Integrand size = 13, antiderivative size = 25 \[ \int x^5 \sqrt [3]{-1+x^3} \, dx=\frac {1}{28} \sqrt [3]{-1+x^3} \left (-3-x^3+4 x^6\right ) \]
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Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int x^5 \sqrt [3]{-1+x^3} \, dx=\frac {1}{7} \left (x^3-1\right )^{7/3}+\frac {1}{4} \left (x^3-1\right )^{4/3} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \sqrt [3]{-1+x} 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 = 0.80 \[ \int x^5 \sqrt [3]{-1+x^3} \, dx=\frac {1}{28} \left (-1+x^3\right )^{4/3} \left (3+4 x^3\right ) \]
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Time = 0.87 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68
method | result | size |
pseudoelliptic | \(\frac {\left (x^{3}-1\right )^{\frac {4}{3}} \left (4 x^{3}+3\right )}{28}\) | \(17\) |
trager | \(\left (\frac {1}{7} x^{6}-\frac {1}{28} x^{3}-\frac {3}{28}\right ) \left (x^{3}-1\right )^{\frac {1}{3}}\) | \(21\) |
risch | \(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (4 x^{6}-x^{3}-3\right )}{28}\) | \(22\) |
gosper | \(\frac {\left (x -1\right ) \left (x^{2}+x +1\right ) \left (4 x^{3}+3\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{28}\) | \(26\) |
meijerg | \(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} x^{6} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, 2\right ], \left [3\right ], x^{3}\right )}{6 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\) | \(33\) |
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Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int x^5 \sqrt [3]{-1+x^3} \, dx=\frac {1}{28} \, {\left (4 \, x^{6} - x^{3} - 3\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} \]
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Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48 \[ \int x^5 \sqrt [3]{-1+x^3} \, dx=\frac {x^{6} \sqrt [3]{x^{3} - 1}}{7} - \frac {x^{3} \sqrt [3]{x^{3} - 1}}{28} - \frac {3 \sqrt [3]{x^{3} - 1}}{28} \]
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Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^5 \sqrt [3]{-1+x^3} \, dx=\frac {1}{7} \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + \frac {1}{4} \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} \]
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Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^5 \sqrt [3]{-1+x^3} \, dx=\frac {1}{7} \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + \frac {1}{4} \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} \]
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Time = 5.12 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int x^5 \sqrt [3]{-1+x^3} \, dx=-{\left (x^3-1\right )}^{1/3}\,\left (-\frac {x^6}{7}+\frac {x^3}{28}+\frac {3}{28}\right ) \]
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