Integrand size = 13, antiderivative size = 34 \[ \int f^{a+b x^3} x^3 \, dx=-\frac {f^a x^4 \Gamma \left (\frac {4}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{4/3}} \]
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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} \[ \int f^{a+b x^3} x^3 \, dx=-\frac {x^4 f^a \Gamma \left (\frac {4}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{4/3}} \]
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Rule 2250
Rubi steps \begin{align*} \text {integral}& = -\frac {f^a x^4 \Gamma \left (\frac {4}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{4/3}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int f^{a+b x^3} x^3 \, dx=-\frac {f^a x^4 \Gamma \left (\frac {4}{3},-b x^3 \log (f)\right )}{3 \left (-b x^3 \log (f)\right )^{4/3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(28)=56\).
Time = 0.04 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.21
method | result | size |
meijerg | \(-\frac {f^{a} \left (-\frac {2 x \left (-b \right )^{\frac {4}{3}} \ln \left (f \right )^{\frac {1}{3}} \pi \sqrt {3}}{9 b \Gamma \left (\frac {2}{3}\right ) \left (-b \,x^{3} \ln \left (f \right )\right )^{\frac {1}{3}}}+\frac {x \left (-b \right )^{\frac {4}{3}} \ln \left (f \right )^{\frac {1}{3}} {\mathrm e}^{b \,x^{3} \ln \left (f \right )}}{b}+\frac {x \left (-b \right )^{\frac {4}{3}} \ln \left (f \right )^{\frac {1}{3}} \Gamma \left (\frac {1}{3}, -b \,x^{3} \ln \left (f \right )\right )}{3 b \left (-b \,x^{3} \ln \left (f \right )\right )^{\frac {1}{3}}}\right )}{3 b \ln \left (f \right )^{\frac {4}{3}} \left (-b \right )^{\frac {1}{3}}}\) | \(109\) |
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none
Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.38 \[ \int f^{a+b x^3} x^3 \, dx=\frac {3 \, b f^{b x^{3} + a} x \log \left (f\right ) - \left (-b \log \left (f\right )\right )^{\frac {2}{3}} f^{a} \Gamma \left (\frac {1}{3}, -b x^{3} \log \left (f\right )\right )}{9 \, b^{2} \log \left (f\right )^{2}} \]
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\[ \int f^{a+b x^3} x^3 \, dx=\int f^{a + b x^{3}} x^{3}\, dx \]
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none
Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int f^{a+b x^3} x^3 \, dx=-\frac {f^{a} x^{4} \Gamma \left (\frac {4}{3}, -b x^{3} \log \left (f\right )\right )}{3 \, \left (-b x^{3} \log \left (f\right )\right )^{\frac {4}{3}}} \]
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\[ \int f^{a+b x^3} x^3 \, dx=\int { f^{b x^{3} + a} x^{3} \,d x } \]
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Time = 0.14 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.21 \[ \int f^{a+b x^3} x^3 \, dx=\frac {f^a\,f^{b\,x^3}\,x}{3\,b\,\ln \left (f\right )}-\frac {f^a\,x^4\,\Gamma \left (\frac {1}{3},-b\,x^3\,\ln \left (f\right )\right )}{9\,{\left (-b\,x^3\,\ln \left (f\right )\right )}^{4/3}}+\frac {2\,\pi \,\sqrt {3}\,f^a\,x^4}{27\,\Gamma \left (\frac {2}{3}\right )\,{\left (-b\,x^3\,\ln \left (f\right )\right )}^{4/3}} \]
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