Integrand size = 13, antiderivative size = 34 \[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\frac {1}{2} f^a x^{11} \Gamma \left (-\frac {11}{2},-\frac {b \log (f)}{x^2}\right ) \left (-\frac {b \log (f)}{x^2}\right )^{11/2} \]
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Time = 0.02 (sec) , antiderivative size = 34, 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} \[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\frac {1}{2} x^{11} f^a \left (-\frac {b \log (f)}{x^2}\right )^{11/2} \Gamma \left (-\frac {11}{2},-\frac {b \log (f)}{x^2}\right ) \]
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Rule 2250
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} f^a x^{11} \Gamma \left (-\frac {11}{2},-\frac {b \log (f)}{x^2}\right ) \left (-\frac {b \log (f)}{x^2}\right )^{11/2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\frac {1}{2} f^a x^{11} \Gamma \left (-\frac {11}{2},-\frac {b \log (f)}{x^2}\right ) \left (-\frac {b \log (f)}{x^2}\right )^{11/2} \]
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Time = 0.37 (sec) , antiderivative size = 124, normalized size of antiderivative = 3.65
method | result | size |
meijerg | \(\frac {f^{a} b^{5} \ln \left (f \right )^{\frac {11}{2}} \sqrt {-b}\, \left (-\frac {2 x^{11} \left (\frac {32 b^{5} \ln \left (f \right )^{5}}{945 x^{10}}+\frac {16 b^{4} \ln \left (f \right )^{4}}{945 x^{8}}+\frac {8 b^{3} \ln \left (f \right )^{3}}{315 x^{6}}+\frac {4 b^{2} \ln \left (f \right )^{2}}{63 x^{4}}+\frac {2 b \ln \left (f \right )}{9 x^{2}}+1\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{11 \left (-b \right )^{\frac {11}{2}} \ln \left (f \right )^{\frac {11}{2}}}+\frac {64 b^{\frac {11}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right )}{10395 \left (-b \right )^{\frac {11}{2}}}\right )}{2}\) | \(124\) |
risch | \(\frac {f^{a} x^{11} f^{\frac {b}{x^{2}}}}{11}+\frac {2 f^{a} \ln \left (f \right ) b \,x^{9} f^{\frac {b}{x^{2}}}}{99}+\frac {4 f^{a} \ln \left (f \right )^{2} b^{2} x^{7} f^{\frac {b}{x^{2}}}}{693}+\frac {8 f^{a} \ln \left (f \right )^{3} b^{3} x^{5} f^{\frac {b}{x^{2}}}}{3465}+\frac {16 f^{a} \ln \left (f \right )^{4} b^{4} x^{3} f^{\frac {b}{x^{2}}}}{10395}+\frac {32 f^{a} \ln \left (f \right )^{5} b^{5} x \,f^{\frac {b}{x^{2}}}}{10395}-\frac {32 f^{a} \ln \left (f \right )^{6} b^{6} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{10395 \sqrt {-b \ln \left (f \right )}}\) | \(155\) |
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Time = 0.31 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.24 \[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\frac {32}{10395} \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} b^{5} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x}\right ) \log \left (f\right )^{5} + \frac {1}{10395} \, {\left (945 \, x^{11} + 210 \, b x^{9} \log \left (f\right ) + 60 \, b^{2} x^{7} \log \left (f\right )^{2} + 24 \, b^{3} x^{5} \log \left (f\right )^{3} + 16 \, b^{4} x^{3} \log \left (f\right )^{4} + 32 \, b^{5} x \log \left (f\right )^{5}\right )} f^{\frac {a x^{2} + b}{x^{2}}} \]
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\[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\int f^{a + \frac {b}{x^{2}}} x^{10}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\frac {1}{2} \, f^{a} x^{11} \left (-\frac {b \log \left (f\right )}{x^{2}}\right )^{\frac {11}{2}} \Gamma \left (-\frac {11}{2}, -\frac {b \log \left (f\right )}{x^{2}}\right ) \]
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\[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\int { f^{a + \frac {b}{x^{2}}} x^{10} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 173, normalized size of antiderivative = 5.09 \[ \int f^{a+\frac {b}{x^2}} x^{10} \, dx=\frac {f^a\,f^{\frac {b}{x^2}}\,x^{11}}{11}-\frac {32\,f^a\,x^{11}\,\sqrt {\pi }\,{\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}^{11/2}}{10395}+\frac {32\,f^a\,x^{11}\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \left (f\right )}{x^2}}\right )\,{\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}^{11/2}}{10395}+\frac {32\,b^5\,f^a\,f^{\frac {b}{x^2}}\,x\,{\ln \left (f\right )}^5}{10395}+\frac {4\,b^2\,f^a\,f^{\frac {b}{x^2}}\,x^7\,{\ln \left (f\right )}^2}{693}+\frac {8\,b^3\,f^a\,f^{\frac {b}{x^2}}\,x^5\,{\ln \left (f\right )}^3}{3465}+\frac {16\,b^4\,f^a\,f^{\frac {b}{x^2}}\,x^3\,{\ln \left (f\right )}^4}{10395}+\frac {2\,b\,f^a\,f^{\frac {b}{x^2}}\,x^9\,\ln \left (f\right )}{99} \]
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