Optimal. Leaf size=144 \[ \frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{x^3-1}+x\right )}{3\ 2^{2/3}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}-x}\right )}{2^{2/3} \sqrt {3}}-\frac {5 \log \left (-\sqrt [3]{2} \sqrt [3]{x^3-1} x+2^{2/3} \left (x^3-1\right )^{2/3}+x^2\right )}{6\ 2^{2/3}}+\frac {\left (x^3-1\right )^{2/3} \left (16 x^6+4 x^3+5\right )}{10 x^8} \]
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Rubi [C] time = 0.61, antiderivative size = 141, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {6725, 271, 264, 277, 239, 430, 429} \begin {gather*} -\frac {15 x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {3 x^3}{2}\right )}{2 \left (1-x^3\right )^{2/3}}+\frac {5}{2} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {5 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (x^3-1\right )^{5/3}}{2 x^8}-\frac {9 \left (x^3-1\right )^{5/3}}{10 x^5}+\frac {5 \left (x^3-1\right )^{2/3}}{2 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 239
Rule 264
Rule 271
Rule 277
Rule 429
Rule 430
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (-4+x^3\right ) \left (-2+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^9 \left (-2+3 x^3\right )} \, dx &=\int \left (-\frac {4 \left (-1+x^3\right )^{2/3}}{x^9}-\frac {3 \left (-1+x^3\right )^{2/3}}{x^6}-\frac {5 \left (-1+x^3\right )^{2/3}}{x^3}+\frac {15 \left (-1+x^3\right )^{2/3}}{-2+3 x^3}\right ) \, dx\\ &=-\left (3 \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx\right )-4 \int \frac {\left (-1+x^3\right )^{2/3}}{x^9} \, dx-5 \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx+15 \int \frac {\left (-1+x^3\right )^{2/3}}{-2+3 x^3} \, dx\\ &=\frac {5 \left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{2 x^8}-\frac {3 \left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {3}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx-5 \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {\left (15 \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{-2+3 x^3} \, dx}{\left (1-x^3\right )^{2/3}}\\ &=\frac {5 \left (-1+x^3\right )^{2/3}}{2 x^2}-\frac {\left (-1+x^3\right )^{5/3}}{2 x^8}-\frac {9 \left (-1+x^3\right )^{5/3}}{10 x^5}-\frac {15 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {3 x^3}{2}\right )}{2 \left (1-x^3\right )^{2/3}}-\frac {5 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {5}{2} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.19, size = 140, normalized size = 0.97 \begin {gather*} \frac {5 \left (2 \log \left (\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+2\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}-1}{\sqrt {3}}\right )-\log \left (-\frac {2^{2/3} x}{\sqrt [3]{1-x^3}}+\frac {\sqrt [3]{2} x^2}{\left (1-x^3\right )^{2/3}}+2\right )\right )}{6\ 2^{2/3}}+\frac {\left (x^3-1\right )^{2/3} \left (16 x^6+4 x^3+5\right )}{10 x^8} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.36, size = 144, normalized size = 1.00 \begin {gather*} \frac {5 \log \left (\sqrt [3]{2} \sqrt [3]{x^3-1}+x\right )}{3\ 2^{2/3}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}-x}\right )}{2^{2/3} \sqrt {3}}-\frac {5 \log \left (-\sqrt [3]{2} \sqrt [3]{x^3-1} x+2^{2/3} \left (x^3-1\right )^{2/3}+x^2\right )}{6\ 2^{2/3}}+\frac {\left (x^3-1\right )^{2/3} \left (16 x^6+4 x^3+5\right )}{10 x^8} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.80, size = 267, normalized size = 1.85 \begin {gather*} \frac {100 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{8} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (3 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \sqrt {3} {\left (27 \, x^{9} - 72 \, x^{6} + 36 \, x^{3} + 8\right )} - 12 \, \sqrt {3} {\left (9 \, x^{8} - 6 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (27 \, x^{9} - 36 \, x^{3} + 8\right )}}\right ) + 50 \cdot 4^{\frac {2}{3}} x^{8} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (3 \, x^{3} - 2\right )} + 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{3 \, x^{3} - 2}\right ) - 25 \cdot 4^{\frac {2}{3}} x^{8} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x - 4^{\frac {1}{3}} {\left (9 \, x^{6} - 6 \, x^{3} - 4\right )} + 6 \, {\left (3 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{9 \, x^{6} - 12 \, x^{3} + 4}\right ) + 36 \, {\left (16 \, x^{6} + 4 \, x^{3} + 5\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{360 \, x^{8}} \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 \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )} {\left (x^{3} - 4\right )}}{{\left (3 \, x^{3} - 2\right )} x^{9}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.50, size = 564, normalized size = 3.92 \begin {gather*} \frac {16 x^{9}-12 x^{6}+x^{3}-5}{10 x^{8} \left (x^{3}-1\right )^{\frac {1}{3}}}+\frac {5 \RootOf \left (\textit {\_Z}^{3}-2\right ) \ln \left (-\frac {18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}-18 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x -18 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}+\RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )-2 \RootOf \left (\textit {\_Z}^{3}-2\right )}{3 x^{3}-2}\right )}{6}+5 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \ln \left (-\frac {18 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} x^{3}+18 \left (x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x +3 \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right ) x^{3}-2 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}+6 x \left (x^{3}-1\right )^{\frac {2}{3}}-12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+36 \textit {\_Z}^{2}\right )+2 \RootOf \left (\textit {\_Z}^{3}-2\right )}{3 x^{3}-2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )} {\left (x^{3} - 4\right )}}{{\left (3 \, x^{3} - 2\right )} x^{9}}\,{d x} \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 \frac {{\left (x^3-1\right )}^{2/3}\,\left (x^3-2\right )\,\left (x^3-4\right )}{x^9\,\left (3\,x^3-2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{3} - 4\right ) \left (x^{3} - 2\right )}{x^{9} \left (3 x^{3} - 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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