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*Ei(b*ln(f)/x)

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Rubi [A]  time = 0.02, antiderivative size = 13, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ -f^a \text {Ei}\left (\frac {b \log (f)}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x)/x,x]

[Out]

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

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fricas [A]  time = 0.41, size = 13, normalized size = 1.00 \[ -f^{a} {\rm Ei}\left (\frac {b \log \relax (f)}{x}\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + \frac {b}{x}}}{x}\,{d x} \]

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)

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maple [A]  time = 0.08, size = 15, normalized size = 1.15 \[ f^{a} \Ei \left (1, -\frac {b \ln \relax (f )}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x)/x,x)

[Out]

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

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maxima [A]  time = 1.59, size = 13, normalized size = 1.00 \[ -f^{a} {\rm Ei}\left (\frac {b \log \relax (f)}{x}\right ) \]

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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mupad [B]  time = 3.49, size = 13, normalized size = 1.00 \[ -f^a\,\mathrm {ei}\left (\frac {b\,\ln \relax (f)}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a + b/x)/x,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {f^{a + \frac {b}{x}}}{x}\, dx \]

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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