Integrand size = 13, antiderivative size = 34 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx=\frac {f^a \Gamma \left (\frac {13}{2},-\frac {b \log (f)}{x^2}\right )}{2 x^{13} \left (-\frac {b \log (f)}{x^2}\right )^{13/2}} \]
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Time = 0.01 (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} \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx=\frac {f^a \Gamma \left (\frac {13}{2},-\frac {b \log (f)}{x^2}\right )}{2 x^{13} \left (-\frac {b \log (f)}{x^2}\right )^{13/2}} \]
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Rule 2250
Rubi steps \begin{align*} \text {integral}& = \frac {f^a \Gamma \left (\frac {13}{2},-\frac {b \log (f)}{x^2}\right )}{2 x^{13} \left (-\frac {b \log (f)}{x^2}\right )^{13/2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx=\frac {f^a \Gamma \left (\frac {13}{2},-\frac {b \log (f)}{x^2}\right )}{2 x^{13} \left (-\frac {b \log (f)}{x^2}\right )^{13/2}} \]
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Time = 0.70 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.74
method | result | size |
meijerg | \(\frac {f^{a} \sqrt {-b}\, \left (-\frac {\left (-b \right )^{\frac {13}{2}} \sqrt {\ln \left (f \right )}\, \left (-\frac {416 b^{5} \ln \left (f \right )^{5}}{x^{10}}+\frac {2288 b^{4} \ln \left (f \right )^{4}}{x^{8}}-\frac {10296 b^{3} \ln \left (f \right )^{3}}{x^{6}}+\frac {36036 b^{2} \ln \left (f \right )^{2}}{x^{4}}-\frac {90090 b \ln \left (f \right )}{x^{2}}+135135\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{416 x \,b^{6}}+\frac {10395 \left (-b \right )^{\frac {13}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right )}{64 b^{\frac {13}{2}}}\right )}{2 b^{7} \ln \left (f \right )^{\frac {13}{2}}}\) | \(127\) |
risch | \(-\frac {f^{a} f^{\frac {b}{x^{2}}}}{2 x^{11} b \ln \left (f \right )}+\frac {11 f^{a} f^{\frac {b}{x^{2}}}}{4 \ln \left (f \right )^{2} b^{2} x^{9}}-\frac {99 f^{a} f^{\frac {b}{x^{2}}}}{8 \ln \left (f \right )^{3} b^{3} x^{7}}+\frac {693 f^{a} f^{\frac {b}{x^{2}}}}{16 \ln \left (f \right )^{4} b^{4} x^{5}}-\frac {3465 f^{a} f^{\frac {b}{x^{2}}}}{32 \ln \left (f \right )^{5} b^{5} x^{3}}+\frac {10395 f^{a} f^{\frac {b}{x^{2}}}}{64 \ln \left (f \right )^{6} b^{6} x}-\frac {10395 f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{128 \ln \left (f \right )^{6} b^{6} \sqrt {-b \ln \left (f \right )}}\) | \(168\) |
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Time = 0.08 (sec) , antiderivative size = 124, normalized size of antiderivative = 3.65 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx=\frac {10395 \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} x^{11} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x}\right ) + 2 \, {\left (10395 \, b x^{10} \log \left (f\right ) - 6930 \, b^{2} x^{8} \log \left (f\right )^{2} + 2772 \, b^{3} x^{6} \log \left (f\right )^{3} - 792 \, b^{4} x^{4} \log \left (f\right )^{4} + 176 \, b^{5} x^{2} \log \left (f\right )^{5} - 32 \, b^{6} \log \left (f\right )^{6}\right )} f^{\frac {a x^{2} + b}{x^{2}}}}{128 \, b^{7} x^{11} \log \left (f\right )^{7}} \]
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Timed out. \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx=\text {Timed out} \]
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none
Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx=\frac {f^{a} \Gamma \left (\frac {13}{2}, -\frac {b \log \left (f\right )}{x^{2}}\right )}{2 \, x^{13} \left (-\frac {b \log \left (f\right )}{x^{2}}\right )^{\frac {13}{2}}} \]
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\[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx=\int { \frac {f^{a + \frac {b}{x^{2}}}}{x^{14}} \,d x } \]
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Time = 0.40 (sec) , antiderivative size = 159, normalized size of antiderivative = 4.68 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^{14}} \, dx=-\frac {\frac {f^a\,\left (\frac {10395\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (f\right )}{x\,\sqrt {b\,\ln \left (f\right )}}\right )}{128}-\frac {10395\,f^{\frac {b}{x^2}}\,\sqrt {b\,\ln \left (f\right )}}{64\,x}\right )}{\sqrt {b\,\ln \left (f\right )}}-\frac {693\,b^2\,f^{a+\frac {b}{x^2}}\,{\ln \left (f\right )}^2}{16\,x^5}+\frac {99\,b^3\,f^{a+\frac {b}{x^2}}\,{\ln \left (f\right )}^3}{8\,x^7}-\frac {11\,b^4\,f^{a+\frac {b}{x^2}}\,{\ln \left (f\right )}^4}{4\,x^9}+\frac {b^5\,f^{a+\frac {b}{x^2}}\,{\ln \left (f\right )}^5}{2\,x^{11}}+\frac {3465\,b\,f^{a+\frac {b}{x^2}}\,\ln \left (f\right )}{32\,x^3}}{b^6\,{\ln \left (f\right )}^6} \]
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