Optimal. Leaf size=21 \[ b^4 f^a \log ^4(f) \Gamma \left (-4,-\frac {b \log (f)}{x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2218} \[ b^4 f^a \log ^4(f) \text {Gamma}\left (-4,-\frac {b \log (f)}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x}} x^3 \, dx &=b^4 f^a \Gamma \left (-4,-\frac {b \log (f)}{x}\right ) \log ^4(f)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 1.00 \[ b^4 f^a \log ^4(f) \Gamma \left (-4,-\frac {b \log (f)}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 68, normalized size = 3.24 \[ -\frac {1}{24} \, b^{4} f^{a} {\rm Ei}\left (\frac {b \log \relax (f)}{x}\right ) \log \relax (f)^{4} + \frac {1}{24} \, {\left (b^{3} x \log \relax (f)^{3} + b^{2} x^{2} \log \relax (f)^{2} + 2 \, b x^{3} \log \relax (f) + 6 \, x^{4}\right )} f^{\frac {a x + b}{x}} \]
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 \[ \int f^{a + \frac {b}{x}} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 99, normalized size = 4.71 \[ \frac {b^{4} f^{a} \Ei \left (1, -\frac {b \ln \relax (f )}{x}\right ) \ln \relax (f )^{4}}{24}+\frac {b^{3} x \,f^{a} f^{\frac {b}{x}} \ln \relax (f )^{3}}{24}+\frac {b^{2} x^{2} f^{a} f^{\frac {b}{x}} \ln \relax (f )^{2}}{24}+\frac {b \,x^{3} f^{a} f^{\frac {b}{x}} \ln \relax (f )}{12}+\frac {x^{4} f^{a} f^{\frac {b}{x}}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.63, size = 21, normalized size = 1.00 \[ b^{4} f^{a} \Gamma \left (-4, -\frac {b \log \relax (f)}{x}\right ) \log \relax (f)^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.63, size = 87, normalized size = 4.14 \[ \frac {b^4\,f^a\,{\ln \relax (f)}^4\,\mathrm {expint}\left (-\frac {b\,\ln \relax (f)}{x}\right )}{24}+b^4\,f^a\,f^{b/x}\,{\ln \relax (f)}^4\,\left (\frac {x^2}{24\,b^2\,{\ln \relax (f)}^2}+\frac {x^3}{12\,b^3\,{\ln \relax (f)}^3}+\frac {x^4}{4\,b^4\,{\ln \relax (f)}^4}+\frac {x}{24\,b\,\ln \relax (f)}\right ) \]
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 \[ \int f^{a + \frac {b}{x}} x^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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