Integrand size = 9, antiderivative size = 28 \[ \int f^{a+\frac {b}{x}} \, dx=f^{a+\frac {b}{x}} x-b f^a \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x}\right ) \log (f) \]
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Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2237, 2241} \[ \int f^{a+\frac {b}{x}} \, dx=x f^{a+\frac {b}{x}}-b f^a \log (f) \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x}\right ) \]
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Rule 2237
Rule 2241
Rubi steps \begin{align*} \text {integral}& = 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*}
Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int f^{a+\frac {b}{x}} \, dx=f^{a+\frac {b}{x}} x-b f^a \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x}\right ) \log (f) \]
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Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11
method | result | size |
risch | \(f^{a} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x}\right ) b \ln \left (f \right )+f^{a} f^{\frac {b}{x}} x\) | \(31\) |
meijerg | \(f^{a} b \ln \left (f \right ) \left (\frac {x}{b \ln \left (f \right )}+1+\ln \left (x \right )-\ln \left (-b \right )-\ln \left (\ln \left (f \right )\right )-\frac {x \left (2+\frac {2 b \ln \left (f \right )}{x}\right )}{2 b \ln \left (f \right )}+\frac {x \,{\mathrm e}^{\frac {b \ln \left (f \right )}{x}}}{b \ln \left (f \right )}+\ln \left (-\frac {b \ln \left (f \right )}{x}\right )+\operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x}\right )\right )\) | \(88\) |
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none
Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int f^{a+\frac {b}{x}} \, dx=-b f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x}\right ) \log \left (f\right ) + f^{\frac {a x + b}{x}} x \]
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\[ \int f^{a+\frac {b}{x}} \, dx=\int f^{a + \frac {b}{x}}\, dx \]
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none
Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.64 \[ \int f^{a+\frac {b}{x}} \, dx=-b f^{a} \Gamma \left (-1, -\frac {b \log \left (f\right )}{x}\right ) \log \left (f\right ) \]
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\[ \int f^{a+\frac {b}{x}} \, dx=\int { f^{a + \frac {b}{x}} \,d x } \]
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Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int f^{a+\frac {b}{x}} \, dx=f^a\,\left (f^{b/x}\,x+b\,\ln \left (f\right )\,\mathrm {expint}\left (-\frac {b\,\ln \left (f\right )}{x}\right )\right ) \]
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