Optimal. Leaf size=84 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}+\frac{3 f^x}{8 a^2 \log (f) \left (a+b f^{2 x}\right )}+\frac{f^x}{4 a \log (f) \left (a+b f^{2 x}\right )^2} \]
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Rubi [A] time = 0.0984949, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{b} \log (f)}+\frac{3 f^x}{8 a^2 \log (f) \left (a+b f^{2 x}\right )}+\frac{f^x}{4 a \log (f) \left (a+b f^{2 x}\right )^2} \]
Antiderivative was successfully verified.
[In] Int[f^x/(a + b*f^(2*x))^3,x]
[Out]
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Rubi in Sympy [A] time = 13.3973, size = 73, normalized size = 0.87 \[ \frac{f^{x}}{4 a \left (a + b f^{2 x}\right )^{2} \log{\left (f \right )}} + \frac{3 f^{x}}{8 a^{2} \left (a + b f^{2 x}\right ) \log{\left (f \right )}} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt{b} f^{x}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} \sqrt{b} \log{\left (f \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(f**x/(a+b*f**(2*x))**3,x)
[Out]
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Mathematica [A] time = 0.0851824, size = 68, normalized size = 0.81 \[ \frac{\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} f^x}{\sqrt{a}}\right )}{\sqrt{b}}+\frac{\sqrt{a} f^x \left (5 a+3 b f^{2 x}\right )}{\left (a+b f^{2 x}\right )^2}}{8 a^{5/2} \log (f)} \]
Antiderivative was successfully verified.
[In] Integrate[f^x/(a + b*f^(2*x))^3,x]
[Out]
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Maple [A] time = 0.061, size = 94, normalized size = 1.1 \[{\frac{ \left ( 3\,b \left ({f}^{x} \right ) ^{2}+5\,a \right ){f}^{x}}{8\,\ln \left ( f \right ){a}^{2} \left ( a+b \left ({f}^{x} \right ) ^{2} \right ) ^{2}}}-{\frac{3}{16\,\ln \left ( f \right ){a}^{2}}\ln \left ({f}^{x}-{a{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{3}{16\,\ln \left ( f \right ){a}^{2}}\ln \left ({f}^{x}+{a{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(f^x/(a+b*f^(2*x))^3,x)
[Out]
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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/(b*f^(2*x) + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.271917, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, \sqrt{-a b} b f^{3 \, x} + 10 \, \sqrt{-a b} a f^{x} + 3 \,{\left (b^{2} f^{4 \, x} + 2 \, a b f^{2 \, x} + a^{2}\right )} \log \left (\frac{2 \, a b f^{x} + \sqrt{-a b} b f^{2 \, x} - \sqrt{-a b} a}{b f^{2 \, x} + a}\right )}{16 \,{\left (\sqrt{-a b} a^{2} b^{2} f^{4 \, x} \log \left (f\right ) + 2 \, \sqrt{-a b} a^{3} b f^{2 \, x} \log \left (f\right ) + \sqrt{-a b} a^{4} \log \left (f\right )\right )}}, \frac{3 \, \sqrt{a b} b f^{3 \, x} + 5 \, \sqrt{a b} a f^{x} - 3 \,{\left (b^{2} f^{4 \, x} + 2 \, a b f^{2 \, x} + a^{2}\right )} \arctan \left (\frac{a}{\sqrt{a b} f^{x}}\right )}{8 \,{\left (\sqrt{a b} a^{2} b^{2} f^{4 \, x} \log \left (f\right ) + 2 \, \sqrt{a b} a^{3} b f^{2 \, x} \log \left (f\right ) + \sqrt{a b} a^{4} \log \left (f\right )\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^x/(b*f^(2*x) + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.689219, size = 85, normalized size = 1.01 \[ \frac{5 a f^{x} + 3 b f^{3 x}}{8 a^{4} \log{\left (f \right )} + 16 a^{3} b f^{2 x} \log{\left (f \right )} + 8 a^{2} b^{2} f^{4 x} \log{\left (f \right )}} + \frac{\operatorname{RootSum}{\left (256 z^{2} a^{5} b + 9, \left ( i \mapsto i \log{\left (\frac{16 i a^{3}}{3} + f^{x} \right )} \right )\right )}}{\log{\left (f \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f**x/(a+b*f**(2*x))**3,x)
[Out]
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GIAC/XCAS [A] time = 0.234863, size = 82, normalized size = 0.98 \[ \frac{3 \, \arctan \left (\frac{b f^{x}}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2}{\rm ln}\left (f\right )} + \frac{3 \, b f^{3 \, x} + 5 \, a f^{x}}{8 \,{\left (b f^{2 \, x} + a\right )}^{2} a^{2}{\rm ln}\left (f\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^x/(b*f^(2*x) + a)^3,x, algorithm="giac")
[Out]