Integrand size = 13, antiderivative size = 58 \[ \int f^{a+\frac {b}{x^2}} x^3 \, dx=\frac {1}{4} f^{a+\frac {b}{x^2}} x^4+\frac {1}{4} b f^{a+\frac {b}{x^2}} x^2 \log (f)-\frac {1}{4} b^2 f^a \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x^2}\right ) \log ^2(f) \]
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Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2245, 2241} \[ \int f^{a+\frac {b}{x^2}} x^3 \, dx=-\frac {1}{4} b^2 f^a \log ^2(f) \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x^2}\right )+\frac {1}{4} b x^2 \log (f) f^{a+\frac {b}{x^2}}+\frac {1}{4} x^4 f^{a+\frac {b}{x^2}} \]
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Rule 2241
Rule 2245
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} f^{a+\frac {b}{x^2}} x^4+\frac {1}{2} (b \log (f)) \int f^{a+\frac {b}{x^2}} x \, dx \\ & = \frac {1}{4} f^{a+\frac {b}{x^2}} x^4+\frac {1}{4} b f^{a+\frac {b}{x^2}} x^2 \log (f)+\frac {1}{2} \left (b^2 \log ^2(f)\right ) \int \frac {f^{a+\frac {b}{x^2}}}{x} \, dx \\ & = \frac {1}{4} f^{a+\frac {b}{x^2}} x^4+\frac {1}{4} b f^{a+\frac {b}{x^2}} x^2 \log (f)-\frac {1}{4} b^2 f^a \text {Ei}\left (\frac {b \log (f)}{x^2}\right ) \log ^2(f) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.76 \[ \int f^{a+\frac {b}{x^2}} x^3 \, dx=\frac {1}{4} f^a \left (-b^2 \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x^2}\right ) \log ^2(f)+f^{\frac {b}{x^2}} x^2 \left (x^2+b \log (f)\right )\right ) \]
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Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98
method | result | size |
risch | \(\frac {f^{a} x^{4} f^{\frac {b}{x^{2}}}}{4}+\frac {f^{a} \ln \left (f \right ) b \,x^{2} f^{\frac {b}{x^{2}}}}{4}+\frac {f^{a} \ln \left (f \right )^{2} b^{2} \operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x^{2}}\right )}{4}\) | \(57\) |
meijerg | \(-\frac {f^{a} b^{2} \ln \left (f \right )^{2} \left (-\frac {x^{4}}{2 b^{2} \ln \left (f \right )^{2}}-\frac {x^{2}}{b \ln \left (f \right )}-\frac {3}{4}-\ln \left (x \right )+\frac {\ln \left (-b \right )}{2}+\frac {\ln \left (\ln \left (f \right )\right )}{2}+\frac {x^{4} \left (\frac {9 b^{2} \ln \left (f \right )^{2}}{x^{4}}+\frac {12 b \ln \left (f \right )}{x^{2}}+6\right )}{12 b^{2} \ln \left (f \right )^{2}}-\frac {x^{4} \left (3+\frac {3 b \ln \left (f \right )}{x^{2}}\right ) {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{6 b^{2} \ln \left (f \right )^{2}}-\frac {\ln \left (-\frac {b \ln \left (f \right )}{x^{2}}\right )}{2}-\frac {\operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x^{2}}\right )}{2}\right )}{2}\) | \(141\) |
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Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.81 \[ \int f^{a+\frac {b}{x^2}} x^3 \, dx=-\frac {1}{4} \, b^{2} f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right )^{2} + \frac {1}{4} \, {\left (x^{4} + b x^{2} \log \left (f\right )\right )} f^{\frac {a x^{2} + b}{x^{2}}} \]
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\[ \int f^{a+\frac {b}{x^2}} x^3 \, dx=\int f^{a + \frac {b}{x^{2}}} x^{3}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.38 \[ \int f^{a+\frac {b}{x^2}} x^3 \, dx=\frac {1}{2} \, b^{2} f^{a} \Gamma \left (-2, -\frac {b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right )^{2} \]
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\[ \int f^{a+\frac {b}{x^2}} x^3 \, dx=\int { f^{a + \frac {b}{x^{2}}} x^{3} \,d x } \]
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Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int f^{a+\frac {b}{x^2}} x^3 \, dx=\frac {b^2\,f^a\,{\ln \left (f\right )}^2\,\left (f^{\frac {b}{x^2}}\,\left (\frac {x^2}{2\,b\,\ln \left (f\right )}+\frac {x^4}{2\,b^2\,{\ln \left (f\right )}^2}\right )+\frac {\mathrm {expint}\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}{2}\right )}{2} \]
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