Optimal. Leaf size=81 \[ -\frac {1}{18} b^3 f^a \log ^3(f) \text {Ei}\left (\frac {b \log (f)}{x^3}\right )+\frac {1}{18} b^2 x^3 \log ^2(f) f^{a+\frac {b}{x^3}}+\frac {1}{9} x^9 f^{a+\frac {b}{x^3}}+\frac {1}{18} b x^6 \log (f) f^{a+\frac {b}{x^3}} \]
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Rubi [A] time = 0.11, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2214, 2210} \[ -\frac {1}{18} b^3 f^a \log ^3(f) \text {Ei}\left (\frac {b \log (f)}{x^3}\right )+\frac {1}{18} b^2 x^3 \log ^2(f) f^{a+\frac {b}{x^3}}+\frac {1}{9} x^9 f^{a+\frac {b}{x^3}}+\frac {1}{18} b x^6 \log (f) f^{a+\frac {b}{x^3}} \]
Antiderivative was successfully verified.
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Rule 2210
Rule 2214
Rubi steps
\begin {align*} \int f^{a+\frac {b}{x^3}} x^8 \, dx &=\frac {1}{9} f^{a+\frac {b}{x^3}} x^9+\frac {1}{3} (b \log (f)) \int f^{a+\frac {b}{x^3}} x^5 \, dx\\ &=\frac {1}{9} f^{a+\frac {b}{x^3}} x^9+\frac {1}{18} b f^{a+\frac {b}{x^3}} x^6 \log (f)+\frac {1}{6} \left (b^2 \log ^2(f)\right ) \int f^{a+\frac {b}{x^3}} x^2 \, dx\\ &=\frac {1}{9} f^{a+\frac {b}{x^3}} x^9+\frac {1}{18} b f^{a+\frac {b}{x^3}} x^6 \log (f)+\frac {1}{18} b^2 f^{a+\frac {b}{x^3}} x^3 \log ^2(f)+\frac {1}{6} \left (b^3 \log ^3(f)\right ) \int \frac {f^{a+\frac {b}{x^3}}}{x} \, dx\\ &=\frac {1}{9} f^{a+\frac {b}{x^3}} x^9+\frac {1}{18} b f^{a+\frac {b}{x^3}} x^6 \log (f)+\frac {1}{18} b^2 f^{a+\frac {b}{x^3}} x^3 \log ^2(f)-\frac {1}{18} b^3 f^a \text {Ei}\left (\frac {b \log (f)}{x^3}\right ) \log ^3(f)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 57, normalized size = 0.70 \[ \frac {1}{18} f^a \left (x^3 f^{\frac {b}{x^3}} \left (b^2 \log ^2(f)+b x^3 \log (f)+2 x^6\right )-b^3 \log ^3(f) \text {Ei}\left (\frac {b \log (f)}{x^3}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 60, normalized size = 0.74 \[ -\frac {1}{18} \, b^{3} f^{a} {\rm Ei}\left (\frac {b \log \relax (f)}{x^{3}}\right ) \log \relax (f)^{3} + \frac {1}{18} \, {\left (2 \, x^{9} + b x^{6} \log \relax (f) + b^{2} x^{3} \log \relax (f)^{2}\right )} f^{\frac {a x^{3} + b}{x^{3}}} \]
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 \[ \int f^{a + \frac {b}{x^{3}}} x^{8}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 177, normalized size = 2.19 \[ \frac {\left (\frac {\left (\frac {4 b \ln \relax (f )}{x^{3}}+\frac {4 b^{2} \ln \relax (f )^{2}}{x^{6}}+8\right ) x^{9} {\mathrm e}^{\frac {b \ln \relax (f )}{x^{3}}}}{24 b^{3} \ln \relax (f )^{3}}-\frac {\left (\frac {36 b \ln \relax (f )}{x^{3}}+\frac {36 b^{2} \ln \relax (f )^{2}}{x^{6}}+\frac {22 b^{3} \ln \relax (f )^{3}}{x^{9}}+24\right ) x^{9}}{72 b^{3} \ln \relax (f )^{3}}+\frac {x^{9}}{3 b^{3} \ln \relax (f )^{3}}+\frac {x^{6}}{2 b^{2} \ln \relax (f )^{2}}+\frac {x^{3}}{2 b \ln \relax (f )}+\frac {\Ei \left (1, -\frac {b \ln \relax (f )}{x^{3}}\right )}{6}+\frac {\ln \relax (x )}{2}-\frac {\ln \left (-b \right )}{6}+\frac {\ln \left (-\frac {b \ln \relax (f )}{x^{3}}\right )}{6}-\frac {\ln \left (\ln \relax (f )\right )}{6}+\frac {11}{36}\right ) b^{3} f^{a} \ln \relax (f )^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 22, normalized size = 0.27 \[ -\frac {1}{3} \, b^{3} f^{a} \Gamma \left (-3, -\frac {b \log \relax (f)}{x^{3}}\right ) \log \relax (f)^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.70, size = 69, normalized size = 0.85 \[ \frac {b^3\,f^a\,{\ln \relax (f)}^3\,\left (f^{\frac {b}{x^3}}\,\left (\frac {x^3}{6\,b\,\ln \relax (f)}+\frac {x^6}{6\,b^2\,{\ln \relax (f)}^2}+\frac {x^9}{3\,b^3\,{\ln \relax (f)}^3}\right )+\frac {\mathrm {expint}\left (-\frac {b\,\ln \relax (f)}{x^3}\right )}{6}\right )}{3} \]
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 \[ \int f^{a + \frac {b}{x^{3}}} x^{8}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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