Integrand size = 13, antiderivative size = 34 \[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\frac {1}{2} f^a x^9 \Gamma \left (-\frac {9}{2},-\frac {b \log (f)}{x^2}\right ) \left (-\frac {b \log (f)}{x^2}\right )^{9/2} \]
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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} \[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\frac {1}{2} x^9 f^a \left (-\frac {b \log (f)}{x^2}\right )^{9/2} \Gamma \left (-\frac {9}{2},-\frac {b \log (f)}{x^2}\right ) \]
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Rule 2250
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} f^a x^9 \Gamma \left (-\frac {9}{2},-\frac {b \log (f)}{x^2}\right ) \left (-\frac {b \log (f)}{x^2}\right )^{9/2} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\frac {1}{2} f^a x^9 \Gamma \left (-\frac {9}{2},-\frac {b \log (f)}{x^2}\right ) \left (-\frac {b \log (f)}{x^2}\right )^{9/2} \]
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Time = 0.24 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.29
method | result | size |
meijerg | \(-\frac {f^{a} b^{4} \ln \left (f \right )^{\frac {9}{2}} \sqrt {-b}\, \left (-\frac {2 x^{9} \left (\frac {16 b^{4} \ln \left (f \right )^{4}}{105 x^{8}}+\frac {8 b^{3} \ln \left (f \right )^{3}}{105 x^{6}}+\frac {4 b^{2} \ln \left (f \right )^{2}}{35 x^{4}}+\frac {2 b \ln \left (f \right )}{7 x^{2}}+1\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{9 \left (-b \right )^{\frac {9}{2}} \ln \left (f \right )^{\frac {9}{2}}}+\frac {32 b^{\frac {9}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right )}{945 \left (-b \right )^{\frac {9}{2}}}\right )}{2}\) | \(112\) |
risch | \(\frac {f^{a} x^{9} f^{\frac {b}{x^{2}}}}{9}+\frac {2 f^{a} \ln \left (f \right ) b \,x^{7} f^{\frac {b}{x^{2}}}}{63}+\frac {4 f^{a} \ln \left (f \right )^{2} b^{2} x^{5} f^{\frac {b}{x^{2}}}}{315}+\frac {8 f^{a} \ln \left (f \right )^{3} b^{3} x^{3} f^{\frac {b}{x^{2}}}}{945}+\frac {16 f^{a} \ln \left (f \right )^{4} b^{4} x \,f^{\frac {b}{x^{2}}}}{945}-\frac {16 f^{a} \ln \left (f \right )^{5} b^{5} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{945 \sqrt {-b \ln \left (f \right )}}\) | \(133\) |
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Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.88 \[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\frac {16}{945} \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} b^{4} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x}\right ) \log \left (f\right )^{4} + \frac {1}{945} \, {\left (105 \, x^{9} + 30 \, b x^{7} \log \left (f\right ) + 12 \, b^{2} x^{5} \log \left (f\right )^{2} + 8 \, b^{3} x^{3} \log \left (f\right )^{3} + 16 \, b^{4} x \log \left (f\right )^{4}\right )} f^{\frac {a x^{2} + b}{x^{2}}} \]
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\[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\int f^{a + \frac {b}{x^{2}}} x^{8}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\frac {1}{2} \, f^{a} x^{9} \left (-\frac {b \log \left (f\right )}{x^{2}}\right )^{\frac {9}{2}} \Gamma \left (-\frac {9}{2}, -\frac {b \log \left (f\right )}{x^{2}}\right ) \]
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\[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\int { f^{a + \frac {b}{x^{2}}} x^{8} \,d x } \]
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Time = 0.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 4.44 \[ \int f^{a+\frac {b}{x^2}} x^8 \, dx=\frac {f^a\,f^{\frac {b}{x^2}}\,x^9}{9}+\frac {16\,f^a\,x^9\,\sqrt {\pi }\,{\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}^{9/2}}{945}-\frac {16\,f^a\,x^9\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\frac {b\,\ln \left (f\right )}{x^2}}\right )\,{\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}^{9/2}}{945}+\frac {16\,b^4\,f^a\,f^{\frac {b}{x^2}}\,x\,{\ln \left (f\right )}^4}{945}+\frac {4\,b^2\,f^a\,f^{\frac {b}{x^2}}\,x^5\,{\ln \left (f\right )}^2}{315}+\frac {8\,b^3\,f^a\,f^{\frac {b}{x^2}}\,x^3\,{\ln \left (f\right )}^3}{945}+\frac {2\,b\,f^a\,f^{\frac {b}{x^2}}\,x^7\,\ln \left (f\right )}{63} \]
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