Integrand size = 13, antiderivative size = 24 \[ \int f^{a+\frac {b}{x^2}} x^9 \, dx=-\frac {1}{2} b^5 f^a \Gamma \left (-5,-\frac {b \log (f)}{x^2}\right ) \log ^5(f) \]
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Time = 0.02 (sec) , antiderivative size = 24, 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 = {2250} \[ \int f^{a+\frac {b}{x^2}} x^9 \, dx=-\frac {1}{2} b^5 f^a \log ^5(f) \Gamma \left (-5,-\frac {b \log (f)}{x^2}\right ) \]
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Rule 2250
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} b^5 f^a \Gamma \left (-5,-\frac {b \log (f)}{x^2}\right ) \log ^5(f) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int f^{a+\frac {b}{x^2}} x^9 \, dx=-\frac {1}{2} b^5 f^a \Gamma \left (-5,-\frac {b \log (f)}{x^2}\right ) \log ^5(f) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(122\) vs. \(2(18)=36\).
Time = 0.31 (sec) , antiderivative size = 123, normalized size of antiderivative = 5.12
method | result | size |
risch | \(\frac {f^{a} x^{10} f^{\frac {b}{x^{2}}}}{10}+\frac {f^{a} \ln \left (f \right ) b \,x^{8} f^{\frac {b}{x^{2}}}}{40}+\frac {f^{a} \ln \left (f \right )^{2} b^{2} x^{6} f^{\frac {b}{x^{2}}}}{120}+\frac {f^{a} \ln \left (f \right )^{3} b^{3} x^{4} f^{\frac {b}{x^{2}}}}{240}+\frac {f^{a} \ln \left (f \right )^{4} b^{4} x^{2} f^{\frac {b}{x^{2}}}}{240}+\frac {f^{a} \ln \left (f \right )^{5} b^{5} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x^{2}}\right )}{240}\) | \(123\) |
meijerg | \(\frac {f^{a} b^{5} \ln \left (f \right )^{5} \left (\frac {x^{10}}{5 b^{5} \ln \left (f \right )^{5}}+\frac {x^{8}}{4 b^{4} \ln \left (f \right )^{4}}+\frac {x^{6}}{6 b^{3} \ln \left (f \right )^{3}}+\frac {x^{4}}{12 b^{2} \ln \left (f \right )^{2}}+\frac {x^{2}}{24 b \ln \left (f \right )}+\frac {137}{7200}+\frac {\ln \left (x \right )}{60}-\frac {\ln \left (-b \right )}{120}-\frac {\ln \left (\ln \left (f \right )\right )}{120}-\frac {x^{10} \left (\frac {137 b^{5} \ln \left (f \right )^{5}}{x^{10}}+\frac {300 b^{4} \ln \left (f \right )^{4}}{x^{8}}+\frac {600 b^{3} \ln \left (f \right )^{3}}{x^{6}}+\frac {1200 b^{2} \ln \left (f \right )^{2}}{x^{4}}+\frac {1800 b \ln \left (f \right )}{x^{2}}+1440\right )}{7200 b^{5} \ln \left (f \right )^{5}}+\frac {x^{10} \left (\frac {6 b^{4} \ln \left (f \right )^{4}}{x^{8}}+\frac {6 b^{3} \ln \left (f \right )^{3}}{x^{6}}+\frac {12 b^{2} \ln \left (f \right )^{2}}{x^{4}}+\frac {36 b \ln \left (f \right )}{x^{2}}+144\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{720 b^{5} \ln \left (f \right )^{5}}+\frac {\ln \left (-\frac {b \ln \left (f \right )}{x^{2}}\right )}{120}+\frac {\operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x^{2}}\right )}{120}\right )}{2}\) | \(249\) |
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (18) = 36\).
Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.50 \[ \int f^{a+\frac {b}{x^2}} x^9 \, dx=-\frac {1}{240} \, b^{5} f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right )^{5} + \frac {1}{240} \, {\left (24 \, x^{10} + 6 \, b x^{8} \log \left (f\right ) + 2 \, b^{2} x^{6} \log \left (f\right )^{2} + b^{3} x^{4} \log \left (f\right )^{3} + b^{4} x^{2} \log \left (f\right )^{4}\right )} f^{\frac {a x^{2} + b}{x^{2}}} \]
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\[ \int f^{a+\frac {b}{x^2}} x^9 \, dx=\int f^{a + \frac {b}{x^{2}}} x^{9}\, dx \]
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none
Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int f^{a+\frac {b}{x^2}} x^9 \, dx=-\frac {1}{2} \, b^{5} f^{a} \Gamma \left (-5, -\frac {b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right )^{5} \]
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\[ \int f^{a+\frac {b}{x^2}} x^9 \, dx=\int { f^{a + \frac {b}{x^{2}}} x^{9} \,d x } \]
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Time = 0.38 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.25 \[ \int f^{a+\frac {b}{x^2}} x^9 \, dx=\frac {b^5\,f^a\,{\ln \left (f\right )}^5\,\mathrm {expint}\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}{240}+\frac {b^5\,f^a\,f^{\frac {b}{x^2}}\,{\ln \left (f\right )}^5\,\left (\frac {x^2}{120\,b\,\ln \left (f\right )}+\frac {x^4}{120\,b^2\,{\ln \left (f\right )}^2}+\frac {x^6}{60\,b^3\,{\ln \left (f\right )}^3}+\frac {x^8}{20\,b^4\,{\ln \left (f\right )}^4}+\frac {x^{10}}{5\,b^5\,{\ln \left (f\right )}^5}\right )}{2} \]
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