Integrand size = 59, antiderivative size = 94 \[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx=\frac {(1+2 x) \sqrt {1+2 x^2+4 x^3+x^4}}{2 \left (-1+2 x^2\right )}-\text {arctanh}\left (\frac {x (2+x) \left (7-x+27 x^2+33 x^3\right )}{\left (2+37 x^2+31 x^3\right ) \sqrt {1+2 x^2+4 x^3+x^4}}\right ) \]
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\[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx=\int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {9}{4 \sqrt {1+2 x^2+4 x^3+x^4}}+\frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}}+\frac {x^2}{2 \sqrt {1+2 x^2+4 x^3+x^4}}+\frac {-13-17 x}{2 \left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}}+\frac {15+2 x}{4 \left (-1+2 x^2\right ) \sqrt {1+2 x^2+4 x^3+x^4}}\right ) \, dx \\ & = \frac {1}{4} \int \frac {15+2 x}{\left (-1+2 x^2\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{2} \int \frac {x^2}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{2} \int \frac {-13-17 x}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {9}{4} \int \frac {1}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx \\ & = \frac {1}{4} \int \left (-\frac {15+\sqrt {2}}{2 \left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}}-\frac {15-\sqrt {2}}{2 \left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}}\right ) \, dx+\frac {1}{2} \int \frac {x^2}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{2} \int \left (-\frac {13}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}}-\frac {17 x}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}}\right ) \, dx+\frac {9}{4} \int \frac {1}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx \\ & = \frac {1}{2} \int \frac {x^2}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {9}{4} \int \frac {1}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{2} \int \frac {1}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {17}{2} \int \frac {x}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15-\sqrt {2}\right ) \int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15+\sqrt {2}\right ) \int \frac {1}{\left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx \\ & = \frac {1}{2} \int \frac {x^2}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {9}{4} \int \frac {1}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{2} \int \left (\frac {1}{2 \left (\sqrt {2}-2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}}+\frac {1}{2 \left (\sqrt {2}+2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}}+\frac {1}{\left (2-4 x^2\right ) \sqrt {1+2 x^2+4 x^3+x^4}}\right ) \, dx-\frac {17}{2} \int \frac {x}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15-\sqrt {2}\right ) \int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15+\sqrt {2}\right ) \int \frac {1}{\left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx \\ & = \frac {1}{2} \int \frac {x^2}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {9}{4} \int \frac {1}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{4} \int \frac {1}{\left (\sqrt {2}-2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{4} \int \frac {1}{\left (\sqrt {2}+2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{2} \int \frac {1}{\left (2-4 x^2\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {17}{2} \int \frac {x}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15-\sqrt {2}\right ) \int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15+\sqrt {2}\right ) \int \frac {1}{\left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx \\ & = \frac {1}{2} \int \frac {x^2}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {9}{4} \int \frac {1}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{4} \int \frac {1}{\left (\sqrt {2}-2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{4} \int \frac {1}{\left (\sqrt {2}+2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{2} \int \left (\frac {1}{4 \left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}}+\frac {1}{4 \left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}}\right ) \, dx-\frac {17}{2} \int \frac {x}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15-\sqrt {2}\right ) \int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15+\sqrt {2}\right ) \int \frac {1}{\left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx \\ & = \frac {1}{2} \int \frac {x^2}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{8} \int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{8} \int \frac {1}{\left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {9}{4} \int \frac {1}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{4} \int \frac {1}{\left (\sqrt {2}-2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {13}{4} \int \frac {1}{\left (\sqrt {2}+2 x\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx-\frac {17}{2} \int \frac {x}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15-\sqrt {2}\right ) \int \frac {1}{\left (1-\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\frac {1}{8} \left (-15+\sqrt {2}\right ) \int \frac {1}{\left (1+\sqrt {2} x\right ) \sqrt {1+2 x^2+4 x^3+x^4}} \, dx+\int \frac {x}{\sqrt {1+2 x^2+4 x^3+x^4}} \, dx \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 16.14 (sec) , antiderivative size = 5141, normalized size of antiderivative = 54.69 \[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx=\text {Result too large to show} \]
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Timed out.
hanged
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Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (88) = 176\).
Time = 0.32 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.90 \[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx=\frac {{\left (2 \, x^{2} - 1\right )} \log \left (\frac {1025 \, x^{10} + 6138 \, x^{9} + 12307 \, x^{8} + 10188 \, x^{7} + 4503 \, x^{6} + 3134 \, x^{5} + 1589 \, x^{4} + 140 \, x^{3} + 176 \, x^{2} - {\left (1023 \, x^{8} + 4104 \, x^{7} + 5084 \, x^{6} + 2182 \, x^{5} + 805 \, x^{4} + 624 \, x^{3} + 10 \, x^{2} + 28 \, x\right )} \sqrt {x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1} + 2}{32 \, x^{10} - 80 \, x^{8} + 80 \, x^{6} - 40 \, x^{4} + 10 \, x^{2} - 1}\right ) + \sqrt {x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1} {\left (2 \, x + 1\right )}}{2 \, {\left (2 \, x^{2} - 1\right )}} \]
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\[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx=\int \frac {2 x^{6} + 4 x^{5} + 7 x^{4} - 3 x^{3} - x^{2} - 8 x - 8}{\sqrt {\left (x + 1\right ) \left (x^{3} + 3 x^{2} - x + 1\right )} \left (2 x^{2} - 1\right )^{2}}\, dx \]
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\[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx=\int { \frac {2 \, x^{6} + 4 \, x^{5} + 7 \, x^{4} - 3 \, x^{3} - x^{2} - 8 \, x - 8}{\sqrt {x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1} {\left (2 \, x^{2} - 1\right )}^{2}} \,d x } \]
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\[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx=\int { \frac {2 \, x^{6} + 4 \, x^{5} + 7 \, x^{4} - 3 \, x^{3} - x^{2} - 8 \, x - 8}{\sqrt {x^{4} + 4 \, x^{3} + 2 \, x^{2} + 1} {\left (2 \, x^{2} - 1\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {-8-8 x-x^2-3 x^3+7 x^4+4 x^5+2 x^6}{\left (-1+2 x^2\right )^2 \sqrt {1+2 x^2+4 x^3+x^4}} \, dx=\int -\frac {-2\,x^6-4\,x^5-7\,x^4+3\,x^3+x^2+8\,x+8}{{\left (2\,x^2-1\right )}^2\,\sqrt {x^4+4\,x^3+2\,x^2+1}} \,d x \]
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