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

Optimal. Leaf size=38 \[ \frac{b f^a \log (f) \text{ExpIntegralEi}\left (b \log (f) x^n\right )}{n}-\frac{x^{-n} f^{a+b x^n}}{n} \]

[Out]

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

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Rubi [A]  time = 0.0772465, antiderivative size = 38, 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 \[ \frac{b f^a \log (f) \text{ExpIntegralEi}\left (b \log (f) x^n\right )}{n}-\frac{x^{-n} f^{a+b x^n}}{n} \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b*x^n)*x^(-1 - n),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.79894, size = 32, normalized size = 0.84 \[ \frac{b f^{a} \log{\left (f \right )} \operatorname{Ei}{\left (b x^{n} \log{\left (f \right )} \right )}}{n} - \frac{f^{a + b x^{n}} x^{- n}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0195919, size = 38, normalized size = 1. \[ \frac{b f^a \log (f) \text{ExpIntegralEi}\left (b \log (f) x^n\right )}{n}-\frac{x^{-n} f^{a+b x^n}}{n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.05, size = 42, normalized size = 1.1 \[ -{\frac{{f}^{a+b{x}^{n}}}{n{x}^{n}}}-{\frac{b\ln \left ( f \right ){f}^{a}{\it Ei} \left ( 1,-b{x}^{n}\ln \left ( f \right ) \right ) }{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b*x^n)*x^(-1-n),x)

[Out]

-f^(a+b*x^n)/n/(x^n)-1/n*ln(f)*b*f^a*Ei(1,-b*x^n*ln(f))

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Maxima [A]  time = 1.07261, size = 27, normalized size = 0.71 \[ \frac{b f^{a} \Gamma \left (-1, -b x^{n} \log \left (f\right )\right ) \log \left (f\right )}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*f^a*gamma(-1, -b*x^n*log(f))*log(f)/n

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Fricas [A]  time = 0.256526, size = 58, normalized size = 1.53 \[ \frac{b f^{a} x^{n}{\rm Ei}\left (b x^{n} \log \left (f\right )\right ) \log \left (f\right ) - e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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 [F]  time = 0., size = 0, normalized size = 0. \[ \int f^{b x^{n} + a} x^{-n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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