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

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

[Out]

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

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Rubi [A]  time = 0.0183578, 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, Rules used = {2210} \[ -f^a \text{Ei}\left (\frac{b \log (f)}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x)/x,x]

[Out]

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

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{f^{a+\frac{b}{x}}}{x} \, dx &=-f^a \text{Ei}\left (\frac{b \log (f)}{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0019576, size = 13, normalized size = 1. \[ -f^a \text{Ei}\left (\frac{b \log (f)}{x}\right ) \]

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 = 0.056, size = 15, normalized size = 1.2 \begin{align*}{f}^{a}{\it Ei} \left ( 1,-{\frac{b\ln \left ( f \right ) }{x}} \right ) \end{align*}

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 = 1.19228, size = 18, normalized size = 1.38 \begin{align*} -f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x}\right ) \end{align*}

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 = 1.73477, size = 28, normalized size = 2.15 \begin{align*} -f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x}\right ) \end{align*}

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. \begin{align*} \int \frac{f^{a + \frac{b}{x}}}{x}\, dx \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f^{a + \frac{b}{x}}}{x}\,{d x} \end{align*}

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)