Optimal. Leaf size=184 \[ \frac{i \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}-\frac{i \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}-\frac{i x \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^2(f)}+\frac{i x \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^2(f)}+\frac{x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \]
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Rubi [A] time = 0.307652, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ \frac{i \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}-\frac{i \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^3(f)}-\frac{i x \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^2(f)}+\frac{i x \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log ^2(f)}+\frac{x^2 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log (f)} \]
Antiderivative was successfully verified.
[In] Int[(f^x*x^2)/(a + b*f^(2*x)),x]
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Rubi in Sympy [A] time = 87.8661, size = 173, normalized size = 0.94 \[ \frac{x^{2} \operatorname{atan}{\left (\frac{\sqrt{b} f^{x}}{\sqrt{a}} \right )}}{\sqrt{a} \sqrt{b} \log{\left (f \right )}} - \frac{i x \operatorname{Li}_{2}\left (- \frac{i \sqrt{b} f^{x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{2}} + \frac{i x \operatorname{Li}_{2}\left (\frac{i \sqrt{b} f^{x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{2}} + \frac{i \operatorname{Li}_{3}\left (- \frac{i \sqrt{b} f^{x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{3}} - \frac{i \operatorname{Li}_{3}\left (\frac{i \sqrt{b} f^{x}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \log{\left (f \right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(f**x*x**2/(a+b*f**(2*x)),x)
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Mathematica [A] time = 0.0446261, size = 168, normalized size = 0.91 \[ \frac{i \left (2 \text{PolyLog}\left (3,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )-2 \text{PolyLog}\left (3,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )-2 x \log (f) \text{PolyLog}\left (2,-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )+2 x \log (f) \text{PolyLog}\left (2,\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )+x^2 \log ^2(f) \log \left (1-\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )-x^2 \log ^2(f) \log \left (1+\frac{i \sqrt{b} f^x}{\sqrt{a}}\right )\right )}{2 \sqrt{a} \sqrt{b} \log ^3(f)} \]
Antiderivative was successfully verified.
[In] Integrate[(f^x*x^2)/(a + b*f^(2*x)),x]
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Maple [F] time = 0.031, size = 0, normalized size = 0. \[ \int{\frac{{f}^{x}{x}^{2}}{a+b{f}^{2\,x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(f^x*x^2/(a+b*f^(2*x)),x)
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^x*x^2/(b*f^(2*x) + a),x, algorithm="maxima")
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Fricas [A] time = 0.274217, size = 338, normalized size = 1.84 \[ \frac{x^{2} \sqrt{-\frac{b}{a}} \log \left (f\right )^{2} \log \left (\frac{b f^{x} + a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}}\right ) - x^{2} \sqrt{-\frac{b}{a}} \log \left (f\right )^{2} \log \left (-\frac{b f^{x} - a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}}\right ) + 2 \, x \sqrt{-\frac{b}{a}}{\rm Li}_2\left (-\frac{b f^{x} + a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}} + 1\right ) \log \left (f\right ) - 2 \, x \sqrt{-\frac{b}{a}}{\rm Li}_2\left (\frac{b f^{x} - a \sqrt{-\frac{b}{a}}}{a \sqrt{-\frac{b}{a}}} + 1\right ) \log \left (f\right ) + 2 \, \sqrt{-\frac{b}{a}}{\rm Li}_{3}(\frac{b f^{x}}{a \sqrt{-\frac{b}{a}}}) - 2 \, \sqrt{-\frac{b}{a}}{\rm Li}_{3}(-\frac{b f^{x}}{a \sqrt{-\frac{b}{a}}})}{2 \, b \log \left (f\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^x*x^2/(b*f^(2*x) + a),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{x} x^{2}}{a + b f^{2 x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f**x*x**2/(a+b*f**(2*x)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{x} x^{2}}{b f^{2 \, x} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^x*x^2/(b*f^(2*x) + a),x, algorithm="giac")
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