3.2.21 \(\int \frac {f^{a+\frac {b}{x}}}{x} \, dx\) [121]

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 = 0.01, 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 = {2241} \begin {gather*} -f^a \text {Ei}\left (\frac {b \log (f)}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x)/x,x]

[Out]

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

Rule 2241

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 \begin {gather*} -f^a \text {Ei}\left (\frac {b \log (f)}{x}\right ) \end {gather*}

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 = 0.06, size = 15, normalized size = 1.15

method	result	size
risch	\(f^{a} \expIntegral \left (1, -\frac {b \ln \left (f \right )}{x}\right )\)	\(15\)
meijerg	\(-f^{a} \left (-\ln \left (-\frac {b \ln \left (f \right )}{x}\right )-\expIntegral \left (1, -\frac {b \ln \left (f \right )}{x}\right )-\ln \left (x \right )+\ln \left (-b \right )+\ln \left (\ln \left (f \right )\right )\right )\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(f^(a+b/x)/x,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.32, size = 13, normalized size = 1.00 \begin {gather*} -f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x}\right ) \end {gather*}

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 = 0.36, size = 13, normalized size = 1.00 \begin {gather*} -f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x}\right ) \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f^{a + \frac {b}{x}}}{x}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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