Integrand size = 13, antiderivative size = 13 \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2}{9} \left (1+x^3\right )^{3/2} \]
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Time = 0.00 (sec) , antiderivative size = 13, 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 = {267} \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2}{9} \left (x^3+1\right )^{3/2} \]
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Rule 267
Rubi steps \begin{align*} \text {integral}& = \frac {2}{9} \left (1+x^3\right )^{3/2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2}{9} \left (1+x^3\right )^{3/2} \]
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Time = 0.75 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\frac {2 \left (x^{3}+1\right )^{\frac {3}{2}}}{9}\) | \(10\) |
default | \(\frac {2 \left (x^{3}+1\right )^{\frac {3}{2}}}{9}\) | \(10\) |
risch | \(\frac {2 \left (x^{3}+1\right )^{\frac {3}{2}}}{9}\) | \(10\) |
pseudoelliptic | \(\frac {2 \left (x^{3}+1\right )^{\frac {3}{2}}}{9}\) | \(10\) |
trager | \(\left (\frac {2}{9}+\frac {2 x^{3}}{9}\right ) \sqrt {x^{3}+1}\) | \(16\) |
gosper | \(\frac {2 \left (1+x \right ) \left (x^{2}-x +1\right ) \sqrt {x^{3}+1}}{9}\) | \(21\) |
meijerg | \(-\frac {\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2 x^{3}+2\right ) \sqrt {x^{3}+1}}{3}}{6 \sqrt {\pi }}\) | \(31\) |
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Time = 0.25 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (10) = 20\).
Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.00 \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2 x^{3} \sqrt {x^{3} + 1}}{9} + \frac {2 \sqrt {x^{3} + 1}}{9} \]
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Time = 0.20 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} \]
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Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} \]
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Time = 0.02 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^2 \sqrt {1+x^3} \, dx=\frac {2\,{\left (x^3+1\right )}^{3/2}}{9} \]
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