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Simplificación de expresiones/ Descomposición en fracciones simples/ (-(sqrt(2*x)+1)/(x-1)^2+sqrt(2)/(2*sqrt(x)*(x-1)))/(1+(sqrt(2*x)+1)^2/(x-1)^2)

¿Cómo vas a descomponer esta (-(sqrt(2*x)+1)/(x-1)^2+sqrt(2)/(2*sqrt(x)*(x-1)))/(1+(sqrt(2*x)+1)^2/(x-1)^2) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    _____              ___     
- \/ 2*x  - 1        \/ 2      
------------- + ---------------
          2         ___        
   (x - 1)      2*\/ x *(x - 1)
-------------------------------
                        2      
           /  _____    \       
           \\/ 2*x  + 1/       
       1 + --------------      
                     2         
              (x - 1)
−2x−1(x−1)2+22x(x−1)1+(2x+1)2(x−1)2\frac{\frac{- \sqrt{2 x} - 1}{\left(x - 1\right)^{2}} + \frac{\sqrt{2}}{2 \sqrt{x} \left(x - 1\right)}}{1 + \frac{\left(\sqrt{2 x} + 1\right)^{2}}{\left(x - 1\right)^{2}}} 
((-sqrt(2*x) - 1)/(x - 1)^2 + sqrt(2)/(((2*sqrt(x))*(x - 1))))/(1 + (sqrt(2*x) + 1)^2/(x - 1)^2)
Simplificación general [src] 
 /  ___       ___       ___\ 
-\\/ 2  + 2*\/ x  + x*\/ 2 / 
-----------------------------
    5/2       ___         ___
 2*x    + 4*\/ x  + 4*x*\/ 2
−2x+2x+22x52+4x+42x- \frac{2 \sqrt{x} + \sqrt{2} x + \sqrt{2}}{2 x^{\frac{5}{2}} + 4 \sqrt{x} + 4 \sqrt{2} x} 
-(sqrt(2) + 2*sqrt(x) + x*sqrt(2))/(2*x^(5/2) + 4*sqrt(x) + 4*x*sqrt(2))
Respuesta numérica [src] 
((-1.0 - 1.4142135623731*x^0.5)/(-1.0 + x)^2 + 0.707106781186548*x^(-0.5)/(-1.0 + x))/(1.0 + 2.0*(0.707106781186547 + x^0.5)^2/(-1.0 + x)^2)
((-1.0 - 1.4142135623731*x^0.5)/(-1.0 + x)^2 + 0.707106781186548*x^(-0.5)/(-1.0 + x))/(1.0 + 2.0*(0.707106781186547 + x^0.5)^2/(-1.0 + x)^2)
Parte trigonométrica [src] 
       ___   ___          ___      
-1 - \/ 2 *\/ x         \/ 2       
---------------- + ----------------
           2           ___         
   (-1 + x)        2*\/ x *(-1 + x)
-----------------------------------
                            2      
           /      ___   ___\       
           \1 + \/ 2 *\/ x /       
       1 + ------------------      
                       2           
               (-1 + x)
−2x−1(x−1)2+22x(x−1)1+(2x+1)2(x−1)2\frac{\frac{- \sqrt{2} \sqrt{x} - 1}{\left(x - 1\right)^{2}} + \frac{\sqrt{2}}{2 \sqrt{x} \left(x - 1\right)}}{1 + \frac{\left(\sqrt{2} \sqrt{x} + 1\right)^{2}}{\left(x - 1\right)^{2}}} 
((-1 - sqrt(2)*sqrt(x))/(-1 + x)^2 + sqrt(2)/(2*sqrt(x)*(-1 + x)))/(1 + (1 + sqrt(2)*sqrt(x))^2/(-1 + x)^2)
Potencias [src] 
       ___   ___          ___      
-1 - \/ 2 *\/ x         \/ 2       
---------------- + ----------------
           2         ___           
   (-1 + x)        \/ x *(-2 + 2*x)
-----------------------------------
                            2      
           /      ___   ___\       
           \1 + \/ 2 *\/ x /       
       1 + ------------------      
                       2           
               (-1 + x)
−2x−1(x−1)2+2x(2x−2)1+(2x+1)2(x−1)2\frac{\frac{- \sqrt{2} \sqrt{x} - 1}{\left(x - 1\right)^{2}} + \frac{\sqrt{2}}{\sqrt{x} \left(2 x - 2\right)}}{1 + \frac{\left(\sqrt{2} \sqrt{x} + 1\right)^{2}}{\left(x - 1\right)^{2}}} 
       ___   ___          ___      
-1 - \/ 2 *\/ x         \/ 2       
---------------- + ----------------
           2           ___         
   (-1 + x)        2*\/ x *(-1 + x)
-----------------------------------
                            2      
           /      ___   ___\       
           \1 + \/ 2 *\/ x /       
       1 + ------------------      
                       2           
               (-1 + x)
−2x−1(x−1)2+22x(x−1)1+(2x+1)2(x−1)2\frac{\frac{- \sqrt{2} \sqrt{x} - 1}{\left(x - 1\right)^{2}} + \frac{\sqrt{2}}{2 \sqrt{x} \left(x - 1\right)}}{1 + \frac{\left(\sqrt{2} \sqrt{x} + 1\right)^{2}}{\left(x - 1\right)^{2}}} 
((-1 - sqrt(2)*sqrt(x))/(-1 + x)^2 + sqrt(2)/(2*sqrt(x)*(-1 + x)))/(1 + (1 + sqrt(2)*sqrt(x))^2/(-1 + x)^2)
Unión de expresiones racionales [src] 
  ___                ___ /       ___   ___\
\/ 2 *(-1 + x) + 2*\/ x *\-1 - \/ 2 *\/ x /
-------------------------------------------
          /                 2            \ 
      ___ |/      ___   ___\            2| 
  2*\/ x *\\1 + \/ 2 *\/ x /  + (-1 + x) /
2x(−2x−1)+2(x−1)2x((x−1)2+(2x+1)2)\frac{2 \sqrt{x} \left(- \sqrt{2} \sqrt{x} - 1\right) + \sqrt{2} \left(x - 1\right)}{2 \sqrt{x} \left(\left(x - 1\right)^{2} + \left(\sqrt{2} \sqrt{x} + 1\right)^{2}\right)} 
(sqrt(2)*(-1 + x) + 2*sqrt(x)*(-1 - sqrt(2)*sqrt(x)))/(2*sqrt(x)*((1 + sqrt(2)*sqrt(x))^2 + (-1 + x)^2))
Abrimos la expresión [src] 
       ___   ___          ___     
-1 - \/ 2 *\/ x         \/ 2      
---------------- + ---------------
           2           ___        
    (x - 1)        2*\/ x *(x - 1)
----------------------------------
                           2      
          /      ___   ___\       
          \1 + \/ 2 *\/ x /       
      1 + ------------------      
                      2           
               (x - 1)
−2x−1(x−1)2+22x(x−1)1+(2x+1)2(x−1)2\frac{\frac{- \sqrt{2} \sqrt{x} - 1}{\left(x - 1\right)^{2}} + \frac{\sqrt{2}}{2 \sqrt{x} \left(x - 1\right)}}{1 + \frac{\left(\sqrt{2} \sqrt{x} + 1\right)^{2}}{\left(x - 1\right)^{2}}} 
((-1 - sqrt(2)*sqrt(x))/(x - 1)^2 + sqrt(2)/(2*sqrt(x)*(x - 1)))/(1 + (1 + sqrt(2)*sqrt(x))^2/(x - 1)^2)
Denominador racional [src] 
      3/2      7/2       ___       5/2       ___         2       ___  4       ___  3       ___         2     ___  2         2
- 12*x    - 2*x    + 4*\/ x  + 10*x    - 4*\/ x *(-1 + x)  - 2*\/ 2 *x  + 2*\/ 2 *x  + 2*\/ 2 *(-1 + x)  + \/ 2 *x *(-1 + x) 
-----------------------------------------------------------------------------------------------------------------------------
                                                ___          /     4            2\                                           
                                            2*\/ x *(-1 + x)*\4 + x  - 8*x + 4*x /
−2x72+10x52−12x32−4x(x−1)2+4x−22x4+22x3+2x2(x−1)2+22(x−1)22x(x−1)(x4+4x2−8x+4)\frac{- 2 x^{\frac{7}{2}} + 10 x^{\frac{5}{2}} - 12 x^{\frac{3}{2}} - 4 \sqrt{x} \left(x - 1\right)^{2} + 4 \sqrt{x} - 2 \sqrt{2} x^{4} + 2 \sqrt{2} x^{3} + \sqrt{2} x^{2} \left(x - 1\right)^{2} + 2 \sqrt{2} \left(x - 1\right)^{2}}{2 \sqrt{x} \left(x - 1\right) \left(x^{4} + 4 x^{2} - 8 x + 4\right)} 
(-12*x^(3/2) - 2*x^(7/2) + 4*sqrt(x) + 10*x^(5/2) - 4*sqrt(x)*(-1 + x)^2 - 2*sqrt(2)*x^4 + 2*sqrt(2)*x^3 + 2*sqrt(2)*(-1 + x)^2 + sqrt(2)*x^2*(-1 + x)^2)/(2*sqrt(x)*(-1 + x)*(4 + x^4 - 8*x + 4*x^2))
Denominador común [src] 
             5/2    9/2      3/2       ___       ___      7/2         ___       ___  2      
  1         x    - x    - 4*x    + 2*\/ 2  + 2*\/ x  + 2*x    - 4*x*\/ 2  + 2*\/ 2 *x       
- - - --------------------------------------------------------------------------------------
  4         3/2      7/2      9/2       ___       5/2        ___  2         ___       ___  3
      - 16*x    - 8*x    + 4*x    + 8*\/ x  + 12*x    - 16*\/ 2 *x  + 8*x*\/ 2  + 8*\/ 2 *x
−−x92+2x72+x52−4x32+2x+22x2−42x+224x92−8x72+12x52−16x32+8x+82x3−162x2+82x−14- \frac{- x^{\frac{9}{2}} + 2 x^{\frac{7}{2}} + x^{\frac{5}{2}} - 4 x^{\frac{3}{2}} + 2 \sqrt{x} + 2 \sqrt{2} x^{2} - 4 \sqrt{2} x + 2 \sqrt{2}}{4 x^{\frac{9}{2}} - 8 x^{\frac{7}{2}} + 12 x^{\frac{5}{2}} - 16 x^{\frac{3}{2}} + 8 \sqrt{x} + 8 \sqrt{2} x^{3} - 16 \sqrt{2} x^{2} + 8 \sqrt{2} x} - \frac{1}{4} 
-1/4 - (x^(5/2) - x^(9/2) - 4*x^(3/2) + 2*sqrt(2) + 2*sqrt(x) + 2*x^(7/2) - 4*x*sqrt(2) + 2*sqrt(2)*x^2)/(-16*x^(3/2) - 8*x^(7/2) + 4*x^(9/2) + 8*sqrt(x) + 12*x^(5/2) - 16*sqrt(2)*x^2 + 8*x*sqrt(2) + 8*sqrt(2)*x^3)
Combinatoria [src] 
  /  ___       ___       ___\   
 -\\/ 2  + 2*\/ x  + x*\/ 2 /   
--------------------------------
    ___ /     2       ___   ___\
2*\/ x *\2 + x  + 2*\/ 2 *\/ x /
−2x+2x+22x(22x+x2+2)- \frac{2 \sqrt{x} + \sqrt{2} x + \sqrt{2}}{2 \sqrt{x} \left(2 \sqrt{2} \sqrt{x} + x^{2} + 2\right)} 
-(sqrt(2) + 2*sqrt(x) + x*sqrt(2))/(2*sqrt(x)*(2 + x^2 + 2*sqrt(2)*sqrt(x)))
Compilar la expresión [src] 
       ___   ___          ___      
-1 - \/ 2 *\/ x         \/ 2       
---------------- + ----------------
           2           ___         
   (-1 + x)        2*\/ x *(-1 + x)
-----------------------------------
                            2      
           /      ___   ___\       
           \1 + \/ 2 *\/ x /       
       1 + ------------------      
                       2           
               (-1 + x)
−2x−1(x−1)2+22x(x−1)1+(2x+1)2(x−1)2\frac{\frac{- \sqrt{2} \sqrt{x} - 1}{\left(x - 1\right)^{2}} + \frac{\sqrt{2}}{2 \sqrt{x} \left(x - 1\right)}}{1 + \frac{\left(\sqrt{2} \sqrt{x} + 1\right)^{2}}{\left(x - 1\right)^{2}}} 
((-1 - sqrt(2)*sqrt(x))/(-1 + x)^2 + sqrt(2)/(2*sqrt(x)*(-1 + x)))/(1 + (1 + sqrt(2)*sqrt(x))^2/(-1 + x)^2)
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