Integrand size = 13, antiderivative size = 82 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{19}} \, dx=\frac {f^{a+\frac {b}{x^3}} \left (120 x^{15}-120 b x^{12} \log (f)+60 b^2 x^9 \log ^2(f)-20 b^3 x^6 \log ^3(f)+5 b^4 x^3 \log ^4(f)-b^5 \log ^5(f)\right )}{3 b^6 x^{15} \log ^6(f)} \]
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Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2249} \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{19}} \, dx=\frac {f^{a+\frac {b}{x^3}} \left (-b^5 \log ^5(f)+5 b^4 x^3 \log ^4(f)-20 b^3 x^6 \log ^3(f)+60 b^2 x^9 \log ^2(f)-120 b x^{12} \log (f)+120 x^{15}\right )}{3 b^6 x^{15} \log ^6(f)} \]
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Rule 2249
Rubi steps \begin{align*} \text {integral}& = \frac {f^{a+\frac {b}{x^3}} \left (120 x^{15}-120 b x^{12} \log (f)+60 b^2 x^9 \log ^2(f)-20 b^3 x^6 \log ^3(f)+5 b^4 x^3 \log ^4(f)-b^5 \log ^5(f)\right )}{3 b^6 x^{15} \log ^6(f)} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.29 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{19}} \, dx=\frac {f^a \Gamma \left (6,-\frac {b \log (f)}{x^3}\right )}{3 b^6 \log ^6(f)} \]
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Time = 0.96 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.01
method | result | size |
meijerg | \(-\frac {f^{a} \left (120-\frac {\left (-\frac {6 b^{5} \ln \left (f \right )^{5}}{x^{15}}+\frac {30 b^{4} \ln \left (f \right )^{4}}{x^{12}}-\frac {120 b^{3} \ln \left (f \right )^{3}}{x^{9}}+\frac {360 b^{2} \ln \left (f \right )^{2}}{x^{6}}-\frac {720 b \ln \left (f \right )}{x^{3}}+720\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{3}}}}{6}\right )}{3 b^{6} \ln \left (f \right )^{6}}\) | \(83\) |
risch | \(-\frac {\left (-120 x^{15}+120 b \,x^{12} \ln \left (f \right )-60 b^{2} x^{9} \ln \left (f \right )^{2}+20 b^{3} x^{6} \ln \left (f \right )^{3}-5 b^{4} x^{3} \ln \left (f \right )^{4}+b^{5} \ln \left (f \right )^{5}\right ) f^{\frac {a \,x^{3}+b}{x^{3}}}}{3 \ln \left (f \right )^{6} b^{6} x^{15}}\) | \(84\) |
parallelrisch | \(\frac {120 f^{a +\frac {b}{x^{3}}} x^{15}-120 f^{a +\frac {b}{x^{3}}} x^{12} b \ln \left (f \right )+60 f^{a +\frac {b}{x^{3}}} x^{9} b^{2} \ln \left (f \right )^{2}-20 f^{a +\frac {b}{x^{3}}} x^{6} b^{3} \ln \left (f \right )^{3}+5 f^{a +\frac {b}{x^{3}}} x^{3} b^{4} \ln \left (f \right )^{4}-f^{a +\frac {b}{x^{3}}} b^{5} \ln \left (f \right )^{5}}{3 x^{15} b^{6} \ln \left (f \right )^{6}}\) | \(126\) |
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Time = 0.31 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.02 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{19}} \, dx=\frac {{\left (120 \, x^{15} - 120 \, b x^{12} \log \left (f\right ) + 60 \, b^{2} x^{9} \log \left (f\right )^{2} - 20 \, b^{3} x^{6} \log \left (f\right )^{3} + 5 \, b^{4} x^{3} \log \left (f\right )^{4} - b^{5} \log \left (f\right )^{5}\right )} f^{\frac {a x^{3} + b}{x^{3}}}}{3 \, b^{6} x^{15} \log \left (f\right )^{6}} \]
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Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.04 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{19}} \, dx=\frac {f^{a + \frac {b}{x^{3}}} \left (- b^{5} \log {\left (f \right )}^{5} + 5 b^{4} x^{3} \log {\left (f \right )}^{4} - 20 b^{3} x^{6} \log {\left (f \right )}^{3} + 60 b^{2} x^{9} \log {\left (f \right )}^{2} - 120 b x^{12} \log {\left (f \right )} + 120 x^{15}\right )}{3 b^{6} x^{15} \log {\left (f \right )}^{6}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.27 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{19}} \, dx=\frac {f^{a} \Gamma \left (6, -\frac {b \log \left (f\right )}{x^{3}}\right )}{3 \, b^{6} \log \left (f\right )^{6}} \]
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\[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{19}} \, dx=\int { \frac {f^{a + \frac {b}{x^{3}}}}{x^{19}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.02 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^{19}} \, dx=-\frac {f^{a+\frac {b}{x^3}}\,\left (\frac {1}{3\,b\,\ln \left (f\right )}-\frac {5\,x^3}{3\,b^2\,{\ln \left (f\right )}^2}+\frac {20\,x^6}{3\,b^3\,{\ln \left (f\right )}^3}-\frac {20\,x^9}{b^4\,{\ln \left (f\right )}^4}+\frac {40\,x^{12}}{b^5\,{\ln \left (f\right )}^5}-\frac {40\,x^{15}}{b^6\,{\ln \left (f\right )}^6}\right )}{x^{15}} \]
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