Integrand size = 39, antiderivative size = 55 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx=\text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x+x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^3}\&\right ] \]
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\[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx=\int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{x+x^3+x^4} \int \frac {\sqrt [3]{x} \left (-2+x^3\right ) \sqrt [3]{1+x^2+x^3}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \\ & = \frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^3 \left (-2+x^9\right ) \sqrt [3]{1+x^6+x^9}}{1+x^6+2 x^9+x^{12}+x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \\ & = \frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \left (-\frac {2 x^3 \sqrt [3]{1+x^6+x^9}}{1+x^6+2 x^9+x^{12}+x^{15}+x^{18}}+\frac {x^{12} \sqrt [3]{1+x^6+x^9}}{1+x^6+2 x^9+x^{12}+x^{15}+x^{18}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \\ & = \frac {\left (3 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^{12} \sqrt [3]{1+x^6+x^9}}{1+x^6+2 x^9+x^{12}+x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}}-\frac {\left (6 \sqrt [3]{x+x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6+x^9}}{1+x^6+2 x^9+x^{12}+x^{15}+x^{18}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \\ \end{align*}
Time = 2.77 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.69 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx=\frac {\sqrt [3]{x+x^3+x^4} \text {RootSum}\left [1-\text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-2 \log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{1+x^2+x^3}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^3}\&\right ]}{\sqrt [3]{x} \sqrt [3]{1+x^2+x^3}} \]
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Time = 9.86 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87
method | result | size |
pseudoelliptic | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-\textit {\_Z}^{3}+1\right )}{\sum }\frac {\textit {\_R} \ln \left (\frac {-\textit {\_R} x +{\left (x \left (x^{3}+x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )}{2 \textit {\_R}^{3}-1}\) | \(48\) |
trager | \(\text {Expression too large to display}\) | \(2244\) |
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Exception generated. \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 1.96 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx=\int \frac {\sqrt [3]{x \left (x^{3} + x^{2} + 1\right )} \left (x^{3} - 2\right )}{x^{6} + x^{5} + x^{4} + 2 x^{3} + x^{2} + 1}\, dx \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx=\int { \frac {{\left (x^{4} + x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{x^{6} + x^{5} + x^{4} + 2 \, x^{3} + x^{2} + 1} \,d x } \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx=\int { \frac {{\left (x^{4} + x^{3} + x\right )}^{\frac {1}{3}} {\left (x^{3} - 2\right )}}{x^{6} + x^{5} + x^{4} + 2 \, x^{3} + x^{2} + 1} \,d x } \]
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Not integrable
Time = 5.86 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71 \[ \int \frac {\left (-2+x^3\right ) \sqrt [3]{x+x^3+x^4}}{1+x^2+2 x^3+x^4+x^5+x^6} \, dx=\int \frac {\left (x^3-2\right )\,{\left (x^4+x^3+x\right )}^{1/3}}{x^6+x^5+x^4+2\,x^3+x^2+1} \,d x \]
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