Integrand size = 30, antiderivative size = 135 \[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx=\frac {4 \sqrt [4]{1-2 x^4+x^5}}{x}-2^{3/4} \arctan \left (\frac {2^{3/4} x \sqrt [4]{1-2 x^4+x^5}}{\sqrt {2} x^2-\sqrt {1-2 x^4+x^5}}\right )-2^{3/4} \text {arctanh}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{1-2 x^4+x^5}}{2 x^2+\sqrt {2} \sqrt {1-2 x^4+x^5}}\right ) \]
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\[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx=\int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt [4]{1-2 x^4+x^5}}{-1-x}-\frac {4 \sqrt [4]{1-2 x^4+x^5}}{x^2}+\frac {\left (1-2 x+3 x^2+x^3\right ) \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4}\right ) \, dx \\ & = -\left (4 \int \frac {\sqrt [4]{1-2 x^4+x^5}}{x^2} \, dx\right )+\int \frac {\sqrt [4]{1-2 x^4+x^5}}{-1-x} \, dx+\int \frac {\left (1-2 x+3 x^2+x^3\right ) \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4} \, dx \\ & = -\left (4 \int \frac {\sqrt [4]{1-2 x^4+x^5}}{x^2} \, dx\right )+\int \frac {\sqrt [4]{1-2 x^4+x^5}}{-1-x} \, dx+\int \left (\frac {\sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4}-\frac {2 x \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4}+\frac {3 x^2 \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4}+\frac {x^3 \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4}\right ) \, dx \\ & = -\left (2 \int \frac {x \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4} \, dx\right )+3 \int \frac {x^2 \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4} \, dx-4 \int \frac {\sqrt [4]{1-2 x^4+x^5}}{x^2} \, dx+\int \frac {\sqrt [4]{1-2 x^4+x^5}}{-1-x} \, dx+\int \frac {\sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4} \, dx+\int \frac {x^3 \sqrt [4]{1-2 x^4+x^5}}{1-x+x^2-x^3+x^4} \, dx \\ \end{align*}
Time = 1.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx=\frac {4 \sqrt [4]{1-2 x^4+x^5}}{x}-2^{3/4} \arctan \left (\frac {2^{3/4} x \sqrt [4]{1-2 x^4+x^5}}{\sqrt {2} x^2-\sqrt {1-2 x^4+x^5}}\right )-2^{3/4} \text {arctanh}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{1-2 x^4+x^5}}{2 x^2+\sqrt {2} \sqrt {1-2 x^4+x^5}}\right ) \]
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Time = 51.30 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.23
method | result | size |
pseudoelliptic | \(\frac {-\ln \left (\frac {\left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}} 2^{\frac {3}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{5}-2 x^{4}+1}}{-\left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}} 2^{\frac {3}{4}} x +\sqrt {2}\, x^{2}+\sqrt {x^{5}-2 x^{4}+1}}\right ) 2^{\frac {3}{4}} x -2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}}+x}{x}\right ) 2^{\frac {3}{4}} x -2 \arctan \left (\frac {2^{\frac {1}{4}} \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}}-x}{x}\right ) 2^{\frac {3}{4}} x +8 \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}}}{2 x}\) | \(166\) |
trager | \(\frac {4 \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}}}{x}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} x^{5}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} x^{4}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}} x^{3}-4 \sqrt {x^{5}-2 x^{4}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right ) x^{2}-4 \left (x^{5}-2 x^{4}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2}\right )}{\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3} x^{5}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3} x^{4}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{2} \left (x^{5}-2 x^{4}+1\right )^{\frac {1}{4}} x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right ) \sqrt {x^{5}-2 x^{4}+1}\, x^{2}-4 \left (x^{5}-2 x^{4}+1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+2\right )^{3}}{\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}\right )\) | \(337\) |
risch | \(\text {Expression too large to display}\) | \(1650\) |
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Result contains complex when optimal does not.
Time = 59.86 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.70 \[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx=-\frac {\left (-2\right )^{\frac {1}{4}} x \log \left (\frac {4 \, \sqrt {-2} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \left (-2\right )^{\frac {1}{4}} \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} - \left (-2\right )^{\frac {3}{4}} {\left (x^{5} - 4 \, x^{4} + 1\right )} + 4 \, {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{5} + 1}\right ) - i \, \left (-2\right )^{\frac {1}{4}} x \log \left (-\frac {4 \, \sqrt {-2} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 i \, \left (-2\right )^{\frac {1}{4}} \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} + i \, \left (-2\right )^{\frac {3}{4}} {\left (x^{5} - 4 \, x^{4} + 1\right )} - 4 \, {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{5} + 1}\right ) + i \, \left (-2\right )^{\frac {1}{4}} x \log \left (-\frac {4 \, \sqrt {-2} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 i \, \left (-2\right )^{\frac {1}{4}} \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} - i \, \left (-2\right )^{\frac {3}{4}} {\left (x^{5} - 4 \, x^{4} + 1\right )} - 4 \, {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{5} + 1}\right ) - \left (-2\right )^{\frac {1}{4}} x \log \left (\frac {4 \, \sqrt {-2} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \, \left (-2\right )^{\frac {1}{4}} \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} + \left (-2\right )^{\frac {3}{4}} {\left (x^{5} - 4 \, x^{4} + 1\right )} + 4 \, {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{5} + 1}\right ) - 8 \, {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{2 \, x} \]
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\[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx=\int \frac {\sqrt [4]{\left (x - 1\right ) \left (x^{4} - x^{3} - x^{2} - x - 1\right )} \left (x^{5} - 4\right )}{x^{2} \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}\, dx \]
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\[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx=\int { \frac {{\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{5} - 4\right )}}{{\left (x^{5} + 1\right )} x^{2}} \,d x } \]
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\[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx=\int { \frac {{\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{5} - 4\right )}}{{\left (x^{5} + 1\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx=\int \frac {\left (x^5-4\right )\,{\left (x^5-2\,x^4+1\right )}^{1/4}}{x^2\,\left (x^5+1\right )} \,d x \]
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