Integrand size = 57, antiderivative size = 63 \[ \int \frac {\sqrt {2-x^3-x^4} \left (4+x^3+2 x^4\right )}{\left (-2-3 x^2+x^3+x^4\right ) \left (-2-x^2+x^3+x^4\right )} \, dx=-\arctan \left (\frac {x}{\sqrt {2-x^3-x^4}}\right )-\sqrt {3} \arctan \left (\frac {\sqrt {3} x \sqrt {2-x^3-x^4}}{-2+x^3+x^4}\right ) \]
Time = 0.88 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {2-x^3-x^4} \left (4+x^3+2 x^4\right )}{\left (-2-3 x^2+x^3+x^4\right ) \left (-2-x^2+x^3+x^4\right )} \, dx=-\arctan \left (\frac {x}{\sqrt {2-x^3-x^4}}\right )+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{\sqrt {2-x^3-x^4}}\right ) \]
Integrate[(Sqrt[2 - x^3 - x^4]*(4 + x^3 + 2*x^4))/((-2 - 3*x^2 + x^3 + x^4 )*(-2 - x^2 + x^3 + x^4)),x]
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.
\(\displaystyle \int \frac {\sqrt {-x^4-x^3+2} \left (2 x^4+x^3+4\right )}{\left (x^4+x^3-3 x^2-2\right ) \left (x^4+x^3-x^2-2\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\sqrt {-x^4-x^3+2} \left (-4 x^2-3 x+2\right )}{2 \left (x^4+x^3-x^2-2\right )}+\frac {\left (4 x^2+3 x-6\right ) \sqrt {-x^4-x^3+2}}{2 \left (x^4+x^3-3 x^2-2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -3 \int \frac {\sqrt {-x^4-x^3+2}}{x^4+x^3-3 x^2-2}dx+\frac {3}{2} \int \frac {x \sqrt {-x^4-x^3+2}}{x^4+x^3-3 x^2-2}dx+2 \int \frac {x^2 \sqrt {-x^4-x^3+2}}{x^4+x^3-3 x^2-2}dx+\int \frac {\sqrt {-x^4-x^3+2}}{x^4+x^3-x^2-2}dx-\frac {3}{2} \int \frac {x \sqrt {-x^4-x^3+2}}{x^4+x^3-x^2-2}dx-2 \int \frac {x^2 \sqrt {-x^4-x^3+2}}{x^4+x^3-x^2-2}dx\) |
Int[(Sqrt[2 - x^3 - x^4]*(4 + x^3 + 2*x^4))/((-2 - 3*x^2 + x^3 + x^4)*(-2 - x^2 + x^3 + x^4)),x]
3.9.34.3.1 Defintions of rubi rules used
Time = 13.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.78
method | result | size |
default | \(-\sqrt {3}\, \arctan \left (\frac {\sqrt {-x^{4}-x^{3}+2}\, \sqrt {3}}{3 x}\right )+\arctan \left (\frac {\sqrt {-x^{4}-x^{3}+2}}{x}\right )\) | \(49\) |
pseudoelliptic | \(-\sqrt {3}\, \arctan \left (\frac {\sqrt {-x^{4}-x^{3}+2}\, \sqrt {3}}{3 x}\right )+\arctan \left (\frac {\sqrt {-x^{4}-x^{3}+2}}{x}\right )\) | \(49\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {-x^{4}-x^{3}+2}\, x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}+x^{3}-x^{2}-2}\right )}{2}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{2}+6 \sqrt {-x^{4}-x^{3}+2}\, x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{x^{4}+x^{3}-3 x^{2}-2}\right )}{2}\) | \(167\) |
elliptic | \(\text {Expression too large to display}\) | \(433723\) |
int((-x^4-x^3+2)^(1/2)*(2*x^4+x^3+4)/(x^4+x^3-3*x^2-2)/(x^4+x^3-x^2-2),x,m ethod=_RETURNVERBOSE)
Time = 0.35 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {2-x^3-x^4} \left (4+x^3+2 x^4\right )}{\left (-2-3 x^2+x^3+x^4\right ) \left (-2-x^2+x^3+x^4\right )} \, dx=-\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {-x^{4} - x^{3} + 2} x}{x^{4} + x^{3} + 3 \, x^{2} - 2}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {-x^{4} - x^{3} + 2} x}{x^{4} + x^{3} + x^{2} - 2}\right ) \]
integrate((-x^4-x^3+2)^(1/2)*(2*x^4+x^3+4)/(x^4+x^3-3*x^2-2)/(x^4+x^3-x^2- 2),x, algorithm="fricas")
-1/2*sqrt(3)*arctan(2*sqrt(3)*sqrt(-x^4 - x^3 + 2)*x/(x^4 + x^3 + 3*x^2 - 2)) + 1/2*arctan(2*sqrt(-x^4 - x^3 + 2)*x/(x^4 + x^3 + x^2 - 2))
Timed out. \[ \int \frac {\sqrt {2-x^3-x^4} \left (4+x^3+2 x^4\right )}{\left (-2-3 x^2+x^3+x^4\right ) \left (-2-x^2+x^3+x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {2-x^3-x^4} \left (4+x^3+2 x^4\right )}{\left (-2-3 x^2+x^3+x^4\right ) \left (-2-x^2+x^3+x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} + x^{3} + 4\right )} \sqrt {-x^{4} - x^{3} + 2}}{{\left (x^{4} + x^{3} - x^{2} - 2\right )} {\left (x^{4} + x^{3} - 3 \, x^{2} - 2\right )}} \,d x } \]
integrate((-x^4-x^3+2)^(1/2)*(2*x^4+x^3+4)/(x^4+x^3-3*x^2-2)/(x^4+x^3-x^2- 2),x, algorithm="maxima")
integrate((2*x^4 + x^3 + 4)*sqrt(-x^4 - x^3 + 2)/((x^4 + x^3 - x^2 - 2)*(x ^4 + x^3 - 3*x^2 - 2)), x)
\[ \int \frac {\sqrt {2-x^3-x^4} \left (4+x^3+2 x^4\right )}{\left (-2-3 x^2+x^3+x^4\right ) \left (-2-x^2+x^3+x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{4} + x^{3} + 4\right )} \sqrt {-x^{4} - x^{3} + 2}}{{\left (x^{4} + x^{3} - x^{2} - 2\right )} {\left (x^{4} + x^{3} - 3 \, x^{2} - 2\right )}} \,d x } \]
integrate((-x^4-x^3+2)^(1/2)*(2*x^4+x^3+4)/(x^4+x^3-3*x^2-2)/(x^4+x^3-x^2- 2),x, algorithm="giac")
integrate((2*x^4 + x^3 + 4)*sqrt(-x^4 - x^3 + 2)/((x^4 + x^3 - x^2 - 2)*(x ^4 + x^3 - 3*x^2 - 2)), x)
Timed out. \[ \int \frac {\sqrt {2-x^3-x^4} \left (4+x^3+2 x^4\right )}{\left (-2-3 x^2+x^3+x^4\right ) \left (-2-x^2+x^3+x^4\right )} \, dx=\int \frac {\sqrt {-x^4-x^3+2}\,\left (2\,x^4+x^3+4\right )}{\left (-x^4-x^3+3\,x^2+2\right )\,\left (-x^4-x^3+x^2+2\right )} \,d x \]
int(((2 - x^4 - x^3)^(1/2)*(x^3 + 2*x^4 + 4))/((3*x^2 - x^3 - x^4 + 2)*(x^ 2 - x^3 - x^4 + 2)),x)