Optimal. Leaf size=30 \[ \frac {1}{140} \sqrt [3]{x^3-1} \left (14 x^9-2 x^6-3 x^3-9\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.33, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^8 \sqrt [3]{-1+x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \sqrt [3]{-1+x} x^2 \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {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*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.83 \begin {gather*} \frac {1}{140} \left (x^3-1\right )^{4/3} \left (14 x^6+12 x^3+9\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 25, normalized size = 0.83 \begin {gather*} \frac {1}{140} \left (-1+x^3\right )^{4/3} \left (9+12 x^3+14 x^6\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 26, normalized size = 0.87 \begin {gather*} \frac {1}{140} \, {\left (14 \, x^{9} - 2 \, x^{6} - 3 \, x^{3} - 9\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 28, normalized size = 0.93 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 26, normalized size = 0.87
method | result | size |
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 | \(\frac {\left (-1+x \right ) \left (x^{2}+x +1\right ) \left (14 x^{6}+12 x^{3}+9\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{140}\) | \(31\) |
meijerg | \(\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {1}{3}, 3\right ], \relax [4], x^{3}\right ) x^{9}}{9 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 28, normalized size = 0.93 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 26, normalized size = 0.87 \begin {gather*} -{\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.82, size = 53, normalized size = 1.77 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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