Integrand size = 41, antiderivative size = 61 \[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\frac {1}{4} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{2 x+x^3+2 x^8}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ] \]
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\[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{2 x+x^3+2 x^8} \int \frac {\sqrt [3]{x} \sqrt [3]{2+x^2+2 x^7} \left (-2+5 x^7\right )}{4+x^4+8 x^7+4 x^{14}} \, dx}{\sqrt [3]{x} \sqrt [3]{2+x^2+2 x^7}} \\ & = \frac {\left (3 \sqrt [3]{2 x+x^3+2 x^8}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{2+x^6+2 x^{21}} \left (-2+5 x^{21}\right )}{4+x^{12}+8 x^{21}+4 x^{42}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{2+x^2+2 x^7}} \\ & = \frac {\left (3 \sqrt [3]{2 x+x^3+2 x^8}\right ) \text {Subst}\left (\int \left (-\frac {2 x^3 \sqrt [3]{2+x^6+2 x^{21}}}{4+x^{12}+8 x^{21}+4 x^{42}}+\frac {5 x^{24} \sqrt [3]{2+x^6+2 x^{21}}}{4+x^{12}+8 x^{21}+4 x^{42}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{2+x^2+2 x^7}} \\ & = -\frac {\left (6 \sqrt [3]{2 x+x^3+2 x^8}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{2+x^6+2 x^{21}}}{4+x^{12}+8 x^{21}+4 x^{42}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{2+x^2+2 x^7}}+\frac {\left (15 \sqrt [3]{2 x+x^3+2 x^8}\right ) \text {Subst}\left (\int \frac {x^{24} \sqrt [3]{2+x^6+2 x^{21}}}{4+x^{12}+8 x^{21}+4 x^{42}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{2+x^2+2 x^7}} \\ \end{align*}
Time = 3.24 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.67 \[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\frac {\sqrt [3]{x \left (2+x^2+2 x^7\right )} \text {RootSum}\left [2-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{2+x^2+2 x^7}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]}{4 \sqrt [3]{x} \sqrt [3]{2+x^2+2 x^7}} \]
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Timed out.
\[\int \frac {\left (5 x^{7}-2\right ) \left (2 x^{8}+x^{3}+2 x \right )^{\frac {1}{3}}}{4 x^{14}+8 x^{7}+x^{4}+4}d x\]
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Exception generated. \[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 3.71 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.61 \[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\int \frac {\sqrt [3]{x \left (2 x^{7} + x^{2} + 2\right )} \left (5 x^{7} - 2\right )}{4 x^{14} + 8 x^{7} + x^{4} + 4}\, dx \]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.67 \[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\int { \frac {{\left (2 \, x^{8} + x^{3} + 2 \, x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} - 2\right )}}{4 \, x^{14} + 8 \, x^{7} + x^{4} + 4} \,d x } \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.67 \[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\int { \frac {{\left (2 \, x^{8} + x^{3} + 2 \, x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} - 2\right )}}{4 \, x^{14} + 8 \, x^{7} + x^{4} + 4} \,d x } \]
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Not integrable
Time = 6.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.67 \[ \int \frac {\left (-2+5 x^7\right ) \sqrt [3]{2 x+x^3+2 x^8}}{4+x^4+8 x^7+4 x^{14}} \, dx=\int \frac {\left (5\,x^7-2\right )\,{\left (2\,x^8+x^3+2\,x\right )}^{1/3}}{4\,x^{14}+8\,x^7+x^4+4} \,d x \]
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