Optimal. Leaf size=143 \[ 3^{2/3} \log \left (3^{2/3} \sqrt [3]{2 x^3+1}-3 x\right )-3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{2 x^3+1}+\sqrt [3]{3} x}\right )+\frac {\left (2 x^3+1\right )^{2/3} \left (23 x^3+4\right )}{10 x^5}-\frac {1}{2} 3^{2/3} \log \left (3^{2/3} \sqrt [3]{2 x^3+1} x+\sqrt [3]{3} \left (2 x^3+1\right )^{2/3}+3 x^2\right ) \]
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Rubi [A] time = 0.17, antiderivative size = 148, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {580, 583, 12, 377, 200, 31, 634, 617, 204, 628} \begin {gather*} 3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{2 x^3+1}}\right )-3 \sqrt [6]{3} \tan ^{-1}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{2 x^3+1}}+\frac {1}{\sqrt {3}}\right )+\frac {2 \left (2 x^3+1\right )^{2/3}}{5 x^5}+\frac {23 \left (2 x^3+1\right )^{2/3}}{10 x^2}-\frac {1}{2} 3^{2/3} \log \left (\frac {\sqrt [3]{3} x}{\sqrt [3]{2 x^3+1}}+\frac {3^{2/3} x^2}{\left (2 x^3+1\right )^{2/3}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 200
Rule 204
Rule 377
Rule 580
Rule 583
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\left (2+x^3\right ) \left (1+2 x^3\right )^{2/3}}{x^6 \left (-1+x^3\right )} \, dx &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}-\frac {1}{5} \int \frac {-23-22 x^3}{x^3 \left (-1+x^3\right ) \sqrt [3]{1+2 x^3}} \, dx\\ &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}+\frac {23 \left (1+2 x^3\right )^{2/3}}{10 x^2}-\frac {1}{10} \int -\frac {90}{\left (-1+x^3\right ) \sqrt [3]{1+2 x^3}} \, dx\\ &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}+\frac {23 \left (1+2 x^3\right )^{2/3}}{10 x^2}+9 \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{1+2 x^3}} \, dx\\ &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}+\frac {23 \left (1+2 x^3\right )^{2/3}}{10 x^2}+9 \operatorname {Subst}\left (\int \frac {1}{-1+3 x^3} \, dx,x,\frac {x}{\sqrt [3]{1+2 x^3}}\right )\\ &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}+\frac {23 \left (1+2 x^3\right )^{2/3}}{10 x^2}+3 \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt [3]{3} x} \, dx,x,\frac {x}{\sqrt [3]{1+2 x^3}}\right )+3 \operatorname {Subst}\left (\int \frac {-2-\sqrt [3]{3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+2 x^3}}\right )\\ &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}+\frac {23 \left (1+2 x^3\right )^{2/3}}{10 x^2}+3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+2 x^3}}\right )-\frac {9}{2} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+2 x^3}}\right )-\frac {1}{2} 3^{2/3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{3}+2\ 3^{2/3} x}{1+\sqrt [3]{3} x+3^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1+2 x^3}}\right )\\ &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}+\frac {23 \left (1+2 x^3\right )^{2/3}}{10 x^2}+3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+2 x^3}}\right )-\frac {1}{2} 3^{2/3} \log \left (1+\frac {3^{2/3} x^2}{\left (1+2 x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{1+2 x^3}}\right )+\left (3\ 3^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{1+2 x^3}}\right )\\ &=\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}+\frac {23 \left (1+2 x^3\right )^{2/3}}{10 x^2}-3 \sqrt [6]{3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{1+2 x^3}}}{\sqrt {3}}\right )+3^{2/3} \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{1+2 x^3}}\right )-\frac {1}{2} 3^{2/3} \log \left (1+\frac {3^{2/3} x^2}{\left (1+2 x^3\right )^{2/3}}+\frac {\sqrt [3]{3} x}{\sqrt [3]{1+2 x^3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.28, size = 128, normalized size = 0.90 \begin {gather*} -3 \sqrt [6]{3} \tan ^{-1}\left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{x^3+2}}+\frac {1}{\sqrt {3}}\right )+\frac {\left (2 x^3+1\right )^{2/3} \left (23 x^3+4\right )}{10 x^5}+\frac {1}{2} 3^{2/3} \left (2 \log \left (1-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}\right )-\log \left (\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+2}}+\frac {3^{2/3} x^2}{\left (x^3+2\right )^{2/3}}+1\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.31, size = 143, normalized size = 1.00 \begin {gather*} \frac {\left (1+2 x^3\right )^{2/3} \left (4+23 x^3\right )}{10 x^5}-3 \sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+2 x^3}}\right )+3^{2/3} \log \left (-3 x+3^{2/3} \sqrt [3]{1+2 x^3}\right )-\frac {1}{2} 3^{2/3} \log \left (3 x^2+3^{2/3} x \sqrt [3]{1+2 x^3}+\sqrt [3]{3} \left (1+2 x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 2.74, size = 281, normalized size = 1.97 \begin {gather*} -\frac {10 \cdot 9^{\frac {1}{3}} \sqrt {3} x^{5} \arctan \left (\frac {2 \cdot 9^{\frac {2}{3}} \sqrt {3} {\left (8 \, x^{7} - 7 \, x^{4} - x\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} - 6 \cdot 9^{\frac {1}{3}} \sqrt {3} {\left (55 \, x^{8} + 25 \, x^{5} + x^{2}\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (377 \, x^{9} + 300 \, x^{6} + 51 \, x^{3} + 1\right )}}{3 \, {\left (487 \, x^{9} + 240 \, x^{6} + 3 \, x^{3} - 1\right )}}\right ) - 10 \cdot 9^{\frac {1}{3}} x^{5} \log \left (\frac {3 \cdot 9^{\frac {2}{3}} {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 9 \, {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} x - 9^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{3} - 1}\right ) + 5 \cdot 9^{\frac {1}{3}} x^{5} \log \left (\frac {9 \cdot 9^{\frac {1}{3}} {\left (8 \, x^{4} + x\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} + 9^{\frac {2}{3}} {\left (55 \, x^{6} + 25 \, x^{3} + 1\right )} + 27 \, {\left (7 \, x^{5} + 2 \, x^{2}\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right ) - 3 \, {\left (23 \, x^{3} + 4\right )} {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}}}{30 \, x^{5}} \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 (2 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} + 2\right )}}{{\left (x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 19.35, size = 935, normalized size = 6.54
method | result | size |
risch | \(\frac {46 x^{6}+31 x^{3}+4}{10 x^{5} \left (2 x^{3}+1\right )^{\frac {1}{3}}}+\RootOf \left (\textit {\_Z}^{3}-9\right ) \ln \left (\frac {6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right )^{3} x^{3}+135 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{3}+21 \left (2 x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) x +8 \left (2 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{2}+9 \left (2 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{2}-4 \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{3}-90 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) x^{3}-3 \left (2 x^{3}+1\right )^{\frac {2}{3}} x -2 \RootOf \left (\textit {\_Z}^{3}-9\right )-45 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )-\ln \left (-\frac {-6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right )^{3} x^{3}+81 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{3}+21 \left (2 x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) x -\left (2 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{2}-72 \left (2 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{2}-10 \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{3}+135 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) x^{3}+24 \left (2 x^{3}+1\right )^{\frac {2}{3}} x -2 \RootOf \left (\textit {\_Z}^{3}-9\right )+27 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right )-9 \ln \left (-\frac {-6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right )^{3} x^{3}+81 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{3}+21 \left (2 x^{3}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) x -\left (2 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-9\right )^{2} x^{2}-72 \left (2 x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{2}-10 \RootOf \left (\textit {\_Z}^{3}-9\right ) x^{3}+135 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right ) x^{3}+24 \left (2 x^{3}+1\right )^{\frac {2}{3}} x -2 \RootOf \left (\textit {\_Z}^{3}-9\right )+27 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right )}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-9\right )^{2}+9 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-9\right )+81 \textit {\_Z}^{2}\right )\) | \(935\) |
trager | \(\text {Expression too large to display}\) | \(1157\) |
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 (2 \, x^{3} + 1\right )}^{\frac {2}{3}} {\left (x^{3} + 2\right )}}{{\left (x^{3} - 1\right )} x^{6}}\,{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+2\right )\,{\left (2\,x^3+1\right )}^{2/3}}{x^6\,\left (x^3-1\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 (x^{3} + 2\right ) \left (2 x^{3} + 1\right )^{\frac {2}{3}}}{x^{6} \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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