3.185 \(\int f^{a+b x^n} x^{-1+n} \, dx\)

Optimal. Leaf size=20 \[ \frac{f^{a+b x^n}}{b n \log (f)} \]

[Out]

f^(a + b*x^n)/(b*n*Log[f])

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Rubi [A]  time = 0.0377487, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{f^{a+b x^n}}{b n \log (f)} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^n)*x^(-1 + n),x]

[Out]

f^(a + b*x^n)/(b*n*Log[f])

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Rubi in Sympy [A]  time = 3.59485, size = 14, normalized size = 0.7 \[ \frac{f^{a + b x^{n}}}{b n \log{\left (f \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b*x**n)*x**(-1+n),x)

[Out]

f**(a + b*x**n)/(b*n*log(f))

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Mathematica [A]  time = 0.00487654, size = 20, normalized size = 1. \[ \frac{f^{a+b x^n}}{b n \log (f)} \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b*x^n)*x^(-1 + n),x]

[Out]

f^(a + b*x^n)/(b*n*Log[f])

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Maple [A]  time = 0.033, size = 25, normalized size = 1.3 \[{\frac{{{\rm e}^{ \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) \ln \left ( f \right ) }}}{\ln \left ( f \right ) bn}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b*x^n)*x^(-1+n),x)

[Out]

1/ln(f)/b/n*exp((a+b*exp(n*ln(x)))*ln(f))

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Maxima [A]  time = 0.794468, size = 27, normalized size = 1.35 \[ \frac{f^{b x^{n} + a}}{b n \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^n + a)*x^(n - 1),x, algorithm="maxima")

[Out]

f^(b*x^n + a)/(b*n*log(f))

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Fricas [A]  time = 0.265244, size = 32, normalized size = 1.6 \[ \frac{e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{b n \log \left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^n + a)*x^(n - 1),x, algorithm="fricas")

[Out]

e^(b*x^n*log(f) + a*log(f))/(b*n*log(f))

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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+n),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224897, size = 27, normalized size = 1.35 \[ \frac{f^{b x^{n} + a}}{b n{\rm ln}\left (f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(b*x^n + a)*x^(n - 1),x, algorithm="giac")

[Out]

f^(b*x^n + a)/(b*n*ln(f))