Integrand size = 13, antiderivative size = 24 \[ \int f^{a+\frac {b}{x^2}} x^7 \, dx=\frac {1}{2} b^4 f^a \Gamma \left (-4,-\frac {b \log (f)}{x^2}\right ) \log ^4(f) \]
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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} \[ \int f^{a+\frac {b}{x^2}} x^7 \, dx=\frac {1}{2} b^4 f^a \log ^4(f) \Gamma \left (-4,-\frac {b \log (f)}{x^2}\right ) \]
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Rule 2250
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} b^4 f^a \Gamma \left (-4,-\frac {b \log (f)}{x^2}\right ) \log ^4(f) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int f^{a+\frac {b}{x^2}} x^7 \, dx=\frac {1}{2} b^4 f^a \Gamma \left (-4,-\frac {b \log (f)}{x^2}\right ) \log ^4(f) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(100\) vs. \(2(18)=36\).
Time = 0.20 (sec) , antiderivative size = 101, normalized size of antiderivative = 4.21
method | result | size |
risch | \(\frac {f^{a} x^{8} f^{\frac {b}{x^{2}}}}{8}+\frac {f^{a} \ln \left (f \right ) b \,x^{6} f^{\frac {b}{x^{2}}}}{24}+\frac {f^{a} \ln \left (f \right )^{2} b^{2} x^{4} f^{\frac {b}{x^{2}}}}{48}+\frac {f^{a} \ln \left (f \right )^{3} b^{3} x^{2} f^{\frac {b}{x^{2}}}}{48}+\frac {f^{a} \ln \left (f \right )^{4} b^{4} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x^{2}}\right )}{48}\) | \(101\) |
meijerg | \(-\frac {f^{a} b^{4} \ln \left (f \right )^{4} \left (-\frac {x^{8}}{4 b^{4} \ln \left (f \right )^{4}}-\frac {x^{6}}{3 b^{3} \ln \left (f \right )^{3}}-\frac {x^{4}}{4 b^{2} \ln \left (f \right )^{2}}-\frac {x^{2}}{6 b \ln \left (f \right )}-\frac {25}{288}-\frac {\ln \left (x \right )}{12}+\frac {\ln \left (-b \right )}{24}+\frac {\ln \left (\ln \left (f \right )\right )}{24}+\frac {x^{8} \left (\frac {125 b^{4} \ln \left (f \right )^{4}}{x^{8}}+\frac {240 b^{3} \ln \left (f \right )^{3}}{x^{6}}+\frac {360 b^{2} \ln \left (f \right )^{2}}{x^{4}}+\frac {480 b \ln \left (f \right )}{x^{2}}+360\right )}{1440 b^{4} \ln \left (f \right )^{4}}-\frac {x^{8} \left (\frac {5 b^{3} \ln \left (f \right )^{3}}{x^{6}}+\frac {5 b^{2} \ln \left (f \right )^{2}}{x^{4}}+\frac {10 b \ln \left (f \right )}{x^{2}}+30\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{120 b^{4} \ln \left (f \right )^{4}}-\frac {\ln \left (-\frac {b \ln \left (f \right )}{x^{2}}\right )}{24}-\frac {\operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x^{2}}\right )}{24}\right )}{2}\) | \(213\) |
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Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (18) = 36\).
Time = 0.28 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.00 \[ \int f^{a+\frac {b}{x^2}} x^7 \, dx=-\frac {1}{48} \, b^{4} f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right )^{4} + \frac {1}{48} \, {\left (6 \, x^{8} + 2 \, b x^{6} \log \left (f\right ) + b^{2} x^{4} \log \left (f\right )^{2} + b^{3} x^{2} \log \left (f\right )^{3}\right )} f^{\frac {a x^{2} + b}{x^{2}}} \]
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\[ \int f^{a+\frac {b}{x^2}} x^7 \, dx=\int f^{a + \frac {b}{x^{2}}} x^{7}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int f^{a+\frac {b}{x^2}} x^7 \, dx=\frac {1}{2} \, b^{4} f^{a} \Gamma \left (-4, -\frac {b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right )^{4} \]
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\[ \int f^{a+\frac {b}{x^2}} x^7 \, dx=\int { f^{a + \frac {b}{x^{2}}} x^{7} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.75 \[ \int f^{a+\frac {b}{x^2}} x^7 \, dx=\frac {b^4\,f^a\,{\ln \left (f\right )}^4\,\mathrm {expint}\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}{48}+\frac {b^4\,f^a\,f^{\frac {b}{x^2}}\,{\ln \left (f\right )}^4\,\left (\frac {x^2}{24\,b\,\ln \left (f\right )}+\frac {x^4}{24\,b^2\,{\ln \left (f\right )}^2}+\frac {x^6}{12\,b^3\,{\ln \left (f\right )}^3}+\frac {x^8}{4\,b^4\,{\ln \left (f\right )}^4}\right )}{2} \]
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