Integrand size = 13, antiderivative size = 81 \[ \int \frac {f^{a+b x^3}}{x^{10}} \, dx=-\frac {f^{a+b x^3}}{9 x^9}-\frac {b f^{a+b x^3} \log (f)}{18 x^6}-\frac {b^2 f^{a+b x^3} \log ^2(f)}{18 x^3}+\frac {1}{18} b^3 f^a \operatorname {ExpIntegralEi}\left (b x^3 \log (f)\right ) \log ^3(f) \]
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Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2245, 2241} \[ \int \frac {f^{a+b x^3}}{x^{10}} \, dx=\frac {1}{18} b^3 f^a \log ^3(f) \operatorname {ExpIntegralEi}\left (b x^3 \log (f)\right )-\frac {b^2 \log ^2(f) f^{a+b x^3}}{18 x^3}-\frac {f^{a+b x^3}}{9 x^9}-\frac {b \log (f) f^{a+b x^3}}{18 x^6} \]
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Rule 2241
Rule 2245
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.73 \[ \int \frac {f^{a+b x^3}}{x^{10}} \, dx=\frac {f^a \left (b^3 x^9 \operatorname {ExpIntegralEi}\left (b x^3 \log (f)\right ) \log ^3(f)-f^{b x^3} \left (2+b x^3 \log (f)+b^2 x^6 \log ^2(f)\right )\right )}{18 x^9} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(176\) vs. \(2(73)=146\).
Time = 0.16 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.19
method | result | size |
meijerg | \(-\frac {f^{a} b^{3} \ln \left (f \right )^{3} \left (\frac {1}{3 b^{3} x^{9} \ln \left (f \right )^{3}}+\frac {1}{2 b^{2} x^{6} \ln \left (f \right )^{2}}+\frac {1}{2 b \,x^{3} \ln \left (f \right )}+\frac {11}{36}-\frac {\ln \left (x \right )}{2}-\frac {\ln \left (-b \right )}{6}-\frac {\ln \left (\ln \left (f \right )\right )}{6}-\frac {22 b^{3} x^{9} \ln \left (f \right )^{3}+36 b^{2} x^{6} \ln \left (f \right )^{2}+36 b \,x^{3} \ln \left (f \right )+24}{72 b^{3} x^{9} \ln \left (f \right )^{3}}+\frac {\left (4 b^{2} x^{6} \ln \left (f \right )^{2}+4 b \,x^{3} \ln \left (f \right )+8\right ) {\mathrm e}^{b \,x^{3} \ln \left (f \right )}}{24 b^{3} x^{9} \ln \left (f \right )^{3}}+\frac {\ln \left (-b \,x^{3} \ln \left (f \right )\right )}{6}+\frac {\operatorname {Ei}_{1}\left (-b \,x^{3} \ln \left (f \right )\right )}{6}\right )}{3}\) | \(177\) |
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Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.73 \[ \int \frac {f^{a+b x^3}}{x^{10}} \, dx=\frac {b^{3} f^{a} x^{9} {\rm Ei}\left (b x^{3} \log \left (f\right )\right ) \log \left (f\right )^{3} - {\left (b^{2} x^{6} \log \left (f\right )^{2} + b x^{3} \log \left (f\right ) + 2\right )} f^{b x^{3} + a}}{18 \, x^{9}} \]
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\[ \int \frac {f^{a+b x^3}}{x^{10}} \, dx=\int \frac {f^{a + b x^{3}}}{x^{10}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.27 \[ \int \frac {f^{a+b x^3}}{x^{10}} \, dx=\frac {1}{3} \, b^{3} f^{a} \Gamma \left (-3, -b x^{3} \log \left (f\right )\right ) \log \left (f\right )^{3} \]
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\[ \int \frac {f^{a+b x^3}}{x^{10}} \, dx=\int { \frac {f^{b x^{3} + a}}{x^{10}} \,d x } \]
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Time = 0.18 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.85 \[ \int \frac {f^{a+b x^3}}{x^{10}} \, dx=-\frac {b^3\,f^a\,{\ln \left (f\right )}^3\,\left (f^{b\,x^3}\,\left (\frac {1}{6\,b\,x^3\,\ln \left (f\right )}+\frac {1}{6\,b^2\,x^6\,{\ln \left (f\right )}^2}+\frac {1}{3\,b^3\,x^9\,{\ln \left (f\right )}^3}\right )+\frac {\mathrm {expint}\left (-b\,x^3\,\ln \left (f\right )\right )}{6}\right )}{3} \]
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