}-\frac {3 i \text {Li}_4\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}+\frac {3 i \text {Li}_4\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b} \log ^4(f)}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 434, normalized size = 0.87 \[ \frac {-\frac {6 i \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {6 i \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {6 i \text {Li}_4\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {6 i \text {Li}_4\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {3 i x \log (f) (x \log (f)-2) \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {3 i x \log (f) (x \log (f)-2) \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {6 i x \log (f) \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {6 i x \log (f) \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {i x^3 \log ^3(f) \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {i x^3 \log ^3(f) \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {3 i x^2 \log ^2(f) \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {3 i x^2 \log ^2(f) \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {2 \sqrt {a} x^3 f^x \log ^3(f)}{a+b f^{2 x}}}{4 a^{3/2} \log ^4(f)} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.44, size = 549, normalized size = 1.10 \[ \frac {2 \, b f^{x} x^{3} \log \relax (f)^{3} + 3 \, {\left ({\left (b x^{2} \log \relax (f)^{2} - 2 \, b x \log \relax (f)\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x^{2} \log \relax (f)^{2} - 2 \, a x \log \relax (f)\right )} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (f^{x} \sqrt {-\frac {b}{a}}\right ) - 3 \, {\left ({\left (b x^{2} \log \relax (f)^{2} - 2 \, b x \log \relax (f)\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x^{2} \log \relax (f)^{2} - 2 \, a x \log \relax (f)\right )} \sqrt {-\frac {b}{a}}\right )} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {b}{a}}\right ) - {\left ({\left (b x^{3} \log \relax (f)^{3} - 3 \, b x^{2} \log \relax (f)^{2}\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x^{3} \log \relax (f)^{3} - 3 \, a x^{2} \log \relax (f)^{2}\right )} \sqrt {-\frac {b}{a}}\right )} \log \left (f^{x} \sqrt {-\frac {b}{a}} + 1\right ) + {\left ({\left (b x^{3} \log \relax (f)^{3} - 3 \, b x^{2} \log \relax (f)^{2}\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x^{3} \log \relax (f)^{3} - 3 \, a x^{2} \log \relax (f)^{2}\right )} \sqrt {-\frac {b}{a}}\right )} \log \left (-f^{x} \sqrt {-\frac {b}{a}} + 1\right ) + 6 \, {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} {\rm polylog}\left (4, f^{x} \sqrt {-\frac {b}{a}}\right ) - 6 \, {\left (b f^{2 \, x} \sqrt {-\frac {b}{a}} + a \sqrt {-\frac {b}{a}}\right )} {\rm polylog}\left (4, -f^{x} \sqrt {-\frac {b}{a}}\right ) - 6 \, {\left ({\left (b x \log \relax (f) - b\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x \log \relax (f) - a\right )} \sqrt {-\frac {b}{a}}\right )} {\rm polylog}\left (3, f^{x} \sqrt {-\frac {b}{a}}\right ) + 6 \, {\left ({\left (b x \log \relax (f) - b\right )} f^{2 \, x} \sqrt {-\frac {b}{a}} + {\left (a x \log \relax (f) - a\right )} \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 \relax (f)^{4} + a^{2} b \log \relax (f)^{4}\right )}} \]
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 \[ \int \frac {f^{x} x^{3}}{{\left (b f^{2 \, x} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} f^{x}}{\left (b \,f^{2 x}+a \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {f^{x} x^{3}}{2 \, {\left (a b f^{2 \, x} \log \relax (f) + a^{2} \log \relax (f)\right )}} + \int \frac {{\left (x^{3} \log \relax (f) - 3 \, x^{2}\right )} f^{x}}{2 \, {\left (a b f^{2 \, x} \log \relax (f) + a^{2} \log \relax (f)\right )}}\,{d x} \]
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 \[ \int \frac {f^x\,x^3}{{\left (a+b\,f^{2\,x}\right )}^2} \,d x \]
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 \[ \frac {f^{x} x^{3}}{2 a^{2} \log {\relax (f )} + 2 a b f^{2 x} \log {\relax (f )}} + \frac {\int \left (- \frac {3 f^{x} x^{2}}{a + b f^{2 x}}\right )\, dx + \int \frac {f^{x} x^{3} \log {\relax (f )}}{a + b f^{2 x}}\, dx}{2 a \log {\relax (f )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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