Integrand size = 14, antiderivative size = 23 \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=-1-3 \left (-3+\left (\frac {1}{3}-x\right )^2\right ) x+(-6-x) x \]
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Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {12} \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=-3 x^3+x^2+\frac {8 x}{3} \]
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Rule 12
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \left (8+6 x-27 x^2\right ) \, dx \\ & = \frac {8 x}{3}+x^2-3 x^3 \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=\frac {8 x}{3}+x^2-3 x^3 \]
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Time = 0.14 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.57
method | result | size |
default | \(-3 x^{3}+x^{2}+\frac {8}{3} x\) | \(13\) |
norman | \(-3 x^{3}+x^{2}+\frac {8}{3} x\) | \(13\) |
risch | \(-3 x^{3}+x^{2}+\frac {8}{3} x\) | \(13\) |
parallelrisch | \(-3 x^{3}+x^{2}+\frac {8}{3} x\) | \(13\) |
parts | \(-3 x^{3}+x^{2}+\frac {8}{3} x\) | \(13\) |
gosper | \(-\frac {x \left (9 x^{2}-3 x -8\right )}{3}\) | \(14\) |
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Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=-3 \, x^{3} + x^{2} + \frac {8}{3} \, x \]
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Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=- 3 x^{3} + x^{2} + \frac {8 x}{3} \]
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none
Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=-3 \, x^{3} + x^{2} + \frac {8}{3} \, x \]
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none
Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=-3 \, x^{3} + x^{2} + \frac {8}{3} \, x \]
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Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.57 \[ \int \frac {1}{3} \left (8+6 x-27 x^2\right ) \, dx=\frac {x\,\left (-9\,x^2+3\,x+8\right )}{3} \]
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