Integrand size = 47, antiderivative size = 170 \[ \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx=\frac {2 \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x}-\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{\sqrt {2} x^2-\sqrt {-2-x^2-2 x^4+2 x^5}}\right )}{\sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{2 x^2+\sqrt {2} \sqrt {-2-x^2-2 x^4+2 x^5}}\right )}{\sqrt [4]{2}} \]
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\[ \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx=\int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2}+\frac {\left (-1+5 x^3\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{-2-x^2+2 x^5}\right ) \, dx \\ & = -\left (2 \int \frac {\sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2} \, dx\right )+\int \frac {\left (-1+5 x^3\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{-2-x^2+2 x^5} \, dx \\ & = -\left (2 \int \frac {\sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2} \, dx\right )+\int \left (\frac {\sqrt [4]{-2-x^2-2 x^4+2 x^5}}{2+x^2-2 x^5}+\frac {5 x^3 \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{-2-x^2+2 x^5}\right ) \, dx \\ & = -\left (2 \int \frac {\sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2} \, dx\right )+5 \int \frac {x^3 \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{-2-x^2+2 x^5} \, dx+\int \frac {\sqrt [4]{-2-x^2-2 x^4+2 x^5}}{2+x^2-2 x^5} \, dx \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.94 \[ \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx=\frac {2 \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x}-\frac {\arctan \left (\frac {2^{3/4} x \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{\sqrt {2} x^2-\sqrt {-2-x^2-2 x^4+2 x^5}}\right )}{\sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {2 x \sqrt [4]{-4-2 x^2-4 x^4+4 x^5}}{2 x^2+\sqrt {-4-2 x^2-4 x^4+4 x^5}}\right )}{\sqrt [4]{2}} \]
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Timed out.
\[\int \frac {\left (x^{5}+x^{2}+4\right ) \left (2 x^{5}-2 x^{4}-x^{2}-2\right )^{\frac {1}{4}}}{x^{2} \left (2 x^{5}-x^{2}-2\right )}d x\]
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Result contains complex when optimal does not.
Time = 49.95 (sec) , antiderivative size = 543, normalized size of antiderivative = 3.19 \[ \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx=\frac {\left (i + 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} x \log \left (\frac {\left (i + 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} x^{2} - 8 i \, \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} x^{3} + 8^{\frac {1}{4}} \sqrt {2} {\left (-\left (2 i - 2\right ) \, x^{5} + \left (4 i - 4\right ) \, x^{4} + \left (i - 1\right ) \, x^{2} + 2 i - 2\right )} - 8 \, {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{5} - x^{2} - 2}\right ) - \left (i - 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} x \log \left (\frac {-\left (i - 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} x^{2} + 8 i \, \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} x^{3} + 8^{\frac {1}{4}} \sqrt {2} {\left (\left (2 i + 2\right ) \, x^{5} - \left (4 i + 4\right ) \, x^{4} - \left (i + 1\right ) \, x^{2} - 2 i - 2\right )} - 8 \, {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{5} - x^{2} - 2}\right ) + \left (i - 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} x \log \left (\frac {\left (i - 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} x^{2} + 8 i \, \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} x^{3} + 8^{\frac {1}{4}} \sqrt {2} {\left (-\left (2 i + 2\right ) \, x^{5} + \left (4 i + 4\right ) \, x^{4} + \left (i + 1\right ) \, x^{2} + 2 i + 2\right )} - 8 \, {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{5} - x^{2} - 2}\right ) - \left (i + 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} x \log \left (\frac {-\left (i + 1\right ) \cdot 8^{\frac {3}{4}} \sqrt {2} \sqrt {2 \, x^{5} - 2 \, x^{4} - x^{2} - 2} x^{2} - 8 i \, \sqrt {2} {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} x^{3} + 8^{\frac {1}{4}} \sqrt {2} {\left (\left (2 i - 2\right ) \, x^{5} - \left (4 i - 4\right ) \, x^{4} - \left (i - 1\right ) \, x^{2} - 2 i + 2\right )} - 8 \, {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {3}{4}} x}{2 \, x^{5} - x^{2} - 2}\right ) + 64 \, {\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}}}{32 \, x} \]
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\[ \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx=\int \frac {\left (x^{5} + x^{2} + 4\right ) \sqrt [4]{2 x^{5} - 2 x^{4} - x^{2} - 2}}{x^{2} \cdot \left (2 x^{5} - x^{2} - 2\right )}\, dx \]
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\[ \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} {\left (x^{5} + x^{2} + 4\right )}}{{\left (2 \, x^{5} - x^{2} - 2\right )} x^{2}} \,d x } \]
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\[ \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{5} - 2 \, x^{4} - x^{2} - 2\right )}^{\frac {1}{4}} {\left (x^{5} + x^{2} + 4\right )}}{{\left (2 \, x^{5} - x^{2} - 2\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (4+x^2+x^5\right ) \sqrt [4]{-2-x^2-2 x^4+2 x^5}}{x^2 \left (-2-x^2+2 x^5\right )} \, dx=\int -\frac {\left (x^5+x^2+4\right )\,{\left (2\,x^5-2\,x^4-x^2-2\right )}^{1/4}}{x^2\,\left (-2\,x^5+x^2+2\right )} \,d x \]
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