Integrand size = 13, antiderivative size = 73 \[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=\frac {1}{3} f^{a+\frac {b}{x^2}} x^3+\frac {2}{3} b f^{a+\frac {b}{x^2}} x \log (f)-\frac {2}{3} b^{3/2} f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right ) \log ^{\frac {3}{2}}(f) \]
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Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2245, 2237, 2242, 2235} \[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=-\frac {2}{3} \sqrt {\pi } b^{3/2} f^a \log ^{\frac {3}{2}}(f) \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )+\frac {2}{3} b x \log (f) f^{a+\frac {b}{x^2}}+\frac {1}{3} x^3 f^{a+\frac {b}{x^2}} \]
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Rule 2235
Rule 2237
Rule 2242
Rule 2245
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} f^{a+\frac {b}{x^2}} x^3+\frac {1}{3} (2 b \log (f)) \int f^{a+\frac {b}{x^2}} \, dx \\ & = \frac {1}{3} f^{a+\frac {b}{x^2}} x^3+\frac {2}{3} b f^{a+\frac {b}{x^2}} x \log (f)+\frac {1}{3} \left (4 b^2 \log ^2(f)\right ) \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx \\ & = \frac {1}{3} f^{a+\frac {b}{x^2}} x^3+\frac {2}{3} b f^{a+\frac {b}{x^2}} x \log (f)-\frac {1}{3} \left (4 b^2 \log ^2(f)\right ) \text {Subst}\left (\int f^{a+b x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{3} f^{a+\frac {b}{x^2}} x^3+\frac {2}{3} b f^{a+\frac {b}{x^2}} x \log (f)-\frac {2}{3} b^{3/2} f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right ) \log ^{\frac {3}{2}}(f) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.82 \[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=\frac {1}{3} f^a \left (-2 b^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right ) \log ^{\frac {3}{2}}(f)+f^{\frac {b}{x^2}} x \left (x^2+2 b \log (f)\right )\right ) \]
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Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92
method | result | size |
risch | \(\frac {f^{a} x^{3} f^{\frac {b}{x^{2}}}}{3}+\frac {2 f^{a} \ln \left (f \right ) b x \,f^{\frac {b}{x^{2}}}}{3}-\frac {2 f^{a} \ln \left (f \right )^{2} b^{2} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{3 \sqrt {-b \ln \left (f \right )}}\) | \(67\) |
meijerg | \(\frac {f^{a} b \ln \left (f \right )^{\frac {3}{2}} \sqrt {-b}\, \left (-\frac {2 x^{3} \left (\frac {2 b \ln \left (f \right )}{x^{2}}+1\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{3 \left (-b \right )^{\frac {3}{2}} \ln \left (f \right )^{\frac {3}{2}}}+\frac {4 b^{\frac {3}{2}} \sqrt {\pi }\, \operatorname {erfi}\left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right )}{3 \left (-b \right )^{\frac {3}{2}}}\right )}{2}\) | \(74\) |
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Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=\frac {2}{3} \, \sqrt {\pi } \sqrt {-b \log \left (f\right )} b f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x}\right ) \log \left (f\right ) + \frac {1}{3} \, {\left (x^{3} + 2 \, b x \log \left (f\right )\right )} f^{\frac {a x^{2} + b}{x^{2}}} \]
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\[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=\int f^{a + \frac {b}{x^{2}}} x^{2}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.38 \[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=\frac {1}{2} \, f^{a} x^{3} \left (-\frac {b \log \left (f\right )}{x^{2}}\right )^{\frac {3}{2}} \Gamma \left (-\frac {3}{2}, -\frac {b \log \left (f\right )}{x^{2}}\right ) \]
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\[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=\int { f^{a + \frac {b}{x^{2}}} x^{2} \,d x } \]
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Time = 0.35 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.97 \[ \int f^{a+\frac {b}{x^2}} x^2 \, dx=x^3\,\left (\frac {f^a\,f^{\frac {b}{x^2}}}{3}+\frac {2\,b\,f^a\,f^{\frac {b}{x^2}}\,\ln \left (f\right )}{3\,x^2}\right )-\frac {2\,b^2\,f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (f\right )}{x\,\sqrt {b\,\ln \left (f\right )}}\right )\,{\ln \left (f\right )}^2}{3\,\sqrt {b\,\ln \left (f\right )}} \]
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