Integrand size = 48, antiderivative size = 35 \[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+3 x+x^2} \] Output:
(-x^4-4*x^3-5*x^2-4*x-1)^(1/2)/(x^2+3*x+1)
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+3 x+x^2} \] Input:
Integrate[((-1 + x^2)*Sqrt[-1 - 4*x - 5*x^2 - 4*x^3 - x^4])/((1 + x + x^2) *(1 + 3*x + x^2)^2),x]
Output:
Sqrt[-1 - 4*x - 5*x^2 - 4*x^3 - x^4]/(1 + 3*x + x^2)
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.
\(\displaystyle \int \frac {\left (x^2-1\right ) \sqrt {-x^4-4 x^3-5 x^2-4 x-1}}{\left (x^2+x+1\right ) \left (x^2+3 x+1\right )^2} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {\sqrt {-x^4-4 x^3-5 x^2-4 x-1} (-2 x-3)}{2 \left (x^2+3 x+1\right )^2}+\frac {(1-x) \sqrt {-x^4-4 x^3-5 x^2-4 x-1}}{4 \left (x^2+x+1\right )}+\frac {(x+1) \sqrt {-x^4-4 x^3-5 x^2-4 x-1}}{4 \left (x^2+3 x+1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{5} \left (3-\sqrt {5}\right ) \int \frac {\sqrt {-x^4-4 x^3-5 x^2-4 x-1}}{\left (-2 x+\sqrt {5}-3\right )^2}dx-\frac {6}{5} \int \frac {\sqrt {-x^4-4 x^3-5 x^2-4 x-1}}{\left (-2 x+\sqrt {5}-3\right )^2}dx-\frac {1}{4} \left (1+i \sqrt {3}\right ) \int \frac {\sqrt {-x^4-4 x^3-5 x^2-4 x-1}}{2 x-i \sqrt {3}+1}dx-\frac {1}{4} \left (1-i \sqrt {3}\right ) \int \frac {\sqrt {-x^4-4 x^3-5 x^2-4 x-1}}{2 x+i \sqrt {3}+1}dx+\frac {1}{20} \left (5-\sqrt {5}\right ) \int \frac {\sqrt {-x^4-4 x^3-5 x^2-4 x-1}}{2 x-\sqrt {5}+3}dx+\frac {2}{5} \left (3+\sqrt {5}\right ) \int \frac {\sqrt {-x^4-4 x^3-5 x^2-4 x-1}}{\left (2 x+\sqrt {5}+3\right )^2}dx-\frac {6}{5} \int \frac {\sqrt {-x^4-4 x^3-5 x^2-4 x-1}}{\left (2 x+\sqrt {5}+3\right )^2}dx+\frac {1}{20} \left (5+\sqrt {5}\right ) \int \frac {\sqrt {-x^4-4 x^3-5 x^2-4 x-1}}{2 x+\sqrt {5}+3}dx\) |
Input:
Int[((-1 + x^2)*Sqrt[-1 - 4*x - 5*x^2 - 4*x^3 - x^4])/((1 + x + x^2)*(1 + 3*x + x^2)^2),x]
Output:
$Aborted
Time = 1.63 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {x^{2}+x +1}{\sqrt {-x^{4}-4 x^{3}-5 x^{2}-4 x -1}}\) | \(31\) |
pseudoelliptic | \(-\frac {x^{2}+x +1}{\sqrt {-x^{4}-4 x^{3}-5 x^{2}-4 x -1}}\) | \(31\) |
default | \(\frac {-x^{2}-x -1}{\sqrt {\left (x^{2}+3 x +1\right ) \left (-x^{2}-x -1\right )}}\) | \(33\) |
elliptic | \(\frac {-x^{2}-x -1}{\sqrt {\left (x^{2}+3 x +1\right ) \left (-x^{2}-x -1\right )}}\) | \(33\) |
gosper | \(\frac {\sqrt {-x^{4}-4 x^{3}-5 x^{2}-4 x -1}}{x^{2}+3 x +1}\) | \(34\) |
trager | \(\frac {\sqrt {-x^{4}-4 x^{3}-5 x^{2}-4 x -1}}{x^{2}+3 x +1}\) | \(34\) |
orering | \(\frac {\left (x^{2}-1\right ) \sqrt {-x^{4}-4 x^{3}-5 x^{2}-4 x -1}}{\left (-1+x \right ) \left (x^{2}+3 x +1\right ) \left (1+x \right )}\) | \(49\) |
Input:
int((x^2-1)*(-x^4-4*x^3-5*x^2-4*x-1)^(1/2)/(x^2+x+1)/(x^2+3*x+1)^2,x,metho d=_RETURNVERBOSE)
Output:
-(x^2+x+1)/(-x^4-4*x^3-5*x^2-4*x-1)^(1/2)
Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\frac {\sqrt {-x^{4} - 4 \, x^{3} - 5 \, x^{2} - 4 \, x - 1}}{x^{2} + 3 \, x + 1} \] Input:
integrate((x^2-1)*(-x^4-4*x^3-5*x^2-4*x-1)^(1/2)/(x^2+x+1)/(x^2+3*x+1)^2,x , algorithm="fricas")
Output:
sqrt(-x^4 - 4*x^3 - 5*x^2 - 4*x - 1)/(x^2 + 3*x + 1)
\[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\int \frac {\sqrt {- \left (x^{2} + x + 1\right ) \left (x^{2} + 3 x + 1\right )} \left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + x + 1\right ) \left (x^{2} + 3 x + 1\right )^{2}}\, dx \] Input:
integrate((x**2-1)*(-x**4-4*x**3-5*x**2-4*x-1)**(1/2)/(x**2+x+1)/(x**2+3*x +1)**2,x)
Output:
Integral(sqrt(-(x**2 + x + 1)*(x**2 + 3*x + 1))*(x - 1)*(x + 1)/((x**2 + x + 1)*(x**2 + 3*x + 1)**2), x)
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\frac {\sqrt {x^{2} + x + 1} \sqrt {-x^{2} - 3 \, x - 1}}{x^{2} + 3 \, x + 1} \] Input:
integrate((x^2-1)*(-x^4-4*x^3-5*x^2-4*x-1)^(1/2)/(x^2+x+1)/(x^2+3*x+1)^2,x , algorithm="maxima")
Output:
sqrt(x^2 + x + 1)*sqrt(-x^2 - 3*x - 1)/(x^2 + 3*x + 1)
\[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\int { \frac {\sqrt {-x^{4} - 4 \, x^{3} - 5 \, x^{2} - 4 \, x - 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 3 \, x + 1\right )}^{2} {\left (x^{2} + x + 1\right )}} \,d x } \] Input:
integrate((x^2-1)*(-x^4-4*x^3-5*x^2-4*x-1)^(1/2)/(x^2+x+1)/(x^2+3*x+1)^2,x , algorithm="giac")
Output:
integrate(sqrt(-x^4 - 4*x^3 - 5*x^2 - 4*x - 1)*(x^2 - 1)/((x^2 + 3*x + 1)^ 2*(x^2 + x + 1)), x)
Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\frac {\sqrt {-x^4-4\,x^3-5\,x^2-4\,x-1}}{x^2+3\,x+1} \] Input:
int(((x^2 - 1)*(- 4*x - 5*x^2 - 4*x^3 - x^4 - 1)^(1/2))/((3*x + x^2 + 1)^2 *(x + x^2 + 1)),x)
Output:
(- 4*x - 5*x^2 - 4*x^3 - x^4 - 1)^(1/2)/(3*x + x^2 + 1)
Time = 0.15 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\frac {\sqrt {-x^{4}-4 x^{3}-5 x^{2}-4 x -1}}{x^{2}+3 x +1} \] Input:
int((x^2-1)*(-x^4-4*x^3-5*x^2-4*x-1)^(1/2)/(x^2+x+1)/(x^2+3*x+1)^2,x)
Output:
sqrt( - x**4 - 4*x**3 - 5*x**2 - 4*x - 1)/(x**2 + 3*x + 1)