3.121 \(\int \frac{f^{a+\frac{b}{x}}}{x} \, dx\)

Optimal. Leaf size=13 \[ -f^a \text{ExpIntegralEi}\left (\frac{b \log (f)}{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0299392, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -f^a \text{ExpIntegralEi}\left (\frac{b \log (f)}{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[f^(a + b/x)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.12915, size = 12, normalized size = 0.92 \[ - f^{a} \operatorname{Ei}{\left (\frac{b \log{\left (f \right )}}{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(f**(a+b/x)/x,x)

[Out]

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

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Mathematica [A]  time = 0.0037518, size = 13, normalized size = 1. \[ -f^a \text{ExpIntegralEi}\left (\frac{b \log (f)}{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[f^(a + b/x)/x,x]

[Out]

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

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Maple [A]  time = 0.019, size = 15, normalized size = 1.2 \[{f}^{a}{\it Ei} \left ( 1,-{\frac{b\ln \left ( f \right ) }{x}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(f^(a+b/x)/x,x)

[Out]

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

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Maxima [A]  time = 0.840594, size = 18, normalized size = 1.38 \[ -f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x)/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.261052, size = 18, normalized size = 1.38 \[ -f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x)/x,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{a + \frac{b}{x}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f**(a+b/x)/x,x)

[Out]

Integral(f**(a + b/x)/x, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f^{a + \frac{b}{x}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(f^(a + b/x)/x,x, algorithm="giac")

[Out]

integrate(f^(a + b/x)/x, x)