Integrand size = 13, antiderivative size = 30 \[ \int x^8 \sqrt [3]{-1+x^3} \, dx=\frac {1}{140} \sqrt [3]{-1+x^3} \left (-9-3 x^3-2 x^6+14 x^9\right ) \]
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Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int x^8 \sqrt [3]{-1+x^3} \, dx=\frac {1}{10} \left (x^3-1\right )^{10/3}+\frac {2}{7} \left (x^3-1\right )^{7/3}+\frac {1}{4} \left (x^3-1\right )^{4/3} \]
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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^2 \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\sqrt [3]{-1+x}+2 (-1+x)^{4/3}+(-1+x)^{7/3}\right ) \, dx,x,x^3\right ) \\ & = \frac {1}{4} \left (-1+x^3\right )^{4/3}+\frac {2}{7} \left (-1+x^3\right )^{7/3}+\frac {1}{10} \left (-1+x^3\right )^{10/3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int x^8 \sqrt [3]{-1+x^3} \, dx=\frac {1}{140} \left (-1+x^3\right )^{4/3} \left (9+12 x^3+14 x^6\right ) \]
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Time = 0.85 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73
method | result | size |
pseudoelliptic | \(\frac {\left (x^{3}-1\right )^{\frac {4}{3}} \left (14 x^{6}+12 x^{3}+9\right )}{140}\) | \(22\) |
trager | \(\left (\frac {1}{10} x^{9}-\frac {1}{70} x^{6}-\frac {3}{140} x^{3}-\frac {9}{140}\right ) \left (x^{3}-1\right )^{\frac {1}{3}}\) | \(26\) |
risch | \(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (14 x^{9}-2 x^{6}-3 x^{3}-9\right )}{140}\) | \(27\) |
gosper | \(\frac {\left (x -1\right ) \left (x^{2}+x +1\right ) \left (14 x^{6}+12 x^{3}+9\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{140}\) | \(31\) |
meijerg | \(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} x^{9} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, 3\right ], \left [4\right ], x^{3}\right )}{9 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\) | \(33\) |
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Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int x^8 \sqrt [3]{-1+x^3} \, dx=\frac {1}{140} \, {\left (14 \, x^{9} - 2 \, x^{6} - 3 \, x^{3} - 9\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).
Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int x^8 \sqrt [3]{-1+x^3} \, dx=\frac {x^{9} \sqrt [3]{x^{3} - 1}}{10} - \frac {x^{6} \sqrt [3]{x^{3} - 1}}{70} - \frac {3 x^{3} \sqrt [3]{x^{3} - 1}}{140} - \frac {9 \sqrt [3]{x^{3} - 1}}{140} \]
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Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int x^8 \sqrt [3]{-1+x^3} \, dx=\frac {1}{10} \, {\left (x^{3} - 1\right )}^{\frac {10}{3}} + \frac {2}{7} \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + \frac {1}{4} \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} \]
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Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int x^8 \sqrt [3]{-1+x^3} \, dx=\frac {1}{10} \, {\left (x^{3} - 1\right )}^{\frac {10}{3}} + \frac {2}{7} \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + \frac {1}{4} \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} \]
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Time = 5.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int x^8 \sqrt [3]{-1+x^3} \, dx=-{\left (x^3-1\right )}^{1/3}\,\left (-\frac {x^9}{10}+\frac {x^6}{70}+\frac {3\,x^3}{140}+\frac {9}{140}\right ) \]
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