Integrand size = 57, antiderivative size = 116 \[ \int \frac {\left (1+x^5\right ) \sqrt [3]{1+x^3+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )} \, dx=\frac {3 \sqrt [3]{1+x^3+x^5}}{2 x}-\frac {1}{4} \text {RootSum}\left [3-6 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-3 \log (x)+3 \log \left (\sqrt [3]{1+x^3+x^5}-x \text {$\#$1}\right )+4 \log (x) \text {$\#$1}^3-4 \log \left (\sqrt [3]{1+x^3+x^5}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ] \]
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\[ \int \frac {\left (1+x^5\right ) \sqrt [3]{1+x^3+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )} \, dx=\int \frac {\left (1+x^5\right ) \sqrt [3]{1+x^3+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 \sqrt [3]{1+x^3+x^5}}{2 x^2}+\frac {x \sqrt [3]{1+x^3+x^5} \left (-6+10 x^2-3 x^3-6 x^5+10 x^7\right )}{2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {x \sqrt [3]{1+x^3+x^5} \left (-6+10 x^2-3 x^3-6 x^5+10 x^7\right )}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}} \, dx-\frac {3}{2} \int \frac {\sqrt [3]{1+x^3+x^5}}{x^2} \, dx \\ & = \frac {1}{2} \int \left (-\frac {6 x \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}}+\frac {10 x^3 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}}-\frac {3 x^4 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}}-\frac {6 x^6 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}}+\frac {10 x^8 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}}\right ) \, dx-\frac {3}{2} \int \frac {\sqrt [3]{1+x^3+x^5}}{x^2} \, dx \\ & = -\left (\frac {3}{2} \int \frac {\sqrt [3]{1+x^3+x^5}}{x^2} \, dx\right )-\frac {3}{2} \int \frac {x^4 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}} \, dx-3 \int \frac {x \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}} \, dx-3 \int \frac {x^6 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}} \, dx+5 \int \frac {x^3 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}} \, dx+5 \int \frac {x^8 \sqrt [3]{1+x^3+x^5}}{2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}} \, dx \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^5\right ) \sqrt [3]{1+x^3+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )} \, dx=\frac {3 \sqrt [3]{1+x^3+x^5}}{2 x}-\frac {1}{4} \text {RootSum}\left [3-6 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-3 \log (x)+3 \log \left (\sqrt [3]{1+x^3+x^5}-x \text {$\#$1}\right )+4 \log (x) \text {$\#$1}^3-4 \log \left (\sqrt [3]{1+x^3+x^5}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+2 \text {$\#$1}^5}\&\right ] \]
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Timed out.
\[\int \frac {\left (x^{5}+1\right ) \left (x^{5}+x^{3}+1\right )^{\frac {1}{3}} \left (2 x^{5}-3\right )}{x^{2} \left (2 x^{10}-2 x^{8}-x^{6}+4 x^{5}-2 x^{3}+2\right )}d x\]
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Exception generated. \[ \int \frac {\left (1+x^5\right ) \sqrt [3]{1+x^3+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 6.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.56 \[ \int \frac {\left (1+x^5\right ) \sqrt [3]{1+x^3+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )} \, dx=\int \frac {\left (x + 1\right ) \left (2 x^{5} - 3\right ) \sqrt [3]{x^{5} + x^{3} + 1} \left (x^{4} - x^{3} + x^{2} - x + 1\right )}{x^{2} \cdot \left (2 x^{10} - 2 x^{8} - x^{6} + 4 x^{5} - 2 x^{3} + 2\right )}\, dx \]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.49 \[ \int \frac {\left (1+x^5\right ) \sqrt [3]{1+x^3+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )} \, dx=\int { \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} + x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{5} + 1\right )}}{{\left (2 \, x^{10} - 2 \, x^{8} - x^{6} + 4 \, x^{5} - 2 \, x^{3} + 2\right )} x^{2}} \,d x } \]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.49 \[ \int \frac {\left (1+x^5\right ) \sqrt [3]{1+x^3+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )} \, dx=\int { \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} + x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{5} + 1\right )}}{{\left (2 \, x^{10} - 2 \, x^{8} - x^{6} + 4 \, x^{5} - 2 \, x^{3} + 2\right )} x^{2}} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.48 \[ \int \frac {\left (1+x^5\right ) \sqrt [3]{1+x^3+x^5} \left (-3+2 x^5\right )}{x^2 \left (2-2 x^3+4 x^5-x^6-2 x^8+2 x^{10}\right )} \, dx=\int -\frac {\left (x^5+1\right )\,\left (2\,x^5-3\right )\,{\left (x^5+x^3+1\right )}^{1/3}}{x^2\,\left (-2\,x^{10}+2\,x^8+x^6-4\,x^5+2\,x^3-2\right )} \,d x \]
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