Integrand size = 30, antiderivative size = 135 \[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx=\frac {4 \sqrt [4]{1-2 x^4+x^5}}{x}-2^{3/4} \arctan \left (\frac {2^{3/4} x \sqrt [4]{1-2 x^4+x^5}}{\sqrt {2} x^2-\sqrt {1-2 x^4+x^5}}\right )-2^{3/4} \text {arctanh}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{1-2 x^4+x^5}}{2 x^2+\sqrt {2} \sqrt {1-2 x^4+x^5}}\right ) \]
4*(x^5-2*x^4+1)^(1/4)/x-2^(3/4)*arctan(2^(3/4)*x*(x^5-2*x^4+1)^(1/4)/(2^(1 /2)*x^2-(x^5-2*x^4+1)^(1/2)))-2^(3/4)*arctanh(2*2^(1/4)*x*(x^5-2*x^4+1)^(1 /4)/(2*x^2+2^(1/2)*(x^5-2*x^4+1)^(1/2)))
Time = 1.14 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx=\frac {4 \sqrt [4]{1-2 x^4+x^5}}{x}-2^{3/4} \arctan \left (\frac {2^{3/4} x \sqrt [4]{1-2 x^4+x^5}}{\sqrt {2} x^2-\sqrt {1-2 x^4+x^5}}\right )-2^{3/4} \text {arctanh}\left (\frac {2 \sqrt [4]{2} x \sqrt [4]{1-2 x^4+x^5}}{2 x^2+\sqrt {2} \sqrt {1-2 x^4+x^5}}\right ) \]
(4*(1 - 2*x^4 + x^5)^(1/4))/x - 2^(3/4)*ArcTan[(2^(3/4)*x*(1 - 2*x^4 + x^5 )^(1/4))/(Sqrt[2]*x^2 - Sqrt[1 - 2*x^4 + x^5])] - 2^(3/4)*ArcTanh[(2*2^(1/ 4)*x*(1 - 2*x^4 + x^5)^(1/4))/(2*x^2 + Sqrt[2]*Sqrt[1 - 2*x^4 + x^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.
| \(\displaystyle \int \frac {\left (x^5-4\right ) \sqrt [4]{x^5-2 x^4+1}}{x^2 \left (x^5+1\right )} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\sqrt [4]{x^5-2 x^4+1}}{-x-1}-\frac {4 \sqrt [4]{x^5-2 x^4+1}}{x^2}+\frac {\sqrt [4]{x^5-2 x^4+1} \left (x^3+3 x^2-2 x+1\right )}{x^4-x^3+x^2-x+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {\sqrt [4]{x^5-2 x^4+1}}{-x-1}dx-4 \int \frac {\sqrt [4]{x^5-2 x^4+1}}{x^2}dx+\int \frac {\sqrt [4]{x^5-2 x^4+1}}{x^4-x^3+x^2-x+1}dx-2 \int \frac {x \sqrt [4]{x^5-2 x^4+1}}{x^4-x^3+x^2-x+1}dx+3 \int \frac {x^2 \sqrt [4]{x^5-2 x^4+1}}{x^4-x^3+x^2-x+1}dx+\int \frac {x^3 \sqrt [4]{x^5-2 x^4+1}}{x^4-x^3+x^2-x+1}dx\) |
3.20.37.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
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 | \(\text {Expression too large to display}\) | \(1650\) |
1/2*(-ln(((x^5-2*x^4+1)^(1/4)*2^(3/4)*x+2^(1/2)*x^2+(x^5-2*x^4+1)^(1/2))/( -(x^5-2*x^4+1)^(1/4)*2^(3/4)*x+2^(1/2)*x^2+(x^5-2*x^4+1)^(1/2)))*2^(3/4)*x -2*arctan((2^(1/4)*(x^5-2*x^4+1)^(1/4)+x)/x)*2^(3/4)*x-2*arctan((2^(1/4)*( x^5-2*x^4+1)^(1/4)-x)/x)*2^(3/4)*x+8*(x^5-2*x^4+1)^(1/4))/x
Result contains complex when optimal does not.
Time = 59.86 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.70 \[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx=-\frac {\left (-2\right )^{\frac {1}{4}} x \log \left (\frac {4 \, \sqrt {-2} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \left (-2\right )^{\frac {1}{4}} \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} - \left (-2\right )^{\frac {3}{4}} {\left (x^{5} - 4 \, x^{4} + 1\right )} + 4 \, {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{5} + 1}\right ) - i \, \left (-2\right )^{\frac {1}{4}} x \log \left (-\frac {4 \, \sqrt {-2} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} + 4 i \, \left (-2\right )^{\frac {1}{4}} \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} + i \, \left (-2\right )^{\frac {3}{4}} {\left (x^{5} - 4 \, x^{4} + 1\right )} - 4 \, {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{5} + 1}\right ) + i \, \left (-2\right )^{\frac {1}{4}} x \log \left (-\frac {4 \, \sqrt {-2} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 i \, \left (-2\right )^{\frac {1}{4}} \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} - i \, \left (-2\right )^{\frac {3}{4}} {\left (x^{5} - 4 \, x^{4} + 1\right )} - 4 \, {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{5} + 1}\right ) - \left (-2\right )^{\frac {1}{4}} x \log \left (\frac {4 \, \sqrt {-2} {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} x^{3} - 4 \, \left (-2\right )^{\frac {1}{4}} \sqrt {x^{5} - 2 \, x^{4} + 1} x^{2} + \left (-2\right )^{\frac {3}{4}} {\left (x^{5} - 4 \, x^{4} + 1\right )} + 4 \, {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {3}{4}} x}{x^{5} + 1}\right ) - 8 \, {\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}}}{2 \, x} \]
-1/2*((-2)^(1/4)*x*log((4*sqrt(-2)*(x^5 - 2*x^4 + 1)^(1/4)*x^3 + 4*(-2)^(1 /4)*sqrt(x^5 - 2*x^4 + 1)*x^2 - (-2)^(3/4)*(x^5 - 4*x^4 + 1) + 4*(x^5 - 2* x^4 + 1)^(3/4)*x)/(x^5 + 1)) - I*(-2)^(1/4)*x*log(-(4*sqrt(-2)*(x^5 - 2*x^ 4 + 1)^(1/4)*x^3 + 4*I*(-2)^(1/4)*sqrt(x^5 - 2*x^4 + 1)*x^2 + I*(-2)^(3/4) *(x^5 - 4*x^4 + 1) - 4*(x^5 - 2*x^4 + 1)^(3/4)*x)/(x^5 + 1)) + I*(-2)^(1/4 )*x*log(-(4*sqrt(-2)*(x^5 - 2*x^4 + 1)^(1/4)*x^3 - 4*I*(-2)^(1/4)*sqrt(x^5 - 2*x^4 + 1)*x^2 - I*(-2)^(3/4)*(x^5 - 4*x^4 + 1) - 4*(x^5 - 2*x^4 + 1)^( 3/4)*x)/(x^5 + 1)) - (-2)^(1/4)*x*log((4*sqrt(-2)*(x^5 - 2*x^4 + 1)^(1/4)* x^3 - 4*(-2)^(1/4)*sqrt(x^5 - 2*x^4 + 1)*x^2 + (-2)^(3/4)*(x^5 - 4*x^4 + 1 ) + 4*(x^5 - 2*x^4 + 1)^(3/4)*x)/(x^5 + 1)) - 8*(x^5 - 2*x^4 + 1)^(1/4))/x
\[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx=\int \frac {\sqrt [4]{\left (x - 1\right ) \left (x^{4} - x^{3} - x^{2} - x - 1\right )} \left (x^{5} - 4\right )}{x^{2} \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}\, dx \]
Integral(((x - 1)*(x**4 - x**3 - x**2 - x - 1))**(1/4)*(x**5 - 4)/(x**2*(x + 1)*(x**4 - x**3 + x**2 - x + 1)), x)
\[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx=\int { \frac {{\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{5} - 4\right )}}{{\left (x^{5} + 1\right )} x^{2}} \,d x } \]
\[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx=\int { \frac {{\left (x^{5} - 2 \, x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{5} - 4\right )}}{{\left (x^{5} + 1\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (-4+x^5\right ) \sqrt [4]{1-2 x^4+x^5}}{x^2 \left (1+x^5\right )} \, dx=\int \frac {\left (x^5-4\right )\,{\left (x^5-2\,x^4+1\right )}^{1/4}}{x^2\,\left (x^5+1\right )} \,d x \]