Optimal. Leaf size=22 \[ -b^5 f^a \log ^5(f) \Gamma \left (-5,-\frac {b \log (f)}{x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 22, 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} \[ -b^5 f^a \log ^5(f) \text {Gamma}\left (-5,-\frac {b \log (f)}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x}} x^4 \, dx &=-b^5 f^a \Gamma \left (-5,-\frac {b \log (f)}{x}\right ) \log ^5(f)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 1.00 \[ -b^5 f^a \log ^5(f) \Gamma \left (-5,-\frac {b \log (f)}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 80, normalized size = 3.64 \[ -\frac {1}{120} \, b^{5} f^{a} {\rm Ei}\left (\frac {b \log \relax (f)}{x}\right ) \log \relax (f)^{5} + \frac {1}{120} \, {\left (b^{4} x \log \relax (f)^{4} + b^{3} x^{2} \log \relax (f)^{3} + 2 \, b^{2} x^{3} \log \relax (f)^{2} + 6 \, b x^{4} \log \relax (f) + 24 \, x^{5}\right )} f^{\frac {a x + b}{x}} \]
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 \[ \int f^{a + \frac {b}{x}} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 121, normalized size = 5.50 \[ \frac {b^{5} f^{a} \Ei \left (1, -\frac {b \ln \relax (f )}{x}\right ) \ln \relax (f )^{5}}{120}+\frac {b^{4} x \,f^{a} f^{\frac {b}{x}} \ln \relax (f )^{4}}{120}+\frac {b^{3} x^{2} f^{a} f^{\frac {b}{x}} \ln \relax (f )^{3}}{120}+\frac {b^{2} x^{3} f^{a} f^{\frac {b}{x}} \ln \relax (f )^{2}}{60}+\frac {b \,x^{4} f^{a} f^{\frac {b}{x}} \ln \relax (f )}{20}+\frac {x^{5} f^{a} f^{\frac {b}{x}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.52, size = 22, normalized size = 1.00 \[ -b^{5} f^{a} \Gamma \left (-5, -\frac {b \log \relax (f)}{x}\right ) \log \relax (f)^{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.69, size = 99, normalized size = 4.50 \[ \frac {b^5\,f^a\,{\ln \relax (f)}^5\,\mathrm {expint}\left (-\frac {b\,\ln \relax (f)}{x}\right )}{120}+b^5\,f^a\,f^{b/x}\,{\ln \relax (f)}^5\,\left (\frac {x^2}{120\,b^2\,{\ln \relax (f)}^2}+\frac {x^3}{60\,b^3\,{\ln \relax (f)}^3}+\frac {x^4}{20\,b^4\,{\ln \relax (f)}^4}+\frac {x^5}{5\,b^5\,{\ln \relax (f)}^5}+\frac {x}{120\,b\,\ln \relax (f)}\right ) \]
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 \[ \int f^{a + \frac {b}{x}} x^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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