Integrand size = 28, antiderivative size = 105 \[ \int \frac {x \left (3+5 x^2\right )}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x \sqrt [3]{1+x^2}}{2+x \sqrt [3]{1+x^2}}\right )+2 \text {arctanh}\left (1-2 x \sqrt [3]{1+x^2}\right )+\frac {1}{2} \log \left (x^2 \left (1+x^2\right )^{2/3}\right )-\frac {1}{2} \log \left (1+x \sqrt [3]{1+x^2}+x^2 \left (1+x^2\right )^{2/3}\right ) \]
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\[ \int \frac {x \left (3+5 x^2\right )}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx=\int \frac {x \left (3+5 x^2\right )}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 x}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )}+\frac {5 x^3}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )}\right ) \, dx \\ & = 3 \int \frac {x}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx+5 \int \frac {x^3}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx \\ \end{align*}
Time = 1.87 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {x \left (3+5 x^2\right )}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} x \sqrt [3]{1+x^2}}{2+x \sqrt [3]{1+x^2}}\right )+2 \text {arctanh}\left (1-2 x \sqrt [3]{1+x^2}\right )+\frac {1}{2} \log \left (x^2 \left (1+x^2\right )^{2/3}\right )-\frac {1}{2} \log \left (1+x \sqrt [3]{1+x^2}+x^2 \left (1+x^2\right )^{2/3}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.08 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.58
method | result | size |
trager | \(\ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{5}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{3}} x -3 x \left (x^{2}+1\right )^{\frac {1}{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2}{x^{5}+x^{3}-1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{5}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {2}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-x^{5}+2 x^{2} \left (x^{2}+1\right )^{\frac {2}{3}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}+1\right )^{\frac {1}{3}} x -x^{3}-x \left (x^{2}+1\right )^{\frac {1}{3}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{5}+x^{3}-1}\right )\) | \(271\) |
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Time = 1.33 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.98 \[ \int \frac {x \left (3+5 x^2\right )}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx=-\sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {2}{3}} x^{2} - 4 \, \sqrt {3} {\left (x^{2} + 1\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (x^{5} + x^{3}\right )}}{x^{5} + x^{3} + 8}\right ) + \frac {1}{2} \, \log \left (\frac {x^{5} + x^{3} - 3 \, {\left (x^{2} + 1\right )}^{\frac {2}{3}} x^{2} + 3 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} x - 1}{x^{5} + x^{3} - 1}\right ) \]
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\[ \int \frac {x \left (3+5 x^2\right )}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx=\int \frac {x \left (5 x^{2} + 3\right )}{\sqrt [3]{x^{2} + 1} \left (x^{5} + x^{3} - 1\right )}\, dx \]
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\[ \int \frac {x \left (3+5 x^2\right )}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx=\int { \frac {{\left (5 \, x^{2} + 3\right )} x}{{\left (x^{5} + x^{3} - 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {x \left (3+5 x^2\right )}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx=\int { \frac {{\left (5 \, x^{2} + 3\right )} x}{{\left (x^{5} + x^{3} - 1\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {x \left (3+5 x^2\right )}{\sqrt [3]{1+x^2} \left (-1+x^3+x^5\right )} \, dx=\int \frac {x\,\left (5\,x^2+3\right )}{{\left (x^2+1\right )}^{1/3}\,\left (x^5+x^3-1\right )} \,d x \]
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