\(\int \frac {(-1+x^2) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{(1+x+x^2) (1+3 x+x^2)^2} \, dx\) [431]

Optimal result

Integrand size = 48, antiderivative size = 35 \[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+3 x+x^2} \]

[Out]

(-x^4-4*x^3-5*x^2-4*x-1)^(1/2)/(x^2+3*x+1)

Rubi [F]

\[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx \]

[In]

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]

[Out]

(-6*Defer[Int][Sqrt[-1 - 4*x - 5*x^2 - 4*x^3 - x^4]/(-3 + Sqrt[5] - 2*x)^2, x])/5 + (2*(3 - Sqrt[5])*Defer[Int
][Sqrt[-1 - 4*x - 5*x^2 - 4*x^3 - x^4]/(-3 + Sqrt[5] - 2*x)^2, x])/5 - ((1 + I*Sqrt[3])*Defer[Int][Sqrt[-1 - 4
*x - 5*x^2 - 4*x^3 - x^4]/(1 - I*Sqrt[3] + 2*x), x])/4 - ((1 - I*Sqrt[3])*Defer[Int][Sqrt[-1 - 4*x - 5*x^2 - 4
*x^3 - x^4]/(1 + I*Sqrt[3] + 2*x), x])/4 + ((5 - Sqrt[5])*Defer[Int][Sqrt[-1 - 4*x - 5*x^2 - 4*x^3 - x^4]/(3 -
Sqrt[5] + 2*x), x])/20 - (6*Defer[Int][Sqrt[-1 - 4*x - 5*x^2 - 4*x^3 - x^4]/(3 + Sqrt[5] + 2*x)^2, x])/5 + (2
*(3 + Sqrt[5])*Defer[Int][Sqrt[-1 - 4*x - 5*x^2 - 4*x^3 - x^4]/(3 + Sqrt[5] + 2*x)^2, x])/5 + ((5 + Sqrt[5])*D
efer[Int][Sqrt[-1 - 4*x - 5*x^2 - 4*x^3 - x^4]/(3 + Sqrt[5] + 2*x), x])/20

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(1-x) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{4 \left (1+x+x^2\right )}+\frac {(-3-2 x) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{2 \left (1+3 x+x^2\right )^2}+\frac {(1+x) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{4 \left (1+3 x+x^2\right )}\right ) \, dx \\ & = \frac {1}{4} \int \frac {(1-x) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+x+x^2} \, dx+\frac {1}{4} \int \frac {(1+x) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+3 x+x^2} \, dx+\frac {1}{2} \int \frac {(-3-2 x) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+3 x+x^2\right )^2} \, dx \\ & = \frac {1}{4} \int \left (\frac {\left (-1-i \sqrt {3}\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1-i \sqrt {3}+2 x}+\frac {\left (-1+i \sqrt {3}\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+i \sqrt {3}+2 x}\right ) \, dx+\frac {1}{4} \int \left (\frac {\left (1-\frac {1}{\sqrt {5}}\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3-\sqrt {5}+2 x}+\frac {\left (1+\frac {1}{\sqrt {5}}\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3+\sqrt {5}+2 x}\right ) \, dx+\frac {1}{2} \int \left (-\frac {3 \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+3 x+x^2\right )^2}-\frac {2 x \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+3 x+x^2\right )^2}\right ) \, dx \\ & = -\left (\frac {3}{2} \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+3 x+x^2\right )^2} \, dx\right )+\frac {1}{4} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1-i \sqrt {3}+2 x} \, dx+\frac {1}{4} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+i \sqrt {3}+2 x} \, dx+\frac {1}{20} \left (5-\sqrt {5}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3-\sqrt {5}+2 x} \, dx+\frac {1}{20} \left (5+\sqrt {5}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3+\sqrt {5}+2 x} \, dx-\int \frac {x \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+3 x+x^2\right )^2} \, dx \\ & = -\left (\frac {3}{2} \int \left (\frac {4 \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \left (-3+\sqrt {5}-2 x\right )^2}+\frac {4 \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \sqrt {5} \left (-3+\sqrt {5}-2 x\right )}+\frac {4 \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \left (3+\sqrt {5}+2 x\right )^2}+\frac {4 \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \sqrt {5} \left (3+\sqrt {5}+2 x\right )}\right ) \, dx\right )+\frac {1}{4} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1-i \sqrt {3}+2 x} \, dx+\frac {1}{4} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+i \sqrt {3}+2 x} \, dx+\frac {1}{20} \left (5-\sqrt {5}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3-\sqrt {5}+2 x} \, dx+\frac {1}{20} \left (5+\sqrt {5}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3+\sqrt {5}+2 x} \, dx-\int \left (\frac {2 \left (-3+\sqrt {5}\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \left (-3+\sqrt {5}-2 x\right )^2}-\frac {6 \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \sqrt {5} \left (-3+\sqrt {5}-2 x\right )}+\frac {2 \left (-3-\sqrt {5}\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \left (3+\sqrt {5}+2 x\right )^2}-\frac {6 \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{5 \sqrt {5} \left (3+\sqrt {5}+2 x\right )}\right ) \, dx \\ & = -\left (\frac {6}{5} \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (-3+\sqrt {5}-2 x\right )^2} \, dx\right )-\frac {6}{5} \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (3+\sqrt {5}+2 x\right )^2} \, dx+\frac {1}{4} \left (-1-i \sqrt {3}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1-i \sqrt {3}+2 x} \, dx+\frac {1}{4} \left (-1+i \sqrt {3}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+i \sqrt {3}+2 x} \, dx+\frac {1}{5} \left (2 \left (3-\sqrt {5}\right )\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (-3+\sqrt {5}-2 x\right )^2} \, dx+\frac {1}{20} \left (5-\sqrt {5}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3-\sqrt {5}+2 x} \, dx+\frac {1}{5} \left (2 \left (3+\sqrt {5}\right )\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (3+\sqrt {5}+2 x\right )^2} \, dx+\frac {1}{20} \left (5+\sqrt {5}\right ) \int \frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{3+\sqrt {5}+2 x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\frac {\sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{1+3 x+x^2} \]

[In]

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]

[Out]

Sqrt[-1 - 4*x - 5*x^2 - 4*x^3 - x^4]/(1 + 3*x + x^2)

Maple [A] (verified)

Time = 6.68 (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	\(\frac {\sqrt {-x^{4}-4 x^{3}-5 x^{2}-4 x -1}}{x^{2}+3 x +1}\)	\(34\)

[In]

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,method=_RETURNVERBOSE)

[Out]

-(x^2+x+1)/(-x^4-4*x^3-5*x^2-4*x-1)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\frac {\sqrt {-x^{4} - 4 \, x^{3} - 5 \, x^{2} - 4 \, x - 1}}{x^{2} + 3 \, x + 1} \]

[In]

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")

[Out]

sqrt(-x^4 - 4*x^3 - 5*x^2 - 4*x - 1)/(x^2 + 3*x + 1)

Sympy [F]

\[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\int \frac {\sqrt {- \left (x^{2} + x + 1\right ) \left (x^{2} + 3 x + 1\right )} \left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + x + 1\right ) \left (x^{2} + 3 x + 1\right )^{2}}\, dx \]

[In]

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)

[Out]

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)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\frac {\sqrt {x^{2} + x + 1} \sqrt {-x^{2} - 3 \, x - 1}}{x^{2} + 3 \, x + 1} \]

[In]

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")

[Out]

sqrt(x^2 + x + 1)*sqrt(-x^2 - 3*x - 1)/(x^2 + 3*x + 1)

Giac [F]

\[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\int { \frac {\sqrt {-x^{4} - 4 \, x^{3} - 5 \, x^{2} - 4 \, x - 1} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 3 \, x + 1\right )}^{2} {\left (x^{2} + x + 1\right )}} \,d x } \]

[In]

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")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int \frac {\left (-1+x^2\right ) \sqrt {-1-4 x-5 x^2-4 x^3-x^4}}{\left (1+x+x^2\right ) \left (1+3 x+x^2\right )^2} \, dx=\frac {\sqrt {-x^4-4\,x^3-5\,x^2-4\,x-1}}{x^2+3\,x+1} \]

[In]

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)

[Out]

(- 4*x - 5*x^2 - 4*x^3 - x^4 - 1)^(1/2)/(3*x + x^2 + 1)