Integrand size = 11, antiderivative size = 34 \[ \int f^{a+\frac {b}{x^3}} x \, dx=\frac {1}{3} f^a x^2 \Gamma \left (-\frac {2}{3},-\frac {b \log (f)}{x^3}\right ) \left (-\frac {b \log (f)}{x^3}\right )^{2/3} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2250} \[ \int f^{a+\frac {b}{x^3}} x \, dx=\frac {1}{3} x^2 f^a \left (-\frac {b \log (f)}{x^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {b \log (f)}{x^3}\right ) \]
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Rule 2250
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} f^a x^2 \Gamma \left (-\frac {2}{3},-\frac {b \log (f)}{x^3}\right ) \left (-\frac {b \log (f)}{x^3}\right )^{2/3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int f^{a+\frac {b}{x^3}} x \, dx=\frac {1}{3} f^a x^2 \Gamma \left (-\frac {2}{3},-\frac {b \log (f)}{x^3}\right ) \left (-\frac {b \log (f)}{x^3}\right )^{2/3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(104\) vs. \(2(28)=56\).
Time = 0.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.09
method | result | size |
meijerg | \(-\frac {f^{a} \left (-b \right )^{\frac {2}{3}} \ln \left (f \right )^{\frac {2}{3}} \left (\frac {\ln \left (f \right )^{\frac {1}{3}} b \pi \sqrt {3}}{x \left (-b \right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right ) \left (-\frac {b \ln \left (f \right )}{x^{3}}\right )^{\frac {1}{3}}}-\frac {3 x^{2} {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{3}}}}{2 \left (-b \right )^{\frac {2}{3}} \ln \left (f \right )^{\frac {2}{3}}}-\frac {3 \ln \left (f \right )^{\frac {1}{3}} b \Gamma \left (\frac {1}{3}, -\frac {b \ln \left (f \right )}{x^{3}}\right )}{2 x \left (-b \right )^{\frac {2}{3}} \left (-\frac {b \ln \left (f \right )}{x^{3}}\right )^{\frac {1}{3}}}\right )}{3}\) | \(105\) |
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none
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.21 \[ \int f^{a+\frac {b}{x^3}} x \, dx=\frac {1}{2} \, f^{\frac {a x^{3} + b}{x^{3}}} x^{2} - \frac {1}{2} \, \left (-b \log \left (f\right )\right )^{\frac {2}{3}} f^{a} \Gamma \left (\frac {1}{3}, -\frac {b \log \left (f\right )}{x^{3}}\right ) \]
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\[ \int f^{a+\frac {b}{x^3}} x \, dx=\int f^{a + \frac {b}{x^{3}}} x\, dx \]
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none
Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int f^{a+\frac {b}{x^3}} x \, dx=\frac {1}{3} \, f^{a} x^{2} \left (-\frac {b \log \left (f\right )}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (-\frac {2}{3}, -\frac {b \log \left (f\right )}{x^{3}}\right ) \]
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\[ \int f^{a+\frac {b}{x^3}} x \, dx=\int { f^{a + \frac {b}{x^{3}}} x \,d x } \]
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Time = 0.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.06 \[ \int f^{a+\frac {b}{x^3}} x \, dx=\frac {f^a\,f^{\frac {b}{x^3}}\,x^2}{2}-\frac {f^a\,x^2\,\Gamma \left (\frac {1}{3},-\frac {b\,\ln \left (f\right )}{x^3}\right )\,{\left (-\frac {b\,\ln \left (f\right )}{x^3}\right )}^{2/3}}{2}+\frac {\pi \,\sqrt {3}\,f^a\,x^2\,{\left (-\frac {b\,\ln \left (f\right )}{x^3}\right )}^{2/3}}{3\,\Gamma \left (\frac {2}{3}\right )} \]
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