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

Optimal. Leaf size=28 \[ x f^{a+\frac{b}{x}}-b f^a \log (f) \text{Ei}\left (\frac{b \log (f)}{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.022346, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2206, 2210} \[ x f^{a+\frac{b}{x}}-b f^a \log (f) \text{Ei}\left (\frac{b \log (f)}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[f^(a + b/x),x]

[Out]

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

Rule 2206

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[((c + d*x)*F^(a + b*(c + d*x)^n))/d, x]
- Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n]
&& ILtQ[n, 0]

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

Mathematica [A]  time = 0.0070814, size = 28, normalized size = 1. \[ x f^{a+\frac{b}{x}}-b f^a \log (f) \text{Ei}\left (\frac{b \log (f)}{x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[f^(a + b/x),x]

[Out]

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

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Maple [A]  time = 0.059, size = 31, normalized size = 1.1 \begin{align*} b\ln \left ( f \right ){f}^{a}{\it Ei} \left ( 1,-{\frac{b\ln \left ( f \right ) }{x}} \right ) +{f}^{a}{f}^{{\frac{b}{x}}}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.36428, size = 24, normalized size = 0.86 \begin{align*} -b f^{a} \Gamma \left (-1, -\frac{b \log \left (f\right )}{x}\right ) \log \left (f\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*f^a*gamma(-1, -b*log(f)/x)*log(f)

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Fricas [A]  time = 1.76561, size = 68, normalized size = 2.43 \begin{align*} -b f^{a}{\rm Ei}\left (\frac{b \log \left (f\right )}{x}\right ) \log \left (f\right ) + f^{\frac{a x + b}{x}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(f**(a+b/x),x)

[Out]

Integral(f**(a + b/x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(f^(a + b/x), x)