Integrand size = 13, antiderivative size = 25 \[ \int x^5 \left (1+x^3\right )^{2/3} \, dx=\frac {1}{40} \left (1+x^3\right )^{2/3} \left (-3+2 x^3+5 x^6\right ) \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int x^5 \left (1+x^3\right )^{2/3} \, dx=\frac {1}{8} \left (x^3+1\right )^{8/3}-\frac {1}{5} \left (x^3+1\right )^{5/3} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int x (1+x)^{2/3} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (-(1+x)^{2/3}+(1+x)^{5/3}\right ) \, dx,x,x^3\right ) \\ & = -\frac {1}{5} \left (1+x^3\right )^{5/3}+\frac {1}{8} \left (1+x^3\right )^{8/3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int x^5 \left (1+x^3\right )^{2/3} \, dx=\frac {1}{40} \left (1+x^3\right )^{5/3} \left (-3+5 x^3\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 2.
Time = 0.84 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68
method | result | size |
meijerg | \(\frac {x^{6} \operatorname {hypergeom}\left (\left [-\frac {2}{3}, 2\right ], \left [3\right ], -x^{3}\right )}{6}\) | \(17\) |
pseudoelliptic | \(\frac {\left (x^{3}+1\right )^{\frac {5}{3}} \left (5 x^{3}-3\right )}{40}\) | \(17\) |
trager | \(\left (\frac {1}{8} x^{6}+\frac {1}{20} x^{3}-\frac {3}{40}\right ) \left (x^{3}+1\right )^{\frac {2}{3}}\) | \(21\) |
risch | \(\frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (5 x^{6}+2 x^{3}-3\right )}{40}\) | \(22\) |
gosper | \(\frac {\left (1+x \right ) \left (x^{2}-x +1\right ) \left (5 x^{3}-3\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{40}\) | \(28\) |
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Time = 0.24 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int x^5 \left (1+x^3\right )^{2/3} \, dx=\frac {1}{40} \, {\left (5 \, x^{6} + 2 \, x^{3} - 3\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} \]
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Time = 0.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int x^5 \left (1+x^3\right )^{2/3} \, dx=\frac {x^{6} \left (x^{3} + 1\right )^{\frac {2}{3}}}{8} + \frac {x^{3} \left (x^{3} + 1\right )^{\frac {2}{3}}}{20} - \frac {3 \left (x^{3} + 1\right )^{\frac {2}{3}}}{40} \]
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Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^5 \left (1+x^3\right )^{2/3} \, dx=\frac {1}{8} \, {\left (x^{3} + 1\right )}^{\frac {8}{3}} - \frac {1}{5} \, {\left (x^{3} + 1\right )}^{\frac {5}{3}} \]
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Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^5 \left (1+x^3\right )^{2/3} \, dx=\frac {1}{8} \, {\left (x^{3} + 1\right )}^{\frac {8}{3}} - \frac {1}{5} \, {\left (x^{3} + 1\right )}^{\frac {5}{3}} \]
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Time = 5.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int x^5 \left (1+x^3\right )^{2/3} \, dx={\left (x^3+1\right )}^{2/3}\,\left (\frac {x^6}{8}+\frac {x^3}{20}-\frac {3}{40}\right ) \]
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