3.184 \(\int f^{a+b x^n} x^{-1+2 n} \, dx\)

Optimal. Leaf size=45 \[ \frac{x^n f^{a+b x^n}}{b n \log (f)}-\frac{f^{a+b x^n}}{b^2 n \log ^2(f)} \]

[Out]

-(f^(a + b*x^n)/(b^2*n*Log[f]^2)) + (f^(a + b*x^n)*x^n)/(b*n*Log[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]

[Out]

-(f^(a + b*x^n)/(b^2*n*Log[f]^2)) + (f^(a + b*x^n)*x^n)/(b*n*Log[f])

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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]

f**(a + b*x**n)*x**n/(b*n*log(f)) - f**(a + b*x**n)/(b**2*n*log(f)**2)

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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]

[Out]

(f^(a + b*x^n)*(-1 + b*x^n*Log[f]))/(b^2*n*Log[f]^2)

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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)

[Out]

1/ln(f)/b/n*exp(n*ln(x))*exp((a+b*exp(n*ln(x)))*ln(f))-1/ln(f)^2/b^2/n*exp((a+b*
exp(n*ln(x)))*ln(f))

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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")

[Out]

(b*f^a*x^n*log(f) - f^a)*f^(b*x^n)/(b^2*n*log(f)^2)

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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]

(b*x^n*log(f) - 1)*e^(b*x^n*log(f) + a*log(f))/(b^2*n*log(f)^2)

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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]

Timed 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]

integrate(f^(b*x^n + a)*x^(2*n - 1), x)