Optimal. Leaf size=268 \[ -\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)}-\frac {3 i \text {Li}_4\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log ^4(f)}+\frac {3 i \text {Li}_4\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log ^4(f)}+\frac {3 i x \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log ^3(f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (f)} \]
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Rubi [A] time = 0.23, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2249, 205, 2245, 12, 5143, 2531, 6609, 2282, 6589} \[ -\frac {3 i x^2 \text {PolyLog}\left (2,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)}+\frac {3 i x^2 \text {PolyLog}\left (2,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {PolyLog}\left (3,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {PolyLog}\left (3,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log ^3(f)}-\frac {3 i \text {PolyLog}\left (4,-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log ^4(f)}+\frac {3 i \text {PolyLog}\left (4,\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log ^4(f)}+\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (f)} \]
Antiderivative was successfully verified.
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Rule 12
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}{a+b f^{2 x}} \, dx &=\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (f)}-3 \int \frac {x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (f)} \, dx\\ &=\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (f)}-\frac {3 \int x^2 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{\sqrt {a} \sqrt {b} \log (f)}\\ &=\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (f)}-\frac {(3 i) \int x^2 \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 \sqrt {a} \sqrt {b} \log (f)}+\frac {(3 i) \int x^2 \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{2 \sqrt {a} \sqrt {b} \log (f)}\\ &=\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (f)}-\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)}+\frac {(3 i) \int x \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{\sqrt {a} \sqrt {b} \log ^2(f)}-\frac {(3 i) \int x \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{\sqrt {a} \sqrt {b} \log ^2(f)}\\ &=\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (f)}-\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log ^3(f)}-\frac {(3 i) \int \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{\sqrt {a} \sqrt {b} \log ^3(f)}+\frac {(3 i) \int \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right ) \, dx}{\sqrt {a} \sqrt {b} \log ^3(f)}\\ &=\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (f)}-\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \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 )}{\sqrt {a} \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 )}{\sqrt {a} \sqrt {b} \log ^4(f)}\\ &=\frac {x^3 \tan ^{-1}\left (\frac {\sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (f)}-\frac {3 i x^2 \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)}+\frac {3 i x^2 \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b} \log ^2(f)}+\frac {3 i x \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log ^3(f)}-\frac {3 i x \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log ^3(f)}-\frac {3 i \text {Li}_4\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log ^4(f)}+\frac {3 i \text {Li}_4\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log ^4(f)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 224, normalized size = 0.84 \[ \frac {i \left (-3 x^2 \log ^2(f) \text {Li}_2\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )+3 x^2 \log ^2(f) \text {Li}_2\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )-6 \text {Li}_4\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )+6 \text {Li}_4\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )+6 x \log (f) \text {Li}_3\left (-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )-6 x \log (f) \text {Li}_3\left (\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )+x^3 \log ^3(f) \log \left (1-\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )-x^3 \log ^3(f) \log \left (1+\frac {i \sqrt {b} f^x}{\sqrt {a}}\right )\right )}{2 \sqrt {a} \sqrt {b} \log ^4(f)} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.46, size = 239, normalized size = 0.89 \[ -\frac {x^{3} \sqrt {-\frac {b}{a}} \log \left (f^{x} \sqrt {-\frac {b}{a}} + 1\right ) \log \relax (f)^{3} - x^{3} \sqrt {-\frac {b}{a}} \log \left (-f^{x} \sqrt {-\frac {b}{a}} + 1\right ) \log \relax (f)^{3} - 3 \, x^{2} \sqrt {-\frac {b}{a}} {\rm Li}_2\left (f^{x} \sqrt {-\frac {b}{a}}\right ) \log \relax (f)^{2} + 3 \, x^{2} \sqrt {-\frac {b}{a}} {\rm Li}_2\left (-f^{x} \sqrt {-\frac {b}{a}}\right ) \log \relax (f)^{2} + 6 \, x \sqrt {-\frac {b}{a}} \log \relax (f) {\rm polylog}\left (3, f^{x} \sqrt {-\frac {b}{a}}\right ) - 6 \, x \sqrt {-\frac {b}{a}} \log \relax (f) {\rm polylog}\left (3, -f^{x} \sqrt {-\frac {b}{a}}\right ) - 6 \, \sqrt {-\frac {b}{a}} {\rm polylog}\left (4, f^{x} \sqrt {-\frac {b}{a}}\right ) + 6 \, \sqrt {-\frac {b}{a}} {\rm polylog}\left (4, -f^{x} \sqrt {-\frac {b}{a}}\right )}{2 \, b \log \relax (f)^{4}} \]
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 \[ \int \frac {f^{x} x^{3}}{b f^{2 \, x} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} f^{x}}{b \,f^{2 x}+a}\, dx \]
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 \[ \int \frac {f^{x} x^{3}}{b f^{2 \, x} + a}\,{d x} \]
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 \[ \int \frac {f^x\,x^3}{a+b\,f^{2\,x}} \,d x \]
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 \[ \int \frac {f^{x} x^{3}}{a + b f^{2 x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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