Optimal. Leaf size=114 \[ \frac {10}{729} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {10 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}+x}\right )}{243 \sqrt {3}}-\frac {5}{729} \log \left (\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right )+\frac {1}{972} \sqrt [3]{x^3-1} \left (81 x^{11}-9 x^8-12 x^5-20 x^2\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 145, normalized size of antiderivative = 1.27, number of steps used = 11, number of rules used = 9, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {279, 321, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} \frac {10}{729} \log \left (1-\frac {x}{\sqrt [3]{x^3-1}}\right )+\frac {10 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{243 \sqrt {3}}+\frac {1}{12} \sqrt [3]{x^3-1} x^{11}-\frac {1}{108} \sqrt [3]{x^3-1} x^8-\frac {1}{81} \sqrt [3]{x^3-1} x^5-\frac {5}{243} \sqrt [3]{x^3-1} x^2-\frac {5}{729} \log \left (\frac {x}{\sqrt [3]{x^3-1}}+\frac {x^2}{\left (x^3-1\right )^{2/3}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 279
Rule 292
Rule 321
Rule 331
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int x^{10} \sqrt [3]{-1+x^3} \, dx &=\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {1}{12} \int \frac {x^{10}}{\left (-1+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {2}{27} \int \frac {x^7}{\left (-1+x^3\right )^{2/3}} \, dx\\ &=-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {5}{81} \int \frac {x^4}{\left (-1+x^3\right )^{2/3}} \, dx\\ &=-\frac {5}{243} x^2 \sqrt [3]{-1+x^3}-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {10}{243} \int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx\\ &=-\frac {5}{243} x^2 \sqrt [3]{-1+x^3}-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {10}{243} \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {5}{243} x^2 \sqrt [3]{-1+x^3}-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {10}{729} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {10}{729} \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {5}{243} x^2 \sqrt [3]{-1+x^3}-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}+\frac {10}{729} \log \left (1-\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {5}{729} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {5}{243} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {5}{243} x^2 \sqrt [3]{-1+x^3}-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}+\frac {10}{729} \log \left (1-\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {5}{729} \log \left (1+\frac {x^2}{\left (-1+x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {10}{243} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {5}{243} x^2 \sqrt [3]{-1+x^3}-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}+\frac {10 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{243 \sqrt {3}}+\frac {10}{729} \log \left (1-\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {5}{729} \log \left (1+\frac {x^2}{\left (-1+x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ \end {align*}
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Mathematica [C] time = 0.03, size = 72, normalized size = 0.63 \begin {gather*} \frac {x^2 \sqrt [3]{x^3-1} \left (20 \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )+\sqrt [3]{1-x^3} \left (27 x^9-3 x^6-4 x^3-20\right )\right )}{324 \sqrt [3]{1-x^3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.32, size = 114, normalized size = 1.00 \begin {gather*} \frac {1}{972} \sqrt [3]{-1+x^3} \left (-20 x^2-12 x^5-9 x^8+81 x^{11}\right )+\frac {10 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{243 \sqrt {3}}+\frac {10}{729} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {5}{729} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 106, normalized size = 0.93 \begin {gather*} -\frac {10}{729} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{972} \, {\left (81 \, x^{11} - 9 \, x^{8} - 12 \, x^{5} - 20 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \frac {10}{729} \, \log \left (-\frac {x - {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {5}{729} \, \log \left (\frac {x^{2} + {\left (x^{3} - 1\right )}^{\frac {1}{3}} x + {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{10}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.59, size = 33, normalized size = 0.29
method | result | size |
meijerg | \(\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} x^{11} \hypergeom \left (\left [-\frac {1}{3}, \frac {11}{3}\right ], \left [\frac {14}{3}\right ], x^{3}\right )}{11 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}\) | \(33\) |
risch | \(\frac {x^{2} \left (81 x^{9}-9 x^{6}-12 x^{3}-20\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{972}-\frac {5 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{2} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{243 \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(63\) |
trager | \(\frac {x^{2} \left (81 x^{9}-9 x^{6}-12 x^{3}-20\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{972}+\frac {10 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{729}-\frac {10 \ln \left (\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+4 x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{729}-\frac {10 \ln \left (\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+4 x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right )}{729}\) | \(325\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 170, normalized size = 1.49 \begin {gather*} -\frac {10}{729} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {\frac {40 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {93 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}}}{x^{4}} - \frac {72 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}}}{x^{7}} + \frac {20 \, {\left (x^{3} - 1\right )}^{\frac {10}{3}}}{x^{10}}}{972 \, {\left (\frac {4 \, {\left (x^{3} - 1\right )}}{x^{3}} - \frac {6 \, {\left (x^{3} - 1\right )}^{2}}{x^{6}} + \frac {4 \, {\left (x^{3} - 1\right )}^{3}}{x^{9}} - \frac {{\left (x^{3} - 1\right )}^{4}}{x^{12}} - 1\right )}} - \frac {5}{729} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {10}{729} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^{10}\,{\left (x^3-1\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.62, size = 32, normalized size = 0.28 \begin {gather*} \frac {x^{11} e^{\frac {i \pi }{3}} \Gamma \left (\frac {11}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {11}{3} \\ \frac {14}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {14}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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