Integrand size = 31, antiderivative size = 132 \[ \int \frac {6+2 x+x^2}{(2+x) \left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {-\frac {2^{2/3} x}{\sqrt {3}}+\frac {\sqrt [3]{2+x+x^2}}{\sqrt {3}}}{\sqrt [3]{2+x+x^2}}\right )}{2^{2/3}}+\frac {\log \left (2^{2/3} x+2 \sqrt [3]{2+x+x^2}\right )}{2^{2/3}}-\frac {\log \left (\sqrt [3]{2} x^2-2^{2/3} x \sqrt [3]{2+x+x^2}+2 \left (2+x+x^2\right )^{2/3}\right )}{2\ 2^{2/3}} \]
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\[ \int \frac {6+2 x+x^2}{(2+x) \left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx=\int \frac {6+2 x+x^2}{(2+x) \left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{(2+x) \sqrt [3]{2+x+x^2}}+\frac {2}{\left (2+x^2\right ) \sqrt [3]{2+x+x^2}}\right ) \, dx \\ & = 2 \int \frac {1}{\left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx+\int \frac {1}{(2+x) \sqrt [3]{2+x+x^2}} \, dx \\ & = 2 \int \frac {1}{\left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx-\frac {\left (\sqrt [3]{\frac {1-i \sqrt {7}+2 x}{2+x}} \sqrt [3]{\frac {1+i \sqrt {7}+2 x}{2+x}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{x} \sqrt [3]{1-\frac {1}{2} \left (3-i \sqrt {7}\right ) x} \sqrt [3]{1-\frac {1}{2} \left (3+i \sqrt {7}\right ) x}} \, dx,x,\frac {1}{2+x}\right )}{2^{2/3} \left (\frac {1}{2+x}\right )^{2/3} \sqrt [3]{2+x+x^2}} \\ & = -\frac {3 \sqrt [3]{\frac {1-i \sqrt {7}+2 x}{2+x}} \sqrt [3]{\frac {1+i \sqrt {7}+2 x}{2+x}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},\frac {3-i \sqrt {7}}{2 (2+x)},\frac {3+i \sqrt {7}}{2 (2+x)}\right )}{2\ 2^{2/3} \sqrt [3]{2+x+x^2}}+2 \int \frac {1}{\left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.82 \[ \int \frac {6+2 x+x^2}{(2+x) \left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {1-\frac {2^{2/3} x}{\sqrt [3]{2+x+x^2}}}{\sqrt {3}}\right )-2 \log \left (2^{2/3} x+2 \sqrt [3]{2+x+x^2}\right )+\log \left (\sqrt [3]{2} x^2-2^{2/3} x \sqrt [3]{2+x+x^2}+2 \left (2+x+x^2\right )^{2/3}\right )}{2\ 2^{2/3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.74 (sec) , antiderivative size = 883, normalized size of antiderivative = 6.69
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Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (102) = 204\).
Time = 6.07 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.99 \[ \int \frac {6+2 x+x^2}{(2+x) \left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx=-\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (x^{7} + x^{6} - x^{5} - 2 \, x^{4} - 10 \, x^{3} - 8 \, x^{2} - 8 \, x\right )} {\left (x^{2} + x + 2\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{9} + 24 \, x^{8} - 36 \, x^{7} - 64 \, x^{6} - 276 \, x^{5} - 168 \, x^{4} - 136 \, x^{3} + 144 \, x^{2} + 96 \, x + 64\right )} + 12 \, {\left (x^{8} - 14 \, x^{7} - 10 \, x^{6} - 20 \, x^{5} + 20 \, x^{4} + 16 \, x^{3} + 16 \, x^{2}\right )} {\left (x^{2} + x + 2\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{9} - 48 \, x^{8} - 36 \, x^{7} - 64 \, x^{6} + 84 \, x^{5} + 120 \, x^{4} + 152 \, x^{3} + 144 \, x^{2} + 96 \, x + 64\right )}}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{2} + x + 2\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} + 2 \, x^{2} + 2 \, x + 4\right )} + 12 \, {\left (x^{2} + x + 2\right )}^{\frac {2}{3}} x}{x^{3} + 2 \, x^{2} + 2 \, x + 4}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{4} - x^{3} - x^{2} - 2 \, x\right )} {\left (x^{2} + x + 2\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{6} - 14 \, x^{5} - 10 \, x^{4} - 20 \, x^{3} + 20 \, x^{2} + 16 \, x + 16\right )} - 6 \, {\left (x^{5} - 4 \, x^{4} - 4 \, x^{3} - 8 \, x^{2}\right )} {\left (x^{2} + x + 2\right )}^{\frac {1}{3}}}{x^{6} + 4 \, x^{5} + 8 \, x^{4} + 16 \, x^{3} + 20 \, x^{2} + 16 \, x + 16}\right ) \]
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\[ \int \frac {6+2 x+x^2}{(2+x) \left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx=\int \frac {x^{2} + 2 x + 6}{\left (x + 2\right ) \left (x^{2} + 2\right ) \sqrt [3]{x^{2} + x + 2}}\, dx \]
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\[ \int \frac {6+2 x+x^2}{(2+x) \left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx=\int { \frac {x^{2} + 2 \, x + 6}{{\left (x^{2} + x + 2\right )}^{\frac {1}{3}} {\left (x^{2} + 2\right )} {\left (x + 2\right )}} \,d x } \]
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\[ \int \frac {6+2 x+x^2}{(2+x) \left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx=\int { \frac {x^{2} + 2 \, x + 6}{{\left (x^{2} + x + 2\right )}^{\frac {1}{3}} {\left (x^{2} + 2\right )} {\left (x + 2\right )}} \,d x } \]
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Timed out. \[ \int \frac {6+2 x+x^2}{(2+x) \left (2+x^2\right ) \sqrt [3]{2+x+x^2}} \, dx=\int \frac {x^2+2\,x+6}{\left (x^2+2\right )\,\left (x+2\right )\,{\left (x^2+x+2\right )}^{1/3}} \,d x \]
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