Optimal. Leaf size=56 \[ -\frac {1}{2} b^2 f^a \log ^2(f) \text {Ei}\left (\frac {b \log (f)}{x}\right )+\frac {1}{2} x^2 f^{a+\frac {b}{x}}+\frac {1}{2} b x \log (f) f^{a+\frac {b}{x}} \]
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Rubi [A] time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2214, 2206, 2210} \[ -\frac {1}{2} b^2 f^a \log ^2(f) \text {Ei}\left (\frac {b \log (f)}{x}\right )+\frac {1}{2} x^2 f^{a+\frac {b}{x}}+\frac {1}{2} b x \log (f) f^{a+\frac {b}{x}} \]
Antiderivative was successfully verified.
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Rule 2206
Rule 2210
Rule 2214
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x}} x \, dx &=\frac {1}{2} f^{a+\frac {b}{x}} x^2+\frac {1}{2} (b \log (f)) \int f^{a+\frac {b}{x}} \, dx\\ &=\frac {1}{2} f^{a+\frac {b}{x}} x^2+\frac {1}{2} b f^{a+\frac {b}{x}} x \log (f)+\frac {1}{2} \left (b^2 \log ^2(f)\right ) \int \frac {f^{a+\frac {b}{x}}}{x} \, dx\\ &=\frac {1}{2} f^{a+\frac {b}{x}} x^2+\frac {1}{2} b f^{a+\frac {b}{x}} x \log (f)-\frac {1}{2} b^2 f^a \text {Ei}\left (\frac {b \log (f)}{x}\right ) \log ^2(f)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 0.71 \[ \frac {1}{2} f^a \left (x f^{b/x} (b \log (f)+x)-b^2 \log ^2(f) \text {Ei}\left (\frac {b \log (f)}{x}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 43, normalized size = 0.77 \[ -\frac {1}{2} \, b^{2} f^{a} {\rm Ei}\left (\frac {b \log \relax (f)}{x}\right ) \log \relax (f)^{2} + \frac {1}{2} \, {\left (b x \log \relax (f) + x^{2}\right )} f^{\frac {a x + b}{x}} \]
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 \[ \int f^{a + \frac {b}{x}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 55, normalized size = 0.98 \[ \frac {b^{2} f^{a} \Ei \left (1, -\frac {b \ln \relax (f )}{x}\right ) \ln \relax (f )^{2}}{2}+\frac {b x \,f^{a} f^{\frac {b}{x}} \ln \relax (f )}{2}+\frac {x^{2} f^{a} f^{\frac {b}{x}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.62, size = 21, normalized size = 0.38 \[ b^{2} f^{a} \Gamma \left (-2, -\frac {b \log \relax (f)}{x}\right ) \log \relax (f)^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.64, size = 54, normalized size = 0.96 \[ b^2\,f^a\,{\ln \relax (f)}^2\,\left (f^{b/x}\,\left (\frac {x^2}{2\,b^2\,{\ln \relax (f)}^2}+\frac {x}{2\,b\,\ln \relax (f)}\right )+\frac {\mathrm {expint}\left (-\frac {b\,\ln \relax (f)}{x}\right )}{2}\right ) \]
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 \[ \int f^{a + \frac {b}{x}} x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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