Optimal. Leaf size=28 \[ x f^{a+\frac {b}{x}}-b f^a \log (f) \text {Ei}\left (\frac {b \log (f)}{x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, 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.
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Rule 2206
Rule 2210
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*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \[ x f^{a+\frac {b}{x}}-b f^a \log (f) \text {Ei}\left (\frac {b \log (f)}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 30, normalized size = 1.07 \[ -b f^{a} {\rm Ei}\left (\frac {b \log \relax (f)}{x}\right ) \log \relax (f) + f^{\frac {a x + b}{x}} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 31, normalized size = 1.11 \[ b \,f^{a} \Ei \left (1, -\frac {b \ln \relax (f )}{x}\right ) \ln \relax (f )+x \,f^{a} f^{\frac {b}{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.58, size = 18, normalized size = 0.64 \[ -b f^{a} \Gamma \left (-1, -\frac {b \log \relax (f)}{x}\right ) \log \relax (f) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.60, size = 27, normalized size = 0.96 \[ f^a\,\left (f^{b/x}\,x+b\,\ln \relax (f)\,\mathrm {expint}\left (-\frac {b\,\ln \relax (f)}{x}\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{a + \frac {b}{x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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