Integrand size = 31, antiderivative size = 70 \[ \int \frac {\left (3+2 x^5\right ) \sqrt {x-2 x^4-x^6}}{\left (-1+x^5\right )^2} \, dx=-\frac {x \sqrt {x-2 x^4-x^6}}{-1+x^5}-\frac {\arctan \left (\frac {\sqrt {2} x \sqrt {x-2 x^4-x^6}}{-1+2 x^3+x^5}\right )}{\sqrt {2}} \]
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\[ \int \frac {\left (3+2 x^5\right ) \sqrt {x-2 x^4-x^6}}{\left (-1+x^5\right )^2} \, dx=\int \frac {\left (3+2 x^5\right ) \sqrt {x-2 x^4-x^6}}{\left (-1+x^5\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {x-2 x^4-x^6} \int \frac {\sqrt {x} \sqrt {1-2 x^3-x^5} \left (3+2 x^5\right )}{\left (-1+x^5\right )^2} \, dx}{\sqrt {x} \sqrt {1-2 x^3-x^5}} \\ & = \frac {\left (2 \sqrt {x-2 x^4-x^6}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {1-2 x^6-x^{10}} \left (3+2 x^{10}\right )}{\left (-1+x^{10}\right )^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-2 x^3-x^5}} \\ & = \frac {\left (2 \sqrt {x-2 x^4-x^6}\right ) \text {Subst}\left (\int \left (\frac {\sqrt {1-2 x^6-x^{10}}}{20 (-1+x)^2}+\frac {\sqrt {1-2 x^6-x^{10}}}{20 (1+x)^2}-\frac {3 \sqrt {1-2 x^6-x^{10}}}{10 \left (-1+x^2\right )}+\frac {\left (-1+3 x-x^2\right ) \sqrt {1-2 x^6-x^{10}}}{4 \left (1-x+x^2-x^3+x^4\right )^2}+\frac {\left (1-x+5 x^2-3 x^3\right ) \sqrt {1-2 x^6-x^{10}}}{20 \left (1-x+x^2-x^3+x^4\right )}+\frac {\left (-1-3 x-x^2\right ) \sqrt {1-2 x^6-x^{10}}}{4 \left (1+x+x^2+x^3+x^4\right )^2}+\frac {\left (1+x+5 x^2+3 x^3\right ) \sqrt {1-2 x^6-x^{10}}}{20 \left (1+x+x^2+x^3+x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1-2 x^3-x^5}} \\ & = \frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{(-1+x)^2} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{(1+x)^2} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {\left (1-x+5 x^2-3 x^3\right ) \sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {\left (1+x+5 x^2+3 x^3\right ) \sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {\left (-1+3 x-x^2\right ) \sqrt {1-2 x^6-x^{10}}}{\left (1-x+x^2-x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {\left (-1-3 x-x^2\right ) \sqrt {1-2 x^6-x^{10}}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \text {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{-1+x^2} \, dx,x,\sqrt {x}\right )}{5 \sqrt {x} \sqrt {1-2 x^3-x^5}} \\ & = \frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{(-1+x)^2} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{(1+x)^2} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \left (\frac {\sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4}-\frac {x \sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4}+\frac {5 x^2 \sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4}-\frac {3 x^3 \sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4}\right ) \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \left (\frac {\sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4}+\frac {x \sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4}+\frac {5 x^2 \sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4}+\frac {3 x^3 \sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4}\right ) \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \left (-\frac {\sqrt {1-2 x^6-x^{10}}}{\left (1-x+x^2-x^3+x^4\right )^2}+\frac {3 x \sqrt {1-2 x^6-x^{10}}}{\left (1-x+x^2-x^3+x^4\right )^2}-\frac {x^2 \sqrt {1-2 x^6-x^{10}}}{\left (1-x+x^2-x^3+x^4\right )^2}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \left (-\frac {\sqrt {1-2 x^6-x^{10}}}{\left (1+x+x^2+x^3+x^4\right )^2}-\frac {3 x \sqrt {1-2 x^6-x^{10}}}{\left (1+x+x^2+x^3+x^4\right )^2}-\frac {x^2 \sqrt {1-2 x^6-x^{10}}}{\left (1+x+x^2+x^3+x^4\right )^2}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \text {Subst}\left (\int \left (\frac {\sqrt {1-2 x^6-x^{10}}}{2 (-1+x)}-\frac {\sqrt {1-2 x^6-x^{10}}}{2 (1+x)}\right ) \, dx,x,\sqrt {x}\right )}{5 \sqrt {x} \sqrt {1-2 x^3-x^5}} \\ & = \frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{(-1+x)^2} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{(1+x)^2} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {x \sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {x \sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \text {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{-1+x} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \text {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{1+x} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4} \, dx,x,\sqrt {x}\right )}{10 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{\left (1-x+x^2-x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {x^2 \sqrt {1-2 x^6-x^{10}}}{\left (1-x+x^2-x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {x^2 \sqrt {1-2 x^6-x^{10}}}{1-x+x^2-x^3+x^4} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {\sqrt {1-2 x^6-x^{10}}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {x^2 \sqrt {1-2 x^6-x^{10}}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\sqrt {x-2 x^4-x^6} \text {Subst}\left (\int \frac {x^2 \sqrt {1-2 x^6-x^{10}}}{1+x+x^2+x^3+x^4} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}+\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \text {Subst}\left (\int \frac {x \sqrt {1-2 x^6-x^{10}}}{\left (1-x+x^2-x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}}-\frac {\left (3 \sqrt {x-2 x^4-x^6}\right ) \text {Subst}\left (\int \frac {x \sqrt {1-2 x^6-x^{10}}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt {1-2 x^3-x^5}} \\ \end{align*}
Time = 4.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.24 \[ \int \frac {\left (3+2 x^5\right ) \sqrt {x-2 x^4-x^6}}{\left (-1+x^5\right )^2} \, dx=\frac {\sqrt {x-2 x^4-x^6} \left (-\frac {2 x^{3/2}}{-1+x^5}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x^{3/2}}{\sqrt {-1+2 x^3+x^5}}\right )}{\sqrt {-1+2 x^3+x^5}}\right )}{2 \sqrt {x}} \]
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Time = 4.60 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\frac {\left (-x^{5}+1\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {-x \left (x^{5}+2 x^{3}-1\right )}\, \sqrt {2}}{2 x^{2}}\right )-2 \sqrt {-x \left (x^{5}+2 x^{3}-1\right )}\, x}{2 x^{5}-2}\) | \(65\) |
trager | \(-\frac {x \sqrt {-x^{6}-2 x^{4}+x}}{x^{5}-1}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{5}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{3}-4 \sqrt {-x^{6}-2 x^{4}+x}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{\left (x -1\right ) \left (x^{4}+x^{3}+x^{2}+x +1\right )}\right )}{4}\) | \(103\) |
risch | \(\frac {x^{2} \left (x^{5}+2 x^{3}-1\right )}{\left (x^{5}-1\right ) \sqrt {-x \left (x^{5}+2 x^{3}-1\right )}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{5}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{3}+4 \sqrt {-x^{6}-2 x^{4}+x}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{\left (x -1\right ) \left (x^{4}+x^{3}+x^{2}+x +1\right )}\right )}{4}\) | \(114\) |
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Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.99 \[ \int \frac {\left (3+2 x^5\right ) \sqrt {x-2 x^4-x^6}}{\left (-1+x^5\right )^2} \, dx=-\frac {\sqrt {2} {\left (x^{5} - 1\right )} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-x^{6} - 2 \, x^{4} + x} x}{x^{5} + 4 \, x^{3} - 1}\right ) + 4 \, \sqrt {-x^{6} - 2 \, x^{4} + x} x}{4 \, {\left (x^{5} - 1\right )}} \]
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Timed out. \[ \int \frac {\left (3+2 x^5\right ) \sqrt {x-2 x^4-x^6}}{\left (-1+x^5\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (3+2 x^5\right ) \sqrt {x-2 x^4-x^6}}{\left (-1+x^5\right )^2} \, dx=\int { \frac {\sqrt {-x^{6} - 2 \, x^{4} + x} {\left (2 \, x^{5} + 3\right )}}{{\left (x^{5} - 1\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (3+2 x^5\right ) \sqrt {x-2 x^4-x^6}}{\left (-1+x^5\right )^2} \, dx=\int { \frac {\sqrt {-x^{6} - 2 \, x^{4} + x} {\left (2 \, x^{5} + 3\right )}}{{\left (x^{5} - 1\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (3+2 x^5\right ) \sqrt {x-2 x^4-x^6}}{\left (-1+x^5\right )^2} \, dx=\int \frac {\left (2\,x^5+3\right )\,\sqrt {-x^6-2\,x^4+x}}{{\left (x^5-1\right )}^2} \,d x \]
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