Integrand size = 13, antiderivative size = 35 \[ \int f^{a+\frac {b}{x^3}} x^2 \, dx=\frac {1}{3} f^{a+\frac {b}{x^3}} x^3-\frac {1}{3} b f^a \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x^3}\right ) \log (f) \]
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Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2245, 2241} \[ \int f^{a+\frac {b}{x^3}} x^2 \, dx=\frac {1}{3} x^3 f^{a+\frac {b}{x^3}}-\frac {1}{3} b f^a \log (f) \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x^3}\right ) \]
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Rule 2241
Rule 2245
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} f^{a+\frac {b}{x^3}} x^3+(b \log (f)) \int \frac {f^{a+\frac {b}{x^3}}}{x} \, dx \\ & = \frac {1}{3} f^{a+\frac {b}{x^3}} x^3-\frac {1}{3} b f^a \text {Ei}\left (\frac {b \log (f)}{x^3}\right ) \log (f) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int f^{a+\frac {b}{x^3}} x^2 \, dx=\frac {1}{3} f^a \left (f^{\frac {b}{x^3}} x^3-b \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x^3}\right ) \log (f)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(96\) vs. \(2(31)=62\).
Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.77
method | result | size |
meijerg | \(\frac {f^{a} b \ln \left (f \right ) \left (\frac {x^{3}}{b \ln \left (f \right )}+1+3 \ln \left (x \right )-\ln \left (-b \right )-\ln \left (\ln \left (f \right )\right )-\frac {x^{3} \left (2+\frac {2 b \ln \left (f \right )}{x^{3}}\right )}{2 b \ln \left (f \right )}+\frac {x^{3} {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{3}}}}{b \ln \left (f \right )}+\ln \left (-\frac {b \ln \left (f \right )}{x^{3}}\right )+\operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x^{3}}\right )\right )}{3}\) | \(97\) |
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none
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int f^{a+\frac {b}{x^3}} x^2 \, dx=\frac {1}{3} \, f^{\frac {a x^{3} + b}{x^{3}}} x^{3} - \frac {1}{3} \, b f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x^{3}}\right ) \log \left (f\right ) \]
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\[ \int f^{a+\frac {b}{x^3}} x^2 \, dx=\int f^{a + \frac {b}{x^{3}}} x^{2}\, dx \]
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none
Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.51 \[ \int f^{a+\frac {b}{x^3}} x^2 \, dx=-\frac {1}{3} \, b f^{a} \Gamma \left (-1, -\frac {b \log \left (f\right )}{x^{3}}\right ) \log \left (f\right ) \]
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\[ \int f^{a+\frac {b}{x^3}} x^2 \, dx=\int { f^{a + \frac {b}{x^{3}}} x^{2} \,d x } \]
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Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int f^{a+\frac {b}{x^3}} x^2 \, dx=\frac {f^a\,f^{\frac {b}{x^3}}\,x^3}{3}+\frac {b\,f^a\,\ln \left (f\right )\,\mathrm {expint}\left (-\frac {b\,\ln \left (f\right )}{x^3}\right )}{3} \]
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