Integrand size = 13, antiderivative size = 23 \[ \int x^5 \sqrt [3]{1+x^3} \, dx=\frac {1}{28} \sqrt [3]{1+x^3} \left (-3+x^3+4 x^6\right ) \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int x^5 \sqrt [3]{1+x^3} \, dx=\frac {1}{7} \left (x^3+1\right )^{7/3}-\frac {1}{4} \left (x^3+1\right )^{4/3} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int x \sqrt [3]{1+x} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (-\sqrt [3]{1+x}+(1+x)^{4/3}\right ) \, dx,x,x^3\right ) \\ & = -\frac {1}{4} \left (1+x^3\right )^{4/3}+\frac {1}{7} \left (1+x^3\right )^{7/3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int x^5 \sqrt [3]{1+x^3} \, dx=\frac {1}{28} \left (1+x^3\right )^{4/3} \left (-3+4 x^3\right ) \]
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Result contains higher order function than in optimal. Order 5 vs. order 2.
Time = 0.80 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74
method | result | size |
meijerg | \(\frac {x^{6} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, 2\right ], \left [3\right ], -x^{3}\right )}{6}\) | \(17\) |
pseudoelliptic | \(\frac {\left (x^{3}+1\right )^{\frac {4}{3}} \left (4 x^{3}-3\right )}{28}\) | \(17\) |
risch | \(\frac {\left (x^{3}+1\right )^{\frac {1}{3}} \left (4 x^{6}+x^{3}-3\right )}{28}\) | \(20\) |
trager | \(\left (\frac {1}{7} x^{6}+\frac {1}{28} x^{3}-\frac {3}{28}\right ) \left (x^{3}+1\right )^{\frac {1}{3}}\) | \(21\) |
gosper | \(\frac {\left (1+x \right ) \left (x^{2}-x +1\right ) \left (4 x^{3}-3\right ) \left (x^{3}+1\right )^{\frac {1}{3}}}{28}\) | \(28\) |
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none
Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int x^5 \sqrt [3]{1+x^3} \, dx=\frac {1}{28} \, {\left (4 \, x^{6} + x^{3} - 3\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} \]
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Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \[ \int x^5 \sqrt [3]{1+x^3} \, dx=\frac {x^{6} \sqrt [3]{x^{3} + 1}}{7} + \frac {x^{3} \sqrt [3]{x^{3} + 1}}{28} - \frac {3 \sqrt [3]{x^{3} + 1}}{28} \]
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none
Time = 0.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int x^5 \sqrt [3]{1+x^3} \, dx=\frac {1}{7} \, {\left (x^{3} + 1\right )}^{\frac {7}{3}} - \frac {1}{4} \, {\left (x^{3} + 1\right )}^{\frac {4}{3}} \]
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none
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int x^5 \sqrt [3]{1+x^3} \, dx=\frac {1}{7} \, {\left (x^{3} + 1\right )}^{\frac {7}{3}} - \frac {1}{4} \, {\left (x^{3} + 1\right )}^{\frac {4}{3}} \]
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Time = 5.47 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int x^5 \sqrt [3]{1+x^3} \, dx={\left (x^3+1\right )}^{1/3}\,\left (\frac {x^6}{7}+\frac {x^3}{28}-\frac {3}{28}\right ) \]
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