Integrand size = 13, antiderivative size = 20 \[ \int f^{a+b x^3} x^2 \, dx=\frac {f^{a+b x^3}}{3 b \log (f)} \]
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Time = 0.01 (sec) , antiderivative size = 20, 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 = {2240} \[ \int f^{a+b x^3} x^2 \, dx=\frac {f^{a+b x^3}}{3 b \log (f)} \]
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Rule 2240
Rubi steps \begin{align*} \text {integral}& = \frac {f^{a+b x^3}}{3 b \log (f)} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int f^{a+b x^3} x^2 \, dx=\frac {f^{a+b x^3}}{3 b \log (f)} \]
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Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
gosper | \(\frac {f^{b \,x^{3}+a}}{3 b \ln \left (f \right )}\) | \(19\) |
derivativedivides | \(\frac {f^{b \,x^{3}+a}}{3 b \ln \left (f \right )}\) | \(19\) |
default | \(\frac {f^{b \,x^{3}+a}}{3 b \ln \left (f \right )}\) | \(19\) |
risch | \(\frac {f^{b \,x^{3}+a}}{3 b \ln \left (f \right )}\) | \(19\) |
parallelrisch | \(\frac {f^{b \,x^{3}+a}}{3 b \ln \left (f \right )}\) | \(19\) |
norman | \(\frac {{\mathrm e}^{\left (b \,x^{3}+a \right ) \ln \left (f \right )}}{3 b \ln \left (f \right )}\) | \(21\) |
meijerg | \(-\frac {f^{a} \left (1-{\mathrm e}^{b \,x^{3} \ln \left (f \right )}\right )}{3 b \ln \left (f \right )}\) | \(25\) |
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none
Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int f^{a+b x^3} x^2 \, dx=\frac {f^{b x^{3} + a}}{3 \, b \log \left (f\right )} \]
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Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int f^{a+b x^3} x^2 \, dx=\begin {cases} \frac {f^{a + b x^{3}}}{3 b \log {\left (f \right )}} & \text {for}\: b \log {\left (f \right )} \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases} \]
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none
Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int f^{a+b x^3} x^2 \, dx=\frac {f^{b x^{3} + a}}{3 \, b \log \left (f\right )} \]
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none
Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int f^{a+b x^3} x^2 \, dx=\frac {f^{b x^{3} + a}}{3 \, b \log \left (f\right )} \]
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Time = 0.11 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int f^{a+b x^3} x^2 \, dx=\frac {f^{b\,x^3+a}}{3\,b\,\ln \left (f\right )} \]
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