Integrand size = 40, antiderivative size = 199 \[ \int \frac {\left (-1+x^2\right ) \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx=\frac {\left (-1-3 x-x^2\right ) \sqrt {1+3 x^2+x^4}}{5 \left (1+x+x^2+x^3+x^4\right )}+\frac {1}{5} \text {RootSum}\left [1-2 \text {$\#$1}-6 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {3 \log (x)-3 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}-2 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-3 \log (x) \text {$\#$1}^2+3 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1-6 \text {$\#$1}+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]
Time = 1.30 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.57 \[ \int \frac {\left (-1+x^2\right ) \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx=\frac {\left (-1-3 x-x^2\right ) \sqrt {1+3 x^2+x^4}}{5 \left (1+x+x^2+x^3+x^4\right )}+\text {RootSum}\left [1-2 \text {$\#$1}-6 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-7 \log (x)+7 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^2+\log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1-6 \text {$\#$1}+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]-\frac {2}{5} \text {RootSum}\left [1-2 \text {$\#$1}-6 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-19 \log (x)+19 \log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}+\log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}-\log (x) \text {$\#$1}^2+\log \left (1+x^2+\sqrt {1+3 x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-1-6 \text {$\#$1}+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]
((-1 - 3*x - x^2)*Sqrt[1 + 3*x^2 + x^4])/(5*(1 + x + x^2 + x^3 + x^4)) + R ootSum[1 - 2*#1 - 6*#1^2 + 2*#1^3 + #1^4 & , (-7*Log[x] + 7*Log[1 + x^2 + Sqrt[1 + 3*x^2 + x^4] - x*#1] - Log[x]*#1^2 + Log[1 + x^2 + Sqrt[1 + 3*x^2 + x^4] - x*#1]*#1^2)/(-1 - 6*#1 + 3*#1^2 + 2*#1^3) & ] - (2*RootSum[1 - 2 *#1 - 6*#1^2 + 2*#1^3 + #1^4 & , (-19*Log[x] + 19*Log[1 + x^2 + Sqrt[1 + 3 *x^2 + x^4] - x*#1] - Log[x]*#1 + Log[1 + x^2 + Sqrt[1 + 3*x^2 + x^4] - x* #1]*#1 - Log[x]*#1^2 + Log[1 + x^2 + Sqrt[1 + 3*x^2 + x^4] - x*#1]*#1^2)/( -1 - 6*#1 + 3*#1^2 + 2*#1^3) & ])/5
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (x^2-1\right ) \left (x^2+x+1\right ) \sqrt {x^4+3 x^2+1}}{\left (x^4+x^3+x^2+x+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\sqrt {x^4+3 x^2+1} \left (-x^2-2 x-2\right )}{\left (x^4+x^3+x^2+x+1\right )^2}+\frac {\sqrt {x^4+3 x^2+1}}{x^4+x^3+x^2+x+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \int \frac {\sqrt {x^4+3 x^2+1}}{\left (x^4+x^3+x^2+x+1\right )^2}dx-2 \int \frac {x \sqrt {x^4+3 x^2+1}}{\left (x^4+x^3+x^2+x+1\right )^2}dx-\int \frac {x^2 \sqrt {x^4+3 x^2+1}}{\left (x^4+x^3+x^2+x+1\right )^2}dx+\int \frac {\sqrt {x^4+3 x^2+1}}{x^4+x^3+x^2+x+1}dx\) |
3.25.54.3.1 Defintions of rubi rules used
Time = 3.08 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.75
| method | result | size |
| risch | \(-\frac {\left (x^{2}+3 x +1\right ) \sqrt {x^{4}+3 x^{2}+1}}{5 \left (x^{4}+x^{3}+x^{2}+x +1\right )}+\frac {\left (5-3 \sqrt {5}\right ) \sqrt {10+2 \sqrt {5}}\, \operatorname {arctanh}\left (\frac {\left (x^{2}+1\right ) \sqrt {5}-\left (-1+x \right )^{2}}{\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {10-2 \sqrt {5}}}\right )}{100}+\frac {3 \,\operatorname {arctanh}\left (\frac {\left (x^{2}+1\right ) \sqrt {5}+\left (-1+x \right )^{2}}{\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {10+2 \sqrt {5}}}\right ) \left (\sqrt {5}+\frac {5}{3}\right ) \sqrt {10-2 \sqrt {5}}}{100}\) | \(150\) |
| default | \(\frac {\sqrt {5}\, \left (\sqrt {10+2 \sqrt {5}}\, \left (x^{4}+x^{3}+x^{2}+x +1\right ) \left (\sqrt {5}-3\right ) \operatorname {arctanh}\left (\frac {\left (x^{2}+1\right ) \sqrt {5}-\left (-1+x \right )^{2}}{\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {10-2 \sqrt {5}}}\right )+\sqrt {10-2 \sqrt {5}}\, \left (x^{4}+x^{3}+x^{2}+x +1\right ) \left (3+\sqrt {5}\right ) \operatorname {arctanh}\left (\frac {\left (x^{2}+1\right ) \sqrt {5}+\left (-1+x \right )^{2}}{\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {10+2 \sqrt {5}}}\right )-4 \sqrt {5}\, \sqrt {x^{4}+3 x^{2}+1}\, \left (x^{2}+3 x +1\right )\right )}{-125 x^{2}+25 \left (2 x^{2}+x +2\right )^{2}}\) | \(183\) |
| pseudoelliptic | \(\frac {\sqrt {5}\, \left (\sqrt {10+2 \sqrt {5}}\, \left (x^{4}+x^{3}+x^{2}+x +1\right ) \left (\sqrt {5}-3\right ) \operatorname {arctanh}\left (\frac {\left (x^{2}+1\right ) \sqrt {5}-\left (-1+x \right )^{2}}{\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {10-2 \sqrt {5}}}\right )+\sqrt {10-2 \sqrt {5}}\, \left (x^{4}+x^{3}+x^{2}+x +1\right ) \left (3+\sqrt {5}\right ) \operatorname {arctanh}\left (\frac {\left (x^{2}+1\right ) \sqrt {5}+\left (-1+x \right )^{2}}{\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {10+2 \sqrt {5}}}\right )-4 \sqrt {5}\, \sqrt {x^{4}+3 x^{2}+1}\, \left (x^{2}+3 x +1\right )\right )}{-125 x^{2}+25 \left (2 x^{2}+x +2\right )^{2}}\) | \(183\) |
| trager | \(-\frac {\left (x^{2}+3 x +1\right ) \sqrt {x^{4}+3 x^{2}+1}}{5 \left (x^{4}+x^{3}+x^{2}+x +1\right )}-\frac {\ln \left (-\frac {-25 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{5} x -20 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3} x^{2}+70 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3} x -20 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3}+44 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right ) x^{2}-33 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right ) x +16 \sqrt {x^{4}+3 x^{2}+1}+44 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )}{5 x \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{2}-4 x^{2}-7 x -4}\right ) \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3}}{2}+\frac {9 \ln \left (-\frac {-25 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{5} x -20 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3} x^{2}+70 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3} x -20 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3}+44 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right ) x^{2}-33 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right ) x +16 \sqrt {x^{4}+3 x^{2}+1}+44 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )}{5 x \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{2}-4 x^{2}-7 x -4}\right ) \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )}{10}+\frac {\operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-25 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{5} x +20 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3} x^{2}+70 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3} x +20 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{3}-28 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right ) x^{2}-49 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right ) x +16 \sqrt {x^{4}+3 x^{2}+1}-28 \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )}{4 x^{2}+5 x \operatorname {RootOf}\left (5 \textit {\_Z}^{4}-10 \textit {\_Z}^{2}+1\right )^{2}-3 x +4}\right )}{5}\) | \(595\) |
| elliptic | \(\frac {\frac {\left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{7}}{5}-\frac {18 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{6}}{5}+\frac {46 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{5}}{5}-4 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{4}-12 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{3}+\frac {66 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{2}}{5}+\frac {12 \sqrt {x^{4}+3 x^{2}+1}}{5}-\frac {12 x^{2}}{5}-\frac {29}{5}}{\left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{8}-2 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{7}+3 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{6}-14 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{5}+50 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{4}-124 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{3}+198 \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}\right )^{2}-172 \sqrt {x^{4}+3 x^{2}+1}+172 x^{2}+61}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{7}+3 \textit {\_Z}^{6}-14 \textit {\_Z}^{5}+50 \textit {\_Z}^{4}-124 \textit {\_Z}^{3}+198 \textit {\_Z}^{2}-172 \textit {\_Z} +61\right )}{\sum }\frac {\left (\textit {\_R}^{6}-34 \textit {\_R}^{5}+100 \textit {\_R}^{4}+80 \textit {\_R}^{3}-790 \textit {\_R}^{2}+1256 \textit {\_R} -634\right ) \ln \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}-\textit {\_R} \right )}{4 \textit {\_R}^{7}-7 \textit {\_R}^{6}+9 \textit {\_R}^{5}-35 \textit {\_R}^{4}+100 \textit {\_R}^{3}-186 \textit {\_R}^{2}+198 \textit {\_R} -86}\right )}{10}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{7}+3 \textit {\_Z}^{6}-14 \textit {\_Z}^{5}+50 \textit {\_Z}^{4}-124 \textit {\_Z}^{3}+198 \textit {\_Z}^{2}-172 \textit {\_Z} +61\right )}{\sum }\frac {\left (\textit {\_R}^{6}-8 \textit {\_R}^{5}+17 \textit {\_R}^{4}+32 \textit {\_R}^{3}-191 \textit {\_R}^{2}+286 \textit {\_R} -142\right ) \ln \left (\sqrt {x^{4}+3 x^{2}+1}-x^{2}-\textit {\_R} \right )}{4 \textit {\_R}^{7}-7 \textit {\_R}^{6}+9 \textit {\_R}^{5}-35 \textit {\_R}^{4}+100 \textit {\_R}^{3}-186 \textit {\_R}^{2}+198 \textit {\_R} -86}\right )}{2}+\frac {\left (\frac {-\frac {\left (x^{4}+3 x^{2}+1\right )^{\frac {3}{2}} \sqrt {2}}{10 x^{3}}+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{4 x}}{\frac {\left (x^{4}+3 x^{2}+1\right )^{2}}{4 x^{4}}-\frac {5 \left (x^{4}+3 x^{2}+1\right )}{4 x^{2}}+\frac {5}{4}}-\frac {\left (5+3 \sqrt {5}\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{x \sqrt {5+\sqrt {5}}}\right )}{25 \sqrt {5+\sqrt {5}}}-\frac {\left (3 \sqrt {5}-5\right ) \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{x \sqrt {5-\sqrt {5}}}\right )}{25 \sqrt {5-\sqrt {5}}}\right ) \sqrt {2}}{2}\) | \(777\) |
-1/5*(x^2+3*x+1)/(x^4+x^3+x^2+x+1)*(x^4+3*x^2+1)^(1/2)+1/100*(5-3*5^(1/2)) *(10+2*5^(1/2))^(1/2)*arctanh(((x^2+1)*5^(1/2)-(-1+x)^2)/(x^4+3*x^2+1)^(1/ 2)/(10-2*5^(1/2))^(1/2))+3/100*arctanh(1/(x^4+3*x^2+1)^(1/2)/(10+2*5^(1/2) )^(1/2)*((x^2+1)*5^(1/2)+(-1+x)^2))*(5^(1/2)+5/3)*(10-2*5^(1/2))^(1/2)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.37 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.69 \[ \int \frac {\left (-1+x^2\right ) \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx=-\frac {\sqrt {5} {\left (x^{4} + x^{3} + x^{2} + x + 1\right )} \sqrt {-8 \, \sqrt {5} + 20} \log \left (-\frac {10 \, \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (2 \, x^{2} + \sqrt {5} x + x + 2\right )} + {\left (5 \, x^{4} + 10 \, x^{3} + 20 \, x^{2} + \sqrt {5} {\left (x^{4} + 6 \, x^{3} + 6 \, x^{2} + 6 \, x + 1\right )} + 10 \, x + 5\right )} \sqrt {-8 \, \sqrt {5} + 20}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) - \sqrt {5} {\left (x^{4} + x^{3} + x^{2} + x + 1\right )} \sqrt {-8 \, \sqrt {5} + 20} \log \left (-\frac {10 \, \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (2 \, x^{2} + \sqrt {5} x + x + 2\right )} - {\left (5 \, x^{4} + 10 \, x^{3} + 20 \, x^{2} + \sqrt {5} {\left (x^{4} + 6 \, x^{3} + 6 \, x^{2} + 6 \, x + 1\right )} + 10 \, x + 5\right )} \sqrt {-8 \, \sqrt {5} + 20}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) - 2 \, \sqrt {5} {\left (x^{4} + x^{3} + x^{2} + x + 1\right )} \sqrt {2 \, \sqrt {5} + 5} \log \left (-\frac {2 \, {\left (5 \, \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (2 \, x^{2} - \sqrt {5} x + x + 2\right )} + {\left (5 \, x^{4} + 10 \, x^{3} + 20 \, x^{2} - \sqrt {5} {\left (x^{4} + 6 \, x^{3} + 6 \, x^{2} + 6 \, x + 1\right )} + 10 \, x + 5\right )} \sqrt {2 \, \sqrt {5} + 5}\right )}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) + 2 \, \sqrt {5} {\left (x^{4} + x^{3} + x^{2} + x + 1\right )} \sqrt {2 \, \sqrt {5} + 5} \log \left (-\frac {2 \, {\left (5 \, \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (2 \, x^{2} - \sqrt {5} x + x + 2\right )} - {\left (5 \, x^{4} + 10 \, x^{3} + 20 \, x^{2} - \sqrt {5} {\left (x^{4} + 6 \, x^{3} + 6 \, x^{2} + 6 \, x + 1\right )} + 10 \, x + 5\right )} \sqrt {2 \, \sqrt {5} + 5}\right )}}{x^{4} + x^{3} + x^{2} + x + 1}\right ) + 20 \, \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + 3 \, x + 1\right )}}{100 \, {\left (x^{4} + x^{3} + x^{2} + x + 1\right )}} \]
-1/100*(sqrt(5)*(x^4 + x^3 + x^2 + x + 1)*sqrt(-8*sqrt(5) + 20)*log(-(10*s qrt(x^4 + 3*x^2 + 1)*(2*x^2 + sqrt(5)*x + x + 2) + (5*x^4 + 10*x^3 + 20*x^ 2 + sqrt(5)*(x^4 + 6*x^3 + 6*x^2 + 6*x + 1) + 10*x + 5)*sqrt(-8*sqrt(5) + 20))/(x^4 + x^3 + x^2 + x + 1)) - sqrt(5)*(x^4 + x^3 + x^2 + x + 1)*sqrt(- 8*sqrt(5) + 20)*log(-(10*sqrt(x^4 + 3*x^2 + 1)*(2*x^2 + sqrt(5)*x + x + 2) - (5*x^4 + 10*x^3 + 20*x^2 + sqrt(5)*(x^4 + 6*x^3 + 6*x^2 + 6*x + 1) + 10 *x + 5)*sqrt(-8*sqrt(5) + 20))/(x^4 + x^3 + x^2 + x + 1)) - 2*sqrt(5)*(x^4 + x^3 + x^2 + x + 1)*sqrt(2*sqrt(5) + 5)*log(-2*(5*sqrt(x^4 + 3*x^2 + 1)* (2*x^2 - sqrt(5)*x + x + 2) + (5*x^4 + 10*x^3 + 20*x^2 - sqrt(5)*(x^4 + 6* x^3 + 6*x^2 + 6*x + 1) + 10*x + 5)*sqrt(2*sqrt(5) + 5))/(x^4 + x^3 + x^2 + x + 1)) + 2*sqrt(5)*(x^4 + x^3 + x^2 + x + 1)*sqrt(2*sqrt(5) + 5)*log(-2* (5*sqrt(x^4 + 3*x^2 + 1)*(2*x^2 - sqrt(5)*x + x + 2) - (5*x^4 + 10*x^3 + 2 0*x^2 - sqrt(5)*(x^4 + 6*x^3 + 6*x^2 + 6*x + 1) + 10*x + 5)*sqrt(2*sqrt(5) + 5))/(x^4 + x^3 + x^2 + x + 1)) + 20*sqrt(x^4 + 3*x^2 + 1)*(x^2 + 3*x + 1))/(x^4 + x^3 + x^2 + x + 1)
Timed out. \[ \int \frac {\left (-1+x^2\right ) \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx=\text {Timed out} \]
Not integrable
Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.20 \[ \int \frac {\left (-1+x^2\right ) \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx=\int { \frac {\sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + x + 1\right )} {\left (x^{2} - 1\right )}}{{\left (x^{4} + x^{3} + x^{2} + x + 1\right )}^{2}} \,d x } \]
Not integrable
Time = 0.35 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.20 \[ \int \frac {\left (-1+x^2\right ) \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx=\int { \frac {\sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + x + 1\right )} {\left (x^{2} - 1\right )}}{{\left (x^{4} + x^{3} + x^{2} + x + 1\right )}^{2}} \,d x } \]
Not integrable
Time = 8.45 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.20 \[ \int \frac {\left (-1+x^2\right ) \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}}{\left (1+x+x^2+x^3+x^4\right )^2} \, dx=\int \frac {\left (x^2-1\right )\,\sqrt {x^4+3\,x^2+1}\,\left (x^2+x+1\right )}{{\left (x^4+x^3+x^2+x+1\right )}^2} \,d x \]