Integrand size = 38, antiderivative size = 123 \[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\frac {2 \sqrt [4]{-1+x^2+x^4} \left (1-x^2+9 x^4\right )}{5 x^5}+\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{-x^2+\sqrt {-1+x^2+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{x^2+\sqrt {-1+x^2+x^4}}\right ) \] Output:
2/5*(x^4+x^2-1)^(1/4)*(9*x^4-x^2+1)/x^5+2^(1/2)*arctan(2^(1/2)*x*(x^4+x^2- 1)^(1/4)/(-x^2+(x^4+x^2-1)^(1/2)))-2^(1/2)*arctanh(2^(1/2)*x*(x^4+x^2-1)^( 1/4)/(x^2+(x^4+x^2-1)^(1/2)))
Time = 0.32 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\frac {2 \sqrt [4]{-1+x^2+x^4} \left (1-x^2+9 x^4\right )}{5 x^5}+\sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{-x^2+\sqrt {-1+x^2+x^4}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^2+x^4}}{x^2+\sqrt {-1+x^2+x^4}}\right ) \] Input:
Integrate[((-2 + x^2)*(-1 + x^2)*(-1 + x^2 + x^4)^(1/4))/(x^6*(-1 + x^2 + 2*x^4)),x]
Output:
(2*(-1 + x^2 + x^4)^(1/4)*(1 - x^2 + 9*x^4))/(5*x^5) + Sqrt[2]*ArcTan[(Sqr t[2]*x*(-1 + x^2 + x^4)^(1/4))/(-x^2 + Sqrt[-1 + x^2 + x^4])] - Sqrt[2]*Ar cTanh[(Sqrt[2]*x*(-1 + x^2 + x^4)^(1/4))/(x^2 + Sqrt[-1 + x^2 + x^4])]
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.
| \(\displaystyle \int \frac {\left (x^2-2\right ) \left (x^2-1\right ) \sqrt [4]{x^4+x^2-1}}{x^6 \left (2 x^4+x^2-1\right )} \, dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {2 \sqrt [4]{x^4+x^2-1}}{x^2+1}+\frac {4 \sqrt [4]{x^4+x^2-1}}{2 x^2-1}-\frac {4 \sqrt [4]{x^4+x^2-1}}{x^2}+\frac {\sqrt [4]{x^4+x^2-1}}{x^4}-\frac {2 \sqrt [4]{x^4+x^2-1}}{x^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {\sqrt [4]{x^4+x^2-1}}{x^2+1}dx+4 \int \frac {\sqrt [4]{x^4+x^2-1}}{2 x^2-1}dx+\frac {4 \sqrt [4]{x^4+x^2-1} \operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{4},-\frac {1}{4},\frac {1}{2},-\frac {2 x^2}{1-\sqrt {5}},-\frac {2 x^2}{1+\sqrt {5}}\right )}{x \sqrt [4]{\frac {2 x^2}{1-\sqrt {5}}+1} \sqrt [4]{\frac {2 x^2}{1+\sqrt {5}}+1}}-\frac {4 \sqrt [4]{x^4+x^2-1} \left (\frac {2 x^2}{1+\sqrt {5}}+1\right ) \left (\left (2 \left (1+\sqrt {5}\right ) x^4-\left (13+3 \sqrt {5}\right ) x^2+3 \left (1+\sqrt {5}\right )\right ) \operatorname {Gamma}\left (-\frac {1}{4}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,-\frac {1}{2},-\frac {2 \sqrt {5} x^2}{2-\left (1+\sqrt {5}\right ) x^2}\right )-4 x^2 \left (2 \sqrt {5} x^2+\sqrt {5}+5\right ) \operatorname {Gamma}\left (\frac {3}{4}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},2,\frac {1}{2},-\frac {2 \sqrt {5} x^2}{2-\left (1+\sqrt {5}\right ) x^2}\right )\right )}{15 \left (3+\sqrt {5}\right ) x^5 \left (2 x^2-\sqrt {5}+1\right ) \operatorname {Gamma}\left (-\frac {1}{4}\right )}-\frac {\sqrt [4]{x^4+x^2-1} \left (\frac {2 x^2}{1+\sqrt {5}}+1\right )^{5/4} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {1}{4},-\frac {1}{2},-\frac {2 \left (\frac {x^2}{1-\sqrt {5}}-\frac {x^2}{1+\sqrt {5}}\right )}{\frac {2 x^2}{1+\sqrt {5}}+1}\right )}{3 x^3 \sqrt [4]{\frac {2 x^2}{1-\sqrt {5}}+1}}\) |
Input:
Int[((-2 + x^2)*(-1 + x^2)*(-1 + x^2 + x^4)^(1/4))/(x^6*(-1 + x^2 + 2*x^4) ),x]
Output:
$Aborted
Time = 6.52 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.23
| method | result | size |
| pseudoelliptic | \(\frac {-5 x^{5} \left (\ln \left (\frac {\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}+\sqrt {x^{4}+x^{2}-1}}{-\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} x \sqrt {2}+x^{2}+\sqrt {x^{4}+x^{2}-1}}\right )+2 \arctan \left (\frac {\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right )+2 \arctan \left (\frac {\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right )\right ) \sqrt {2}+4 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \left (9 x^{4}-x^{2}+1\right )}{10 x^{5}}\) | \(151\) |
| trager | \(\frac {2 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \left (9 x^{4}-x^{2}+1\right )}{5 x^{5}}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-2 \sqrt {x^{4}+x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \left (x^{4}+x^{2}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\left (x^{2}+1\right ) \left (2 x^{2}-1\right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \sqrt {x^{4}+x^{2}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \left (x^{4}+x^{2}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{\left (x^{2}+1\right ) \left (2 x^{2}-1\right )}\right )\) | \(239\) |
| risch | \(\frac {\frac {18}{5} x^{8}+\frac {16}{5} x^{6}-\frac {18}{5} x^{4}+\frac {4}{5} x^{2}-\frac {2}{5}}{x^{5} \left (x^{4}+x^{2}-1\right )^{\frac {3}{4}}}+\frac {\left (-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{9}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{10}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{7}+x^{8} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{5}-3 x^{6} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {3}{4}} x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (x^{4}+x^{2}-1\right )^{2} \left (x^{2}+1\right ) \left (2 x^{2}-1\right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{10}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{9}-x^{8} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {3}{4}} x^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{7}+3 x^{6} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{4}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+2 \sqrt {x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{12}+3 x^{10}-5 x^{6}+3 x^{2}-1\right )^{\frac {1}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (x^{4}+x^{2}-1\right )^{2} \left (x^{2}+1\right ) \left (2 x^{2}-1\right )}\right )\right ) {\left (\left (x^{4}+x^{2}-1\right )^{3}\right )}^{\frac {1}{4}}}{\left (x^{4}+x^{2}-1\right )^{\frac {3}{4}}}\) | \(849\) |
Input:
int((x^2-2)*(x^2-1)*(x^4+x^2-1)^(1/4)/x^6/(2*x^4+x^2-1),x,method=_RETURNVE RBOSE)
Output:
1/10*(-5*x^5*(ln(((x^4+x^2-1)^(1/4)*x*2^(1/2)+x^2+(x^4+x^2-1)^(1/2))/(-(x^ 4+x^2-1)^(1/4)*x*2^(1/2)+x^2+(x^4+x^2-1)^(1/2)))+2*arctan(((x^4+x^2-1)^(1/ 4)*2^(1/2)-x)/x)+2*arctan(((x^4+x^2-1)^(1/4)*2^(1/2)+x)/x))*2^(1/2)+4*(x^4 +x^2-1)^(1/4)*(9*x^4-x^2+1))/x^5
Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (104) = 208\).
Time = 5.40 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.97 \[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\frac {20 \, \sqrt {2} x^{5} \arctan \left (\frac {\sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x}{x^{2} - 1}\right ) - 5 \, \sqrt {2} x^{5} \log \left (\frac {2 \, x^{4} + 2 \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{4} + x^{2} - 1} x^{2} + 2 \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x + x^{2} - 1}{2 \, x^{4} + x^{2} - 1}\right ) + 5 \, \sqrt {2} x^{5} \log \left (\frac {2 \, x^{4} - 2 \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{4} + x^{2} - 1} x^{2} - 2 \, \sqrt {2} {\left (x^{4} + x^{2} - 1\right )}^{\frac {3}{4}} x + x^{2} - 1}{2 \, x^{4} + x^{2} - 1}\right ) + 8 \, {\left (9 \, x^{4} - x^{2} + 1\right )} {\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}}}{20 \, x^{5}} \] Input:
integrate((x^2-2)*(x^2-1)*(x^4+x^2-1)^(1/4)/x^6/(2*x^4+x^2-1),x, algorithm ="fricas")
Output:
1/20*(20*sqrt(2)*x^5*arctan((sqrt(2)*(x^4 + x^2 - 1)^(1/4)*x^3 + sqrt(2)*( x^4 + x^2 - 1)^(3/4)*x)/(x^2 - 1)) - 5*sqrt(2)*x^5*log((2*x^4 + 2*sqrt(2)* (x^4 + x^2 - 1)^(1/4)*x^3 + 4*sqrt(x^4 + x^2 - 1)*x^2 + 2*sqrt(2)*(x^4 + x ^2 - 1)^(3/4)*x + x^2 - 1)/(2*x^4 + x^2 - 1)) + 5*sqrt(2)*x^5*log((2*x^4 - 2*sqrt(2)*(x^4 + x^2 - 1)^(1/4)*x^3 + 4*sqrt(x^4 + x^2 - 1)*x^2 - 2*sqrt( 2)*(x^4 + x^2 - 1)^(3/4)*x + x^2 - 1)/(2*x^4 + x^2 - 1)) + 8*(9*x^4 - x^2 + 1)*(x^4 + x^2 - 1)^(1/4))/x^5
\[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} - 2\right ) \sqrt [4]{x^{4} + x^{2} - 1}}{x^{6} \left (x^{2} + 1\right ) \left (2 x^{2} - 1\right )}\, dx \] Input:
integrate((x**2-2)*(x**2-1)*(x**4+x**2-1)**(1/4)/x**6/(2*x**4+x**2-1),x)
Output:
Integral((x - 1)*(x + 1)*(x**2 - 2)*(x**4 + x**2 - 1)**(1/4)/(x**6*(x**2 + 1)*(2*x**2 - 1)), x)
\[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\int { \frac {{\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )} {\left (x^{2} - 2\right )}}{{\left (2 \, x^{4} + x^{2} - 1\right )} x^{6}} \,d x } \] Input:
integrate((x^2-2)*(x^2-1)*(x^4+x^2-1)^(1/4)/x^6/(2*x^4+x^2-1),x, algorithm ="maxima")
Output:
integrate((x^4 + x^2 - 1)^(1/4)*(x^2 - 1)*(x^2 - 2)/((2*x^4 + x^2 - 1)*x^6 ), x)
\[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\int { \frac {{\left (x^{4} + x^{2} - 1\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )} {\left (x^{2} - 2\right )}}{{\left (2 \, x^{4} + x^{2} - 1\right )} x^{6}} \,d x } \] Input:
integrate((x^2-2)*(x^2-1)*(x^4+x^2-1)^(1/4)/x^6/(2*x^4+x^2-1),x, algorithm ="giac")
Output:
integrate((x^4 + x^2 - 1)^(1/4)*(x^2 - 1)*(x^2 - 2)/((2*x^4 + x^2 - 1)*x^6 ), x)
Timed out. \[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\int \frac {\left (x^2-1\right )\,\left (x^2-2\right )\,{\left (x^4+x^2-1\right )}^{1/4}}{x^6\,\left (2\,x^4+x^2-1\right )} \,d x \] Input:
int(((x^2 - 1)*(x^2 - 2)*(x^2 + x^4 - 1)^(1/4))/(x^6*(x^2 + 2*x^4 - 1)),x)
Output:
int(((x^2 - 1)*(x^2 - 2)*(x^2 + x^4 - 1)^(1/4))/(x^6*(x^2 + 2*x^4 - 1)), x )
\[ \int \frac {\left (-2+x^2\right ) \left (-1+x^2\right ) \sqrt [4]{-1+x^2+x^4}}{x^6 \left (-1+x^2+2 x^4\right )} \, dx=\frac {217612 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} x^{4}-75918 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} x^{2}+1368 \left (x^{4}+x^{2}-1\right )^{\frac {1}{4}}-223650 \left (\int \frac {\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}}}{2 x^{12}+3 x^{10}-2 x^{8}-2 x^{6}+x^{4}}d x \right ) x^{5}+615325 \left (\int \frac {\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}}}{2 x^{10}+3 x^{8}-2 x^{6}-2 x^{4}+x^{2}}d x \right ) x^{5}+102075 \left (\int \frac {\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}}}{2 x^{8}+3 x^{6}-2 x^{4}-2 x^{2}+1}d x \right ) x^{5}-86060 \left (\int \frac {\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} x^{4}}{2 x^{8}+3 x^{6}-2 x^{4}-2 x^{2}+1}d x \right ) x^{5}-843480 \left (\int \frac {\left (x^{4}+x^{2}-1\right )^{\frac {1}{4}} x^{2}}{2 x^{8}+3 x^{6}-2 x^{4}-2 x^{2}+1}d x \right ) x^{5}}{3420 x^{5}} \] Input:
int((x^2-2)*(x^2-1)*(x^4+x^2-1)^(1/4)/x^6/(2*x^4+x^2-1),x)
Output:
(217612*(x**4 + x**2 - 1)**(1/4)*x**4 - 75918*(x**4 + x**2 - 1)**(1/4)*x** 2 + 1368*(x**4 + x**2 - 1)**(1/4) - 223650*int((x**4 + x**2 - 1)**(1/4)/(2 *x**12 + 3*x**10 - 2*x**8 - 2*x**6 + x**4),x)*x**5 + 615325*int((x**4 + x* *2 - 1)**(1/4)/(2*x**10 + 3*x**8 - 2*x**6 - 2*x**4 + x**2),x)*x**5 + 10207 5*int((x**4 + x**2 - 1)**(1/4)/(2*x**8 + 3*x**6 - 2*x**4 - 2*x**2 + 1),x)* x**5 - 86060*int(((x**4 + x**2 - 1)**(1/4)*x**4)/(2*x**8 + 3*x**6 - 2*x**4 - 2*x**2 + 1),x)*x**5 - 843480*int(((x**4 + x**2 - 1)**(1/4)*x**2)/(2*x** 8 + 3*x**6 - 2*x**4 - 2*x**2 + 1),x)*x**5)/(3420*x**5)