Optimal. Leaf size=111 \[ \frac {1}{3} \log \left (\sqrt [3]{2 x^3+1}-x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2 x^3+1}+x}\right )}{\sqrt {3}}+\frac {\left (2 x^3+1\right )^{2/3} \left (-3 x^3-4\right )}{10 x^5}-\frac {1}{6} \log \left (\sqrt [3]{2 x^3+1} x+\left (2 x^3+1\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 123, normalized size of antiderivative = 1.11, 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, 618, 204, 628} \begin {gather*} \frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{2 x^3+1}}\right )-\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{2 x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2 \left (2 x^3+1\right )^{2/3}}{5 x^5}-\frac {3 \left (2 x^3+1\right )^{2/3}}{10 x^2}-\frac {1}{6} \log \left (\frac {x}{\sqrt [3]{2 x^3+1}}+\frac {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 618
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 {3-2 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 {3 \left (1+2 x^3\right )^{2/3}}{10 x^2}-\frac {1}{10} \int \frac {10}{\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 {3 \left (1+2 x^3\right )^{2/3}}{10 x^2}-\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 {3 \left (1+2 x^3\right )^{2/3}}{10 x^2}-\operatorname {Subst}\left (\int \frac {1}{1-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 {3 \left (1+2 x^3\right )^{2/3}}{10 x^2}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x}{\sqrt [3]{1+2 x^3}}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {2+x}{1+x+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 {3 \left (1+2 x^3\right )^{2/3}}{10 x^2}+\frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{1+2 x^3}}\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+2 x^3}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x+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 {3 \left (1+2 x^3\right )^{2/3}}{10 x^2}+\frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{1+2 x^3}}\right )-\frac {1}{6} \log \left (1+\frac {x^2}{\left (1+2 x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{1+2 x^3}}\right )+\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{1+2 x^3}}\right )\\ &=-\frac {2 \left (1+2 x^3\right )^{2/3}}{5 x^5}-\frac {3 \left (1+2 x^3\right )^{2/3}}{10 x^2}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+2 x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{1+2 x^3}}\right )-\frac {1}{6} \log \left (1+\frac {x^2}{\left (1+2 x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{1+2 x^3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.22, size = 102, normalized size = 0.92 \begin {gather*} \frac {1}{30} \left (10 \log \left (1-\frac {x}{\sqrt [3]{x^3+2}}\right )-10 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )-\frac {3 \left (2 x^3+1\right )^{2/3} \left (3 x^3+4\right )}{x^5}-5 \log \left (\frac {x}{\sqrt [3]{x^3+2}}+\frac {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 = 0.24, size = 111, normalized size = 1.00 \begin {gather*} \frac {\left (-4-3 x^3\right ) \left (1+2 x^3\right )^{2/3}}{10 x^5}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+2 x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (-x+\sqrt [3]{1+2 x^3}\right )-\frac {1}{6} \log \left (x^2+x \sqrt [3]{1+2 x^3}+\left (1+2 x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.04, size = 133, normalized size = 1.20 \begin {gather*} -\frac {10 \, \sqrt {3} x^{5} \arctan \left (-\frac {4 \, \sqrt {3} {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 2 \, \sqrt {3} {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (2 \, x^{3} + 1\right )}}{10 \, x^{3} + 1}\right ) - 5 \, x^{5} \log \left (\frac {x^{3} + 3 \, {\left (2 \, x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (2 \, x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{x^{3} + 1}\right ) + 3 \, {\left (3 \, 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 = 4.51, size = 364, normalized size = 3.28
method | result | size |
risch | \(-\frac {6 x^{6}+11 x^{3}+4}{10 x^{5} \left (2 x^{3}+1\right )^{\frac {1}{3}}}-\frac {\ln \left (-\frac {9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-3 \left (2 x^{3}+1\right )^{\frac {2}{3}} x -3 \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}-4 x^{3}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{3}-\ln \left (-\frac {9 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-3 \left (2 x^{3}+1\right )^{\frac {2}{3}} x -3 \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}-4 x^{3}-3 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-2}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+\RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {18 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}+15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}} x +15 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+27 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+4 \left (2 x^{3}+1\right )^{\frac {2}{3}} x +4 \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+9 x^{3}+6 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+3}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )\) | \(364\) |
trager | \(-\frac {\left (3 x^{3}+4\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}}}{10 x^{5}}+\frac {\ln \left (\frac {12321 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-48096 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}} x +99369 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}-48225 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-33123 \left (2 x^{3}+1\right )^{\frac {2}{3}} x +17091 \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+33786 x^{3}-12321 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-5166 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+15016}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{3}-\frac {\ln \left (-\frac {46107 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-48096 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}} x -51273 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+164952 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+17091 \left (2 x^{3}+1\right )^{\frac {2}{3}} x -33123 \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+37540 x^{3}-46107 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+34845 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+11262}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )}{3}-\ln \left (-\frac {46107 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-48096 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {2}{3}} x -51273 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+164952 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}+17091 \left (2 x^{3}+1\right )^{\frac {2}{3}} x -33123 \left (2 x^{3}+1\right )^{\frac {1}{3}} x^{2}+37540 x^{3}-46107 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+34845 \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )+11262}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right ) \RootOf \left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )\) | \(522\) |
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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