Optimal. Leaf size=141 \[ -2^{2/3} \log \left (2^{2/3} \sqrt [3]{3 x^3+x+1}-2 x\right )+2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{3 x^3+x+1}+x}\right )-\frac {3 \left (3 x^3+x+1\right )^{2/3}}{2 x^2}+\frac {\log \left (2^{2/3} \sqrt [3]{3 x^3+x+1} x+\sqrt [3]{2} \left (3 x^3+x+1\right )^{2/3}+2 x^2\right )}{\sqrt [3]{2}} \]
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Rubi [F] time = 1.76, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {(3+2 x) \left (1+x+3 x^3\right )^{2/3}}{x^3 \left (1+x+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(3+2 x) \left (1+x+3 x^3\right )^{2/3}}{x^3 \left (1+x+x^3\right )} \, dx &=\int \left (\frac {3 \left (1+x+3 x^3\right )^{2/3}}{x^3}-\frac {\left (1+x+3 x^3\right )^{2/3}}{x^2}+\frac {\left (1+x+3 x^3\right )^{2/3}}{x}+\frac {\left (-4+x-x^2\right ) \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3}\right ) \, dx\\ &=3 \int \frac {\left (1+x+3 x^3\right )^{2/3}}{x^3} \, dx-\int \frac {\left (1+x+3 x^3\right )^{2/3}}{x^2} \, dx+\int \frac {\left (1+x+3 x^3\right )^{2/3}}{x} \, dx+\int \frac {\left (-4+x-x^2\right ) \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3} \, dx\\ &=-\frac {\left (1+x+3 x^3\right )^{2/3} \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x^2} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\frac {\left (1+x+3 x^3\right )^{2/3} \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\frac {\left (3 \left (1+x+3 x^3\right )^{2/3}\right ) \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x^3} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\int \left (-\frac {4 \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3}+\frac {x \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3}-\frac {x^2 \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3}\right ) \, dx\\ &=-\left (4 \int \frac {\left (1+x+3 x^3\right )^{2/3}}{1+x+x^3} \, dx\right )-\frac {\left (1+x+3 x^3\right )^{2/3} \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x^2} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\frac {\left (1+x+3 x^3\right )^{2/3} \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\frac {\left (3 \left (1+x+3 x^3\right )^{2/3}\right ) \int \frac {\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}{x^3} \, dx}{\left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}+3 x\right )^{2/3} \left (1+\left (\frac {2}{-9+\sqrt {85}}\right )^{2/3}+\left (\frac {1}{2} \left (-9+\sqrt {85}\right )\right )^{2/3}-3 \left (\sqrt [3]{\frac {2}{-9+\sqrt {85}}}-\sqrt [3]{\frac {1}{2} \left (-9+\sqrt {85}\right )}\right ) x+9 x^2\right )^{2/3}}+\int \frac {x \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3} \, dx-\int \frac {x^2 \left (1+x+3 x^3\right )^{2/3}}{1+x+x^3} \, dx\\ \end {align*}
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Mathematica [F] time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(3+2 x) \left (1+x+3 x^3\right )^{2/3}}{x^3 \left (1+x+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.46, size = 141, normalized size = 1.00 \begin {gather*} -\frac {3 \left (1+x+3 x^3\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x+3 x^3}}\right )-2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x+3 x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x+3 x^3}+\sqrt [3]{2} \left (1+x+3 x^3\right )^{2/3}\right )}{\sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 11.63, size = 380, normalized size = 2.70 \begin {gather*} \frac {2 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{2} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (7 \, x^{7} + 8 \, x^{5} + 8 \, x^{4} + x^{3} + 2 \, x^{2} + x\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} - 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (55 \, x^{8} + 20 \, x^{6} + 20 \, x^{5} + x^{4} + 2 \, x^{3} + x^{2}\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (433 \, x^{9} + 255 \, x^{7} + 255 \, x^{6} + 39 \, x^{5} + 78 \, x^{4} + 40 \, x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}}{3 \, {\left (323 \, x^{9} + 105 \, x^{7} + 105 \, x^{6} - 3 \, x^{5} - 6 \, x^{4} - 4 \, x^{3} - 3 \, x^{2} - 3 \, x - 1\right )}}\right ) + 2 \, \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (3 \, x^{3} + x + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} x - \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + x + 1\right )}}{x^{3} + x + 1}\right ) - \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x^{2} + x\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (55 \, x^{6} + 20 \, x^{4} + 20 \, x^{3} + x^{2} + 2 \, x + 1\right )} - 24 \, {\left (4 \, x^{5} + x^{3} + x^{2}\right )} {\left (3 \, x^{3} + x + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{4} + 2 \, x^{3} + x^{2} + 2 \, x + 1}\right ) - 9 \, {\left (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \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 (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} {\left (2 \, x + 3\right )}}{{\left (x^{3} + x + 1\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 19.56, size = 695, normalized size = 4.93
method | result | size |
risch | \(-\frac {3 \left (3 x^{3}+x +1\right )^{\frac {2}{3}}}{2 x^{2}}+\RootOf \left (\textit {\_Z}^{3}+4\right ) \ln \left (-\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}+5 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+4 \left (3 x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +4 \left (3 x^{3}+x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+\left (3 x^{3}+x +1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}-5 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}-25 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) x^{3}-7 \left (3 x^{3}+x +1\right )^{\frac {2}{3}} x -\RootOf \left (\textit {\_Z}^{3}+4\right ) x -5 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) x -\RootOf \left (\textit {\_Z}^{3}+4\right )-5 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right )}{x^{3}+x +1}\right )+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \ln \left (\frac {5 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+8 \left (3 x^{3}+x +1\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +8 \left (3 x^{3}+x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+14 \left (3 x^{3}+x +1\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}+15 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}+12 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) x^{3}-2 \left (3 x^{3}+x +1\right )^{\frac {2}{3}} x +5 \RootOf \left (\textit {\_Z}^{3}+4\right ) x +4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) x +5 \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right )}{x^{3}+x +1}\right )\) | \(695\) |
trager | \(\text {Expression too large to display}\) | \(1455\) |
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 (3 \, x^{3} + x + 1\right )}^{\frac {2}{3}} {\left (2 \, x + 3\right )}}{{\left (x^{3} + x + 1\right )} x^{3}}\,{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 (2\,x+3\right )\,{\left (3\,x^3+x+1\right )}^{2/3}}{x^3\,\left (x^3+x+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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