Optimal. Leaf size=81 \[ \frac {1}{18} b^3 f^a \log ^3(f) \text {Ei}\left (b x^3 \log (f)\right )-\frac {b^2 \log ^2(f) f^{a+b x^3}}{18 x^3}-\frac {f^{a+b x^3}}{9 x^9}-\frac {b \log (f) f^{a+b x^3}}{18 x^6} \]
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Rubi [A] time = 0.09, 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} \[ \frac {1}{18} b^3 f^a \log ^3(f) \text {Ei}\left (b x^3 \log (f)\right )-\frac {b^2 \log ^2(f) f^{a+b x^3}}{18 x^3}-\frac {f^{a+b x^3}}{9 x^9}-\frac {b \log (f) f^{a+b x^3}}{18 x^6} \]
Antiderivative was successfully verified.
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Rule 2210
Rule 2214
Rubi steps
\begin {align*} \int \frac {f^{a+b x^3}}{x^{10}} \, dx &=-\frac {f^{a+b x^3}}{9 x^9}+\frac {1}{3} (b \log (f)) \int \frac {f^{a+b x^3}}{x^7} \, dx\\ &=-\frac {f^{a+b x^3}}{9 x^9}-\frac {b f^{a+b x^3} \log (f)}{18 x^6}+\frac {1}{6} \left (b^2 \log ^2(f)\right ) \int \frac {f^{a+b x^3}}{x^4} \, dx\\ &=-\frac {f^{a+b x^3}}{9 x^9}-\frac {b f^{a+b x^3} \log (f)}{18 x^6}-\frac {b^2 f^{a+b x^3} \log ^2(f)}{18 x^3}+\frac {1}{6} \left (b^3 \log ^3(f)\right ) \int \frac {f^{a+b x^3}}{x} \, dx\\ &=-\frac {f^{a+b x^3}}{9 x^9}-\frac {b f^{a+b x^3} \log (f)}{18 x^6}-\frac {b^2 f^{a+b x^3} \log ^2(f)}{18 x^3}+\frac {1}{18} b^3 f^a \text {Ei}\left (b x^3 \log (f)\right ) \log ^3(f)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 59, normalized size = 0.73 \[ \frac {f^a \left (b^3 x^9 \log ^3(f) \text {Ei}\left (b x^3 \log (f)\right )-f^{b x^3} \left (b^2 x^6 \log ^2(f)+b x^3 \log (f)+2\right )\right )}{18 x^9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 59, normalized size = 0.73 \[ \frac {b^{3} f^{a} x^{9} {\rm Ei}\left (b x^{3} \log \relax (f)\right ) \log \relax (f)^{3} - {\left (b^{2} x^{6} \log \relax (f)^{2} + b x^{3} \log \relax (f) + 2\right )} f^{b x^{3} + a}}{18 \, x^{9}} \]
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 \[ \int \frac {f^{b x^{3} + a}}{x^{10}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 177, normalized size = 2.19 \[ -\frac {\left (\frac {\Ei \left (1, -b \,x^{3} \ln \relax (f )\right )}{6}-\frac {\ln \relax (x )}{2}-\frac {\ln \left (-b \right )}{6}+\frac {\ln \left (-b \,x^{3} \ln \relax (f )\right )}{6}-\frac {\ln \left (\ln \relax (f )\right )}{6}+\frac {1}{2 b \,x^{3} \ln \relax (f )}+\frac {1}{2 b^{2} x^{6} \ln \relax (f )^{2}}+\frac {\left (4 b^{2} x^{6} \ln \relax (f )^{2}+4 b \,x^{3} \ln \relax (f )+8\right ) {\mathrm e}^{b \,x^{3} \ln \relax (f )}}{24 b^{3} x^{9} \ln \relax (f )^{3}}-\frac {22 b^{3} x^{9} \ln \relax (f )^{3}+36 b^{2} x^{6} \ln \relax (f )^{2}+36 b \,x^{3} \ln \relax (f )+24}{72 b^{3} x^{9} \ln \relax (f )^{3}}+\frac {1}{3 b^{3} x^{9} \ln \relax (f )^{3}}+\frac {11}{36}\right ) b^{3} f^{a} \ln \relax (f )^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 22, normalized size = 0.27 \[ \frac {1}{3} \, b^{3} f^{a} \Gamma \left (-3, -b x^{3} \log \relax (f)\right ) \log \relax (f)^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.53, size = 69, normalized size = 0.85 \[ -\frac {b^3\,f^a\,{\ln \relax (f)}^3\,\left (f^{b\,x^3}\,\left (\frac {1}{6\,b\,x^3\,\ln \relax (f)}+\frac {1}{6\,b^2\,x^6\,{\ln \relax (f)}^2}+\frac {1}{3\,b^3\,x^9\,{\ln \relax (f)}^3}\right )+\frac {\mathrm {expint}\left (-b\,x^3\,\ln \relax (f)\right )}{6}\right )}{3} \]
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 \[ \int \frac {f^{a + b x^{3}}}{x^{10}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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