Integrand size = 31, antiderivative size = 78 \[ \int \frac {(8+3 x) \sqrt [4]{-2-x+2 x^4}}{x^2 \left (2+x+x^4\right )} \, dx=-\frac {4 \sqrt [4]{-2-x+2 x^4}}{x}-2 \sqrt [4]{3} \arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-2-x+2 x^4}}\right )+2 \sqrt [4]{3} \text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-2-x+2 x^4}}\right ) \]
-4*(2*x^4-x-2)^(1/4)/x-2*3^(1/4)*arctan(3^(1/4)*x/(2*x^4-x-2)^(1/4))+2*3^( 1/4)*arctanh(3^(1/4)*x/(2*x^4-x-2)^(1/4))
Time = 0.42 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00 \[ \int \frac {(8+3 x) \sqrt [4]{-2-x+2 x^4}}{x^2 \left (2+x+x^4\right )} \, dx=-\frac {4 \sqrt [4]{-2-x+2 x^4}}{x}-2 \sqrt [4]{3} \arctan \left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-2-x+2 x^4}}\right )+2 \sqrt [4]{3} \text {arctanh}\left (\frac {\sqrt [4]{3} x}{\sqrt [4]{-2-x+2 x^4}}\right ) \]
(-4*(-2 - x + 2*x^4)^(1/4))/x - 2*3^(1/4)*ArcTan[(3^(1/4)*x)/(-2 - x + 2*x ^4)^(1/4)] + 2*3^(1/4)*ArcTanh[(3^(1/4)*x)/(-2 - x + 2*x^4)^(1/4)]
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.
| \(\displaystyle \int \frac {(3 x+8) \sqrt [4]{2 x^4-x-2}}{x^2 \left (x^4+x+2\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\sqrt [4]{2 x^4-x-2}}{2 x}+\frac {4 \sqrt [4]{2 x^4-x-2}}{x^2}+\frac {\sqrt [4]{2 x^4-x-2} \left (x^3-8 x^2+1\right )}{2 \left (x^4+x+2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{2} \int \frac {\sqrt [4]{2 x^4-x-2}}{x}dx+\frac {1}{2} \int \frac {\sqrt [4]{2 x^4-x-2}}{x^4+x+2}dx+\frac {1}{2} \int \frac {x^3 \sqrt [4]{2 x^4-x-2}}{x^4+x+2}dx+4 \int \frac {\sqrt [4]{2 x^4-x-2}}{x^2}dx-4 \int \frac {x^2 \sqrt [4]{2 x^4-x-2}}{x^4+x+2}dx\) |
3.11.31.3.1 Defintions of rubi rules used
Time = 11.62 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.15
| method | result | size |
| pseudoelliptic | \(\frac {x \left (\ln \left (\frac {3^{\frac {1}{4}} x +\left (2 x^{4}-x -2\right )^{\frac {1}{4}}}{-3^{\frac {1}{4}} x +\left (2 x^{4}-x -2\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {3^{\frac {3}{4}} \left (2 x^{4}-x -2\right )^{\frac {1}{4}}}{3 x}\right )\right ) 3^{\frac {1}{4}}-4 \left (2 x^{4}-x -2\right )^{\frac {1}{4}}}{x}\) | \(90\) |
| trager | \(-\frac {4 \left (2 x^{4}-x -2\right )^{\frac {1}{4}}}{x}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{3} x^{4}+6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2} \left (2 x^{4}-x -2\right )^{\frac {1}{4}} x^{3}+6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right ) \sqrt {2 x^{4}-x -2}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{3} x +6 \left (2 x^{4}-x -2\right )^{\frac {3}{4}} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{3}}{x^{4}+x +2}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2}\right ) \ln \left (-\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2}\right ) x^{4}+6 \operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2} \left (2 x^{4}-x -2\right )^{\frac {1}{4}} x^{3}+6 \sqrt {2 x^{4}-x -2}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2}\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2}\right ) x -6 \left (2 x^{4}-x -2\right )^{\frac {3}{4}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-3\right )^{2}\right )}{x^{4}+x +2}\right )\) | \(303\) |
| risch | \(\text {Expression too large to display}\) | \(1594\) |
(x*(ln((3^(1/4)*x+(2*x^4-x-2)^(1/4))/(-3^(1/4)*x+(2*x^4-x-2)^(1/4)))+2*arc tan(1/3*3^(3/4)/x*(2*x^4-x-2)^(1/4)))*3^(1/4)-4*(2*x^4-x-2)^(1/4))/x
Result contains complex when optimal does not.
Time = 8.67 (sec) , antiderivative size = 368, normalized size of antiderivative = 4.72 \[ \int \frac {(8+3 x) \sqrt [4]{-2-x+2 x^4}}{x^2 \left (2+x+x^4\right )} \, dx=\frac {3^{\frac {1}{4}} x \log \left (\frac {6 \, \sqrt {3} {\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}} x^{3} + 6 \cdot 3^{\frac {1}{4}} \sqrt {2 \, x^{4} - x - 2} x^{2} + 3^{\frac {3}{4}} {\left (5 \, x^{4} - x - 2\right )} + 6 \, {\left (2 \, x^{4} - x - 2\right )}^{\frac {3}{4}} x}{x^{4} + x + 2}\right ) - i \cdot 3^{\frac {1}{4}} x \log \left (-\frac {6 \, \sqrt {3} {\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}} x^{3} + 6 i \cdot 3^{\frac {1}{4}} \sqrt {2 \, x^{4} - x - 2} x^{2} - 3^{\frac {3}{4}} {\left (5 i \, x^{4} - i \, x - 2 i\right )} - 6 \, {\left (2 \, x^{4} - x - 2\right )}^{\frac {3}{4}} x}{x^{4} + x + 2}\right ) + i \cdot 3^{\frac {1}{4}} x \log \left (-\frac {6 \, \sqrt {3} {\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}} x^{3} - 6 i \cdot 3^{\frac {1}{4}} \sqrt {2 \, x^{4} - x - 2} x^{2} - 3^{\frac {3}{4}} {\left (-5 i \, x^{4} + i \, x + 2 i\right )} - 6 \, {\left (2 \, x^{4} - x - 2\right )}^{\frac {3}{4}} x}{x^{4} + x + 2}\right ) - 3^{\frac {1}{4}} x \log \left (\frac {6 \, \sqrt {3} {\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}} x^{3} - 6 \cdot 3^{\frac {1}{4}} \sqrt {2 \, x^{4} - x - 2} x^{2} - 3^{\frac {3}{4}} {\left (5 \, x^{4} - x - 2\right )} + 6 \, {\left (2 \, x^{4} - x - 2\right )}^{\frac {3}{4}} x}{x^{4} + x + 2}\right ) - 8 \, {\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}}}{2 \, x} \]
1/2*(3^(1/4)*x*log((6*sqrt(3)*(2*x^4 - x - 2)^(1/4)*x^3 + 6*3^(1/4)*sqrt(2 *x^4 - x - 2)*x^2 + 3^(3/4)*(5*x^4 - x - 2) + 6*(2*x^4 - x - 2)^(3/4)*x)/( x^4 + x + 2)) - I*3^(1/4)*x*log(-(6*sqrt(3)*(2*x^4 - x - 2)^(1/4)*x^3 + 6* I*3^(1/4)*sqrt(2*x^4 - x - 2)*x^2 - 3^(3/4)*(5*I*x^4 - I*x - 2*I) - 6*(2*x ^4 - x - 2)^(3/4)*x)/(x^4 + x + 2)) + I*3^(1/4)*x*log(-(6*sqrt(3)*(2*x^4 - x - 2)^(1/4)*x^3 - 6*I*3^(1/4)*sqrt(2*x^4 - x - 2)*x^2 - 3^(3/4)*(-5*I*x^ 4 + I*x + 2*I) - 6*(2*x^4 - x - 2)^(3/4)*x)/(x^4 + x + 2)) - 3^(1/4)*x*log ((6*sqrt(3)*(2*x^4 - x - 2)^(1/4)*x^3 - 6*3^(1/4)*sqrt(2*x^4 - x - 2)*x^2 - 3^(3/4)*(5*x^4 - x - 2) + 6*(2*x^4 - x - 2)^(3/4)*x)/(x^4 + x + 2)) - 8* (2*x^4 - x - 2)^(1/4))/x
Timed out. \[ \int \frac {(8+3 x) \sqrt [4]{-2-x+2 x^4}}{x^2 \left (2+x+x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {(8+3 x) \sqrt [4]{-2-x+2 x^4}}{x^2 \left (2+x+x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}} {\left (3 \, x + 8\right )}}{{\left (x^{4} + x + 2\right )} x^{2}} \,d x } \]
\[ \int \frac {(8+3 x) \sqrt [4]{-2-x+2 x^4}}{x^2 \left (2+x+x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} - x - 2\right )}^{\frac {1}{4}} {\left (3 \, x + 8\right )}}{{\left (x^{4} + x + 2\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {(8+3 x) \sqrt [4]{-2-x+2 x^4}}{x^2 \left (2+x+x^4\right )} \, dx=\int \frac {\left (3\,x+8\right )\,{\left (2\,x^4-x-2\right )}^{1/4}}{x^2\,\left (x^4+x+2\right )} \,d x \]