Optimal. Leaf size=100 \[ \log \left (\sqrt [3]{x^2+2}+x-1\right )-\frac {1}{2} \log \left (x^2+\left (x^2+2\right )^{2/3}+(1-x) \sqrt [3]{x^2+2}-2 x+1\right )-\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2+2}}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}+\frac {2}{\sqrt {3}}}{\sqrt [3]{x^2+2}}\right ) \]
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Rubi [F] time = 0.59, 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 {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx &=\int \left (\frac {6}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )}+\frac {2 x}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )}+\frac {x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )}\right ) \, dx\\ &=2 \int \frac {x}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx+6 \int \frac {1}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx+\int \frac {x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx\\ \end {align*}
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Mathematica [F] time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {6+2 x+x^2}{\sqrt [3]{2+x^2} \left (1+3 x-2 x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.14, size = 100, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\frac {2}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}+\frac {\sqrt [3]{2+x^2}}{\sqrt {3}}}{\sqrt [3]{2+x^2}}\right )+\log \left (-1+x+\sqrt [3]{2+x^2}\right )-\frac {1}{2} \log \left (1-2 x+x^2+(1-x) \sqrt [3]{2+x^2}+\left (2+x^2\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.24, size = 138, normalized size = 1.38 \begin {gather*} -\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{2} + 2\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x^{2} - 2 \, x + 1\right )} {\left (x^{2} + 2\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{3} - 2 \, x^{2} + 3 \, x + 1\right )}}{3 \, {\left (x^{3} - 4 \, x^{2} + 3 \, x - 3\right )}}\right ) + \frac {1}{2} \, \log \left (\frac {x^{3} - 2 \, x^{2} + 3 \, {\left (x^{2} + 2\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 3 \, {\left (x^{2} - 2 \, x + 1\right )} {\left (x^{2} + 2\right )}^{\frac {1}{3}} + 3 \, x + 1}{x^{3} - 2 \, x^{2} + 3 \, x + 1}\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 \frac {x^{2} + 2 \, x + 6}{{\left (x^{3} - 2 \, x^{2} + 3 \, x + 1\right )} {\left (x^{2} + 2\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.23, size = 772, normalized size = 7.72
method | result | size |
trager | \(\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {188 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}} x -573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}-470 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+676 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}}+891 \left (x^{2}+2\right )^{\frac {2}{3}} x +1146 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x -891 \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}+564 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -2019 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+591 x^{3}-891 \left (x^{2}+2\right )^{\frac {2}{3}}-573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}}+1782 \left (x^{2}+2\right )^{\frac {1}{3}} x +2028 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -2167 x^{2}-891 \left (x^{2}+2\right )^{\frac {1}{3}}-658 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1773 x -1379}{x^{3}-2 x^{2}+3 x +1}\right )-\ln \left (\frac {-188 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}} x -573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}+470 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+300 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}}-318 \left (x^{2}+2\right )^{\frac {2}{3}} x +1146 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x +318 \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}-564 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -1079 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-103 x^{3}+318 \left (x^{2}+2\right )^{\frac {2}{3}}-573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}}-636 \left (x^{2}+2\right )^{\frac {1}{3}} x +900 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +618 x^{2}+318 \left (x^{2}+2\right )^{\frac {1}{3}}-658 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-309 x +721}{x^{3}-2 x^{2}+3 x +1}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {-188 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}} x -573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}+470 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+300 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {2}{3}}-318 \left (x^{2}+2\right )^{\frac {2}{3}} x +1146 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}} x +318 \left (x^{2}+2\right )^{\frac {1}{3}} x^{2}-564 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -1079 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-103 x^{3}+318 \left (x^{2}+2\right )^{\frac {2}{3}}-573 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+2\right )^{\frac {1}{3}}-636 \left (x^{2}+2\right )^{\frac {1}{3}} x +900 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +618 x^{2}+318 \left (x^{2}+2\right )^{\frac {1}{3}}-658 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-309 x +721}{x^{3}-2 x^{2}+3 x +1}\right )\) | \(772\) |
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 {x^{2} + 2 \, x + 6}{{\left (x^{3} - 2 \, x^{2} + 3 \, x + 1\right )} {\left (x^{2} + 2\right )}^{\frac {1}{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 {x^2+2\,x+6}{{\left (x^2+2\right )}^{1/3}\,\left (x^3-2\,x^2+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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