Integrand size = 13, antiderivative size = 34 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^3} \, dx=\frac {f^a \Gamma \left (\frac {2}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^2 \left (-\frac {b \log (f)}{x^3}\right )^{2/3}} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2250} \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^3} \, dx=\frac {f^a \Gamma \left (\frac {2}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^2 \left (-\frac {b \log (f)}{x^3}\right )^{2/3}} \]
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Rule 2250
Rubi steps \begin{align*} \text {integral}& = \frac {f^a \Gamma \left (\frac {2}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^2 \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 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^3} \, dx=\frac {f^a \Gamma \left (\frac {2}{3},-\frac {b \log (f)}{x^3}\right )}{3 x^2 \left (-\frac {b \log (f)}{x^3}\right )^{2/3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(77\) vs. \(2(28)=56\).
Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.29
method | result | size |
meijerg | \(\frac {f^{a} \left (-b \right )^{\frac {1}{3}} \left (\frac {\left (-b \right )^{\frac {2}{3}} \ln \left (f \right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right )}{x^{2} \left (-\frac {b \ln \left (f \right )}{x^{3}}\right )^{\frac {2}{3}}}-\frac {\left (-b \right )^{\frac {2}{3}} \ln \left (f \right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}, -\frac {b \ln \left (f \right )}{x^{3}}\right )}{x^{2} \left (-\frac {b \ln \left (f \right )}{x^{3}}\right )^{\frac {2}{3}}}\right )}{3 b \ln \left (f \right )^{\frac {2}{3}}}\) | \(78\) |
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none
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^3} \, dx=-\frac {\left (-b \log \left (f\right )\right )^{\frac {1}{3}} f^{a} \Gamma \left (\frac {2}{3}, -\frac {b \log \left (f\right )}{x^{3}}\right )}{3 \, b \log \left (f\right )} \]
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\[ \int \frac {f^{a+\frac {b}{x^3}}}{x^3} \, dx=\int \frac {f^{a + \frac {b}{x^{3}}}}{x^{3}}\, dx \]
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Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^3} \, dx=\frac {f^{a} \Gamma \left (\frac {2}{3}, -\frac {b \log \left (f\right )}{x^{3}}\right )}{3 \, x^{2} \left (-\frac {b \log \left (f\right )}{x^{3}}\right )^{\frac {2}{3}}} \]
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\[ \int \frac {f^{a+\frac {b}{x^3}}}{x^3} \, dx=\int { \frac {f^{a + \frac {b}{x^{3}}}}{x^{3}} \,d x } \]
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Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {f^{a+\frac {b}{x^3}}}{x^3} \, dx=-\frac {f^a\,\left (\Gamma \left (\frac {2}{3}\right )-\Gamma \left (\frac {2}{3},-\frac {b\,\ln \left (f\right )}{x^3}\right )\right )}{3\,x^2\,{\left (-\frac {b\,\ln \left (f\right )}{x^3}\right )}^{2/3}} \]
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