Optimal. Leaf size=56 \[ \frac {1}{2} f^{a+\frac {b}{x}} x^2+\frac {1}{2} b f^{a+\frac {b}{x}} x \log (f)-\frac {1}{2} b^2 f^a \text {Ei}\left (\frac {b \log (f)}{x}\right ) \log ^2(f) \]
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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 = {2245, 2237,
2241} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2237
Rule 2241
Rule 2245
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 = 0.01, size = 40, normalized size = 0.71 \begin {gather*} \frac {1}{2} f^a \left (-b^2 \text {Ei}\left (\frac {b \log (f)}{x}\right ) \log ^2(f)+f^{b/x} x (x+b \log (f))\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 55, normalized size = 0.98
method | result | size |
risch | \(\frac {f^{a} f^{\frac {b}{x}} x^{2}}{2}+\frac {b \ln \left (f \right ) f^{a} f^{\frac {b}{x}} x}{2}+\frac {b^{2} \ln \left (f \right )^{2} f^{a} \expIntegral \left (1, -\frac {b \ln \left (f \right )}{x}\right )}{2}\) | \(55\) |
meijerg | \(-f^{a} b^{2} \ln \left (f \right )^{2} \left (\frac {x^{2} \left (\frac {9 b^{2} \ln \left (f \right )^{2}}{x^{2}}+\frac {12 b \ln \left (f \right )}{x}+6\right )}{12 b^{2} \ln \left (f \right )^{2}}-\frac {x^{2} \left (3+\frac {3 b \ln \left (f \right )}{x}\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x}}}{6 b^{2} \ln \left (f \right )^{2}}-\frac {\ln \left (-\frac {b \ln \left (f \right )}{x}\right )}{2}-\frac {\expIntegral \left (1, -\frac {b \ln \left (f \right )}{x}\right )}{2}-\frac {3}{4}-\frac {\ln \left (x \right )}{2}+\frac {\ln \left (-b \right )}{2}+\frac {\ln \left (\ln \left (f \right )\right )}{2}-\frac {x^{2}}{2 b^{2} \ln \left (f \right )^{2}}-\frac {x}{\ln \left (f \right ) b}\right )\) | \(139\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 21, normalized size = 0.38 \begin {gather*} b^{2} f^{a} \Gamma \left (-2, -\frac {b \log \left (f\right )}{x}\right ) \log \left (f\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 43, normalized size = 0.77 \begin {gather*} -\frac {1}{2} \, b^{2} f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x}\right ) \log \left (f\right )^{2} + \frac {1}{2} \, {\left (b x \log \left (f\right ) + x^{2}\right )} f^{\frac {a x + b}{x}} \end {gather*}
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 \begin {gather*} \int f^{a + \frac {b}{x}} x\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} b^2\,f^a\,{\ln \left (f\right )}^2\,\left (f^{b/x}\,\left (\frac {x^2}{2\,b^2\,{\ln \left (f\right )}^2}+\frac {x}{2\,b\,\ln \left (f\right )}\right )+\frac {\mathrm {expint}\left (-\frac {b\,\ln \left (f\right )}{x}\right )}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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