Optimal. Leaf size=58 \[ -\frac {1}{6} b^2 f^a \log ^2(f) \text {Ei}\left (\frac {b \log (f)}{x^3}\right )+\frac {1}{6} b x^3 \log (f) f^{a+\frac {b}{x^3}}+\frac {1}{6} x^6 f^{a+\frac {b}{x^3}} \]
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Rubi [A] time = 0.08, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2214, 2210} \[ -\frac {1}{6} b^2 f^a \log ^2(f) \text {Ei}\left (\frac {b \log (f)}{x^3}\right )+\frac {1}{6} x^6 f^{a+\frac {b}{x^3}}+\frac {1}{6} b x^3 \log (f) f^{a+\frac {b}{x^3}} \]
Antiderivative was successfully verified.
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Rule 2210
Rule 2214
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x^3}} x^5 \, dx &=\frac {1}{6} f^{a+\frac {b}{x^3}} x^6+\frac {1}{2} (b \log (f)) \int f^{a+\frac {b}{x^3}} x^2 \, dx\\ &=\frac {1}{6} f^{a+\frac {b}{x^3}} x^6+\frac {1}{6} b f^{a+\frac {b}{x^3}} x^3 \log (f)+\frac {1}{2} \left (b^2 \log ^2(f)\right ) \int \frac {f^{a+\frac {b}{x^3}}}{x} \, dx\\ &=\frac {1}{6} f^{a+\frac {b}{x^3}} x^6+\frac {1}{6} b f^{a+\frac {b}{x^3}} x^3 \log (f)-\frac {1}{6} b^2 f^a \text {Ei}\left (\frac {b \log (f)}{x^3}\right ) \log ^2(f)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.76 \[ \frac {1}{6} f^a \left (x^3 f^{\frac {b}{x^3}} \left (b \log (f)+x^3\right )-b^2 \log ^2(f) \text {Ei}\left (\frac {b \log (f)}{x^3}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 47, normalized size = 0.81 \[ -\frac {1}{6} \, b^{2} f^{a} {\rm Ei}\left (\frac {b \log \relax (f)}{x^{3}}\right ) \log \relax (f)^{2} + \frac {1}{6} \, {\left (x^{6} + b x^{3} \log \relax (f)\right )} f^{\frac {a x^{3} + b}{x^{3}}} \]
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 \[ \int f^{a + \frac {b}{x^{3}}} x^{5}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 141, normalized size = 2.43 \[ -\frac {\left (-\frac {\left (\frac {3 b \ln \relax (f )}{x^{3}}+3\right ) x^{6} {\mathrm e}^{\frac {b \ln \relax (f )}{x^{3}}}}{6 b^{2} \ln \relax (f )^{2}}+\frac {\left (\frac {12 b \ln \relax (f )}{x^{3}}+\frac {9 b^{2} \ln \relax (f )^{2}}{x^{6}}+6\right ) x^{6}}{12 b^{2} \ln \relax (f )^{2}}-\frac {x^{6}}{2 b^{2} \ln \relax (f )^{2}}-\frac {x^{3}}{b \ln \relax (f )}-\frac {\Ei \left (1, -\frac {b \ln \relax (f )}{x^{3}}\right )}{2}-\frac {3 \ln \relax (x )}{2}+\frac {\ln \left (-b \right )}{2}-\frac {\ln \left (-\frac {b \ln \relax (f )}{x^{3}}\right )}{2}+\frac {\ln \left (\ln \relax (f )\right )}{2}-\frac {3}{4}\right ) b^{2} f^{a} \ln \relax (f )^{2}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 22, normalized size = 0.38 \[ \frac {1}{3} \, b^{2} f^{a} \Gamma \left (-2, -\frac {b \log \relax (f)}{x^{3}}\right ) \log \relax (f)^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.65, size = 57, normalized size = 0.98 \[ \frac {b^2\,f^a\,{\ln \relax (f)}^2\,\left (f^{\frac {b}{x^3}}\,\left (\frac {x^3}{2\,b\,\ln \relax (f)}+\frac {x^6}{2\,b^2\,{\ln \relax (f)}^2}\right )+\frac {\mathrm {expint}\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}{2}\right )}{3} \]
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 \[ \int f^{a + \frac {b}{x^{3}}} x^{5}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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