Integrand size = 13, antiderivative size = 34 \[ \int \frac {f^{a+b x^3}}{x^2} \, dx=-\frac {f^a \Gamma \left (-\frac {1}{3},-b x^3 \log (f)\right ) \sqrt [3]{-b x^3 \log (f)}}{3 x} \]
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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} \[ \int \frac {f^{a+b x^3}}{x^2} \, dx=-\frac {f^a \sqrt [3]{-b x^3 \log (f)} \Gamma \left (-\frac {1}{3},-b x^3 \log (f)\right )}{3 x} \]
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Rule 2250
Rubi steps \begin{align*} \text {integral}& = -\frac {f^a \Gamma \left (-\frac {1}{3},-b x^3 \log (f)\right ) \sqrt [3]{-b x^3 \log (f)}}{3 x} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {f^{a+b x^3}}{x^2} \, dx=-\frac {f^a \Gamma \left (-\frac {1}{3},-b x^3 \log (f)\right ) \sqrt [3]{-b x^3 \log (f)}}{3 x} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(99\) vs. \(2(28)=56\).
Time = 0.02 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.94
method | result | size |
meijerg | \(\frac {f^{a} \left (-b \right )^{\frac {1}{3}} \ln \left (f \right )^{\frac {1}{3}} \left (\frac {3 x^{2} \ln \left (f \right )^{\frac {2}{3}} b \Gamma \left (\frac {2}{3}\right )}{\left (-b \right )^{\frac {1}{3}} \left (-b \,x^{3} \ln \left (f \right )\right )^{\frac {2}{3}}}-\frac {3 \,{\mathrm e}^{b \,x^{3} \ln \left (f \right )}}{x \left (-b \right )^{\frac {1}{3}} \ln \left (f \right )^{\frac {1}{3}}}-\frac {3 x^{2} \ln \left (f \right )^{\frac {2}{3}} b \Gamma \left (\frac {2}{3}, -b \,x^{3} \ln \left (f \right )\right )}{\left (-b \right )^{\frac {1}{3}} \left (-b \,x^{3} \ln \left (f \right )\right )^{\frac {2}{3}}}\right )}{3}\) | \(100\) |
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none
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.12 \[ \int \frac {f^{a+b x^3}}{x^2} \, dx=\frac {\left (-b \log \left (f\right )\right )^{\frac {1}{3}} f^{a} x \Gamma \left (\frac {2}{3}, -b x^{3} \log \left (f\right )\right ) - f^{b x^{3} + a}}{x} \]
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\[ \int \frac {f^{a+b x^3}}{x^2} \, dx=\int \frac {f^{a + b x^{3}}}{x^{2}}\, dx \]
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none
Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int \frac {f^{a+b x^3}}{x^2} \, dx=-\frac {\left (-b x^{3} \log \left (f\right )\right )^{\frac {1}{3}} f^{a} \Gamma \left (-\frac {1}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, x} \]
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\[ \int \frac {f^{a+b x^3}}{x^2} \, dx=\int { \frac {f^{b x^{3} + a}}{x^{2}} \,d x } \]
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Time = 0.18 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.85 \[ \int \frac {f^{a+b x^3}}{x^2} \, dx=\frac {f^a\,\Gamma \left (\frac {2}{3},-b\,x^3\,\ln \left (f\right )\right )\,{\left (-b\,x^3\,\ln \left (f\right )\right )}^{1/3}}{x}-\frac {f^a\,\Gamma \left (\frac {2}{3}\right )\,{\left (-b\,x^3\,\ln \left (f\right )\right )}^{1/3}}{x}-\frac {f^a\,f^{b\,x^3}}{x} \]
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