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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
                                 2
          2     /         x     \ 
         x      |1 + -----------| 
  -1 + ------   |       ________| 
            2   |      /      2 | 
       1 + x    \    \/  1 + x  / 
- ----------- - ------------------
     ________           ________  
    /      2           /      2   
  \/  1 + x      x + \/  1 + x    
----------------------------------
                ________          
               /      2           
         x + \/  1 + x
$$\frac{- \frac{\frac{x^{2}}{x^{2} + 1} - 1}{\sqrt{x^{2} + 1}} - \frac{\left(\frac{x}{\sqrt{x^{2} + 1}} + 1\right)^{2}}{x + \sqrt{x^{2} + 1}}}{x + \sqrt{x^{2} + 1}}$$ 
(-(-1 + x^2/(1 + x^2))/sqrt(1 + x^2) - (1 + x/sqrt(1 + x^2))^2/(x + sqrt(1 + x^2)))/(x + sqrt(1 + x^2))
Simplificación general [src] 
                 3/2 /       ________\
     2   /     2\    |      /      2 |
1 + x  - \1 + x /   *\x + \/  1 + x  /
--------------------------------------
            5/2 /       ________\     
    /     2\    |      /      2 |     
    \1 + x /   *\x + \/  1 + x  /
$$\frac{x^{2} - \left(x + \sqrt{x^{2} + 1}\right) \left(x^{2} + 1\right)^{\frac{3}{2}} + 1}{\left(x + \sqrt{x^{2} + 1}\right) \left(x^{2} + 1\right)^{\frac{5}{2}}}$$ 
(1 + x^2 - (1 + x^2)^(3/2)*(x + sqrt(1 + x^2)))/((1 + x^2)^(5/2)*(x + sqrt(1 + x^2)))
Denominador común [src] 
                    /                   ________\                    
                    |       3      2   /      2 |                    
                   -\x + 2*x  + 2*x *\/  1 + x  /                    
---------------------------------------------------------------------
   ________                               ________           ________
  /      2             5      3      4   /      2       2   /      2 
\/  1 + x   + 2*x + 2*x  + 4*x  + 2*x *\/  1 + x   + 3*x *\/  1 + x
$$- \frac{2 x^{3} + 2 x^{2} \sqrt{x^{2} + 1} + x}{2 x^{5} + 2 x^{4} \sqrt{x^{2} + 1} + 4 x^{3} + 3 x^{2} \sqrt{x^{2} + 1} + 2 x + \sqrt{x^{2} + 1}}$$ 
-(x + 2*x^3 + 2*x^2*sqrt(1 + x^2))/(sqrt(1 + x^2) + 2*x + 2*x^5 + 4*x^3 + 2*x^4*sqrt(1 + x^2) + 3*x^2*sqrt(1 + x^2))
Potencias [src] 
                               2
        2     /         x     \ 
       x      |1 + -----------| 
 1 - ------   |       ________| 
          2   |      /      2 | 
     1 + x    \    \/  1 + x  / 
----------- - ------------------
   ________           ________  
  /      2           /      2   
\/  1 + x      x + \/  1 + x    
--------------------------------
               ________         
              /      2          
        x + \/  1 + x
$$\frac{\frac{- \frac{x^{2}}{x^{2} + 1} + 1}{\sqrt{x^{2} + 1}} - \frac{\left(\frac{x}{\sqrt{x^{2} + 1}} + 1\right)^{2}}{x + \sqrt{x^{2} + 1}}}{x + \sqrt{x^{2} + 1}}$$ 
((1 - x^2/(1 + x^2))/sqrt(1 + x^2) - (1 + x/sqrt(1 + x^2))^2/(x + sqrt(1 + x^2)))/(x + sqrt(1 + x^2))
Denominador racional [src] 
                 2 /                                                                            2\
/       ________\  |                     ________         ________           2 /       ________\ |
|      /      2 |  |     4      2       /      2     3   /      2    /     2\  |      /      2 | |
\x - \/  1 + x  / *\1 + x  + 2*x  + x*\/  1 + x   + x *\/  1 + x   - \1 + x / *\x + \/  1 + x  / /
--------------------------------------------------------------------------------------------------
                                            6      2      4                                       
                                       1 + x  + 3*x  + 3*x
$$\frac{\left(x - \sqrt{x^{2} + 1}\right)^{2} \left(x^{4} + x^{3} \sqrt{x^{2} + 1} + 2 x^{2} + x \sqrt{x^{2} + 1} - \left(x + \sqrt{x^{2} + 1}\right)^{2} \left(x^{2} + 1\right)^{2} + 1\right)}{x^{6} + 3 x^{4} + 3 x^{2} + 1}$$ 
(x - sqrt(1 + x^2))^2*(1 + x^4 + 2*x^2 + x*sqrt(1 + x^2) + x^3*sqrt(1 + x^2) - (1 + x^2)^2*(x + sqrt(1 + x^2))^2)/(1 + x^6 + 3*x^2 + 3*x^4)
Respuesta numérica [src] 
(-(1.0 + x^2)^(-0.5)*(-1.0 + x^2/(1.0 + x^2)) - (1.0 + x*(1.0 + x^2)^(-0.5))^2/(x + (1.0 + x^2)^0.5))/(x + (1.0 + x^2)^0.5)
(-(1.0 + x^2)^(-0.5)*(-1.0 + x^2/(1.0 + x^2)) - (1.0 + x*(1.0 + x^2)^(-0.5))^2/(x + (1.0 + x^2)^0.5))/(x + (1.0 + x^2)^0.5)
Unión de expresiones racionales [src] 
                 3/2 /       ________\
     2   /     2\    |      /      2 |
1 + x  - \1 + x /   *\x + \/  1 + x  /
--------------------------------------
            5/2 /       ________\     
    /     2\    |      /      2 |     
    \1 + x /   *\x + \/  1 + x  /
$$\frac{x^{2} - \left(x + \sqrt{x^{2} + 1}\right) \left(x^{2} + 1\right)^{\frac{3}{2}} + 1}{\left(x + \sqrt{x^{2} + 1}\right) \left(x^{2} + 1\right)^{\frac{5}{2}}}$$ 
(1 + x^2 - (1 + x^2)^(3/2)*(x + sqrt(1 + x^2)))/((1 + x^2)^(5/2)*(x + sqrt(1 + x^2)))
Combinatoria [src] 
    -x     
-----------
        3/2
/     2\   
\1 + x /
$$- \frac{x}{\left(x^{2} + 1\right)^{\frac{3}{2}}}$$ 
-x/(1 + x^2)^(3/2)
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