Optimal. Leaf size=45 \[ \frac{x^n f^{a+b x^n}}{b n \log (f)}-\frac{f^{a+b x^n}}{b^2 n \log ^2(f)} \]
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Rubi [A] time = 0.0805026, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x^n f^{a+b x^n}}{b n \log (f)}-\frac{f^{a+b x^n}}{b^2 n \log ^2(f)} \]
Antiderivative was successfully verified.
[In] Int[f^(a + b*x^n)*x^(-1 + 2*n),x]
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Rubi in Sympy [A] time = 7.2798, size = 36, normalized size = 0.8 \[ \frac{f^{a + b x^{n}} x^{n}}{b n \log{\left (f \right )}} - \frac{f^{a + b x^{n}}}{b^{2} n \log{\left (f \right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(f**(a+b*x**n)*x**(-1+2*n),x)
[Out]
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Mathematica [A] time = 0.0157486, size = 29, normalized size = 0.64 \[ \frac{f^{a+b x^n} \left (b \log (f) x^n-1\right )}{b^2 n \log ^2(f)} \]
Antiderivative was successfully verified.
[In] Integrate[f^(a + b*x^n)*x^(-1 + 2*n),x]
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Maple [A] time = 0.05, size = 56, normalized size = 1.2 \[{\frac{{{\rm e}^{n\ln \left ( x \right ) }}{{\rm e}^{ \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) \ln \left ( f \right ) }}}{\ln \left ( f \right ) bn}}-{\frac{{{\rm e}^{ \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) \ln \left ( f \right ) }}}{{b}^{2}n \left ( \ln \left ( f \right ) \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(f^(a+b*x^n)*x^(-1+2*n),x)
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Maxima [A] time = 0.814474, size = 46, normalized size = 1.02 \[ \frac{{\left (b f^{a} x^{n} \log \left (f\right ) - f^{a}\right )} f^{b x^{n}}}{b^{2} n \log \left (f\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(b*x^n + a)*x^(2*n - 1),x, algorithm="maxima")
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Fricas [A] time = 0.256017, size = 45, normalized size = 1. \[ \frac{{\left (b x^{n} \log \left (f\right ) - 1\right )} e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{b^{2} n \log \left (f\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(b*x^n + a)*x^(2*n - 1),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f**(a+b*x**n)*x**(-1+2*n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int f^{b x^{n} + a} x^{2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(f^(b*x^n + a)*x^(2*n - 1),x, algorithm="giac")
[Out]