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Simplificación de expresiones/ Descomposición en fracciones simples/ sqrt(x/(x+4))*(x+4)*(1/(2*(x+4))-x/(2*(x+4)^2))/(x*sqrt(1-x/(x+4)))

¿Cómo vas a descomponer esta sqrt(x/(x+4))*(x+4)*(1/(2*(x+4))-x/(2*(x+4)^2))/(x*sqrt(1-x/(x+4))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    _______                                 
   /   x            /    1           x     \
  /  ----- *(x + 4)*|--------- - ----------|
\/   x + 4          |2*(x + 4)            2|
                    \            2*(x + 4) /
--------------------------------------------
                   ___________              
                  /       x                 
             x*  /  1 - -----               
               \/       x + 4
xx+4(x+4)(−x2(x+4)2+12(x+4))x−xx+4+1\frac{\sqrt{\frac{x}{x + 4}} \left(x + 4\right) \left(- \frac{x}{2 \left(x + 4\right)^{2}} + \frac{1}{2 \left(x + 4\right)}\right)}{x \sqrt{- \frac{x}{x + 4} + 1}} 
((sqrt(x/(x + 4))*(x + 4))*(1/(2*(x + 4)) - x/(2*(x + 4)^2)))/((x*sqrt(1 - x/(x + 4))))
Simplificación general [src] 
         _______     
        /   x        
       /  -----      
     \/   4 + x      
---------------------
      _______        
     /   1           
x*  /  ----- *(4 + x)
  \/   4 + x
xx+4x(x+4)1x+4\frac{\sqrt{\frac{x}{x + 4}}}{x \left(x + 4\right) \sqrt{\frac{1}{x + 4}}} 
sqrt(x/(4 + x))/(x*sqrt(1/(4 + x))*(4 + x))
Respuesta numérica [src] 
(x/(4.0 + x))^0.5*(1.0 - x/(4.0 + x))^(-0.5)*(4.0 + x)*(1/(8.0 + 2.0*x) - 0.03125*x/(1 + 0.25*x)^2)/x
(x/(4.0 + x))^0.5*(1.0 - x/(4.0 + x))^(-0.5)*(4.0 + x)*(1/(8.0 + 2.0*x) - 0.03125*x/(1 + 0.25*x)^2)/x
Abrimos la expresión [src] 
    _______                                 
   /   1            /    1           x     \
  /  ----- *(x + 4)*|--------- - ----------|
\/   x + 4          |2*(x + 4)            2|
                    \            2*(x + 4) /
--------------------------------------------
                     ___________            
             ___    /       x               
           \/ x *  /  1 - -----             
                 \/       x + 4
(x+4)(−x2(x+4)2+12(x+4))1x+4x−xx+4+1\frac{\left(x + 4\right) \left(- \frac{x}{2 \left(x + 4\right)^{2}} + \frac{1}{2 \left(x + 4\right)}\right) \sqrt{\frac{1}{x + 4}}}{\sqrt{x} \sqrt{- \frac{x}{x + 4} + 1}} 
sqrt(1/(x + 4))*(x + 4)*(1/(2*(x + 4)) - x/(2*(x + 4)^2))/(sqrt(x)*sqrt(1 - x/(x + 4)))
Combinatoria [src] 
         _______     
        /   x        
       /  -----      
     \/   4 + x      
---------------------
      _______        
     /   1           
x*  /  ----- *(4 + x)
  \/   4 + x
xx+4x(x+4)1x+4\frac{\sqrt{\frac{x}{x + 4}}}{x \left(x + 4\right) \sqrt{\frac{1}{x + 4}}} 
sqrt(x/(4 + x))/(x*sqrt(1/(4 + x))*(4 + x))
Parte trigonométrica [src] 
    _______                               
   /   x            /   1          x     \
  /  ----- *(4 + x)*|------- - ----------|
\/   4 + x          |8 + 2*x            2|
                    \          2*(4 + x) /
------------------------------------------
                  ___________             
                 /       x                
            x*  /  1 - -----              
              \/       4 + x
xx+4(x+4)(−x2(x+4)2+12x+8)x−xx+4+1\frac{\sqrt{\frac{x}{x + 4}} \left(x + 4\right) \left(- \frac{x}{2 \left(x + 4\right)^{2}} + \frac{1}{2 x + 8}\right)}{x \sqrt{- \frac{x}{x + 4} + 1}} 
sqrt(x/(4 + x))*(4 + x)*(1/(8 + 2*x) - x/(2*(4 + x)^2))/(x*sqrt(1 - x/(4 + x)))
Denominador común [src] 
    _______     ___________
   /   x       /       x   
  /  ----- *  /  1 - ----- 
\/   4 + x  \/       4 + x 
---------------------------
            2*x
xx+4−xx+4+12x\frac{\sqrt{\frac{x}{x + 4}} \sqrt{- \frac{x}{x + 4} + 1}}{2 x} 
sqrt(x/(4 + x))*sqrt(1 - x/(4 + x))/(2*x)
Compilar la expresión [src] 
    _______                               
   /   x            /   1          x     \
  /  ----- *(4 + x)*|------- - ----------|
\/   4 + x          |8 + 2*x            2|
                    \          2*(4 + x) /
------------------------------------------
                  ___________             
                 /       x                
            x*  /  1 - -----              
              \/       4 + x
xx+4(x+4)(−x2(x+4)2+12x+8)x−xx+4+1\frac{\sqrt{\frac{x}{x + 4}} \left(x + 4\right) \left(- \frac{x}{2 \left(x + 4\right)^{2}} + \frac{1}{2 x + 8}\right)}{x \sqrt{- \frac{x}{x + 4} + 1}} 
sqrt(x/(4 + x))*(4 + x)*(1/(8 + 2*x) - x/(2*(4 + x)^2))/(x*sqrt(1 - x/(4 + x)))
Potencias [src] 
    _______                               
   /   x            /   1          x     \
  /  ----- *(4 + x)*|------- - ----------|
\/   4 + x          |8 + 2*x            2|
                    \          2*(4 + x) /
------------------------------------------
                  ___________             
                 /       x                
            x*  /  1 - -----              
              \/       4 + x
xx+4(x+4)(−x2(x+4)2+12x+8)x−xx+4+1\frac{\sqrt{\frac{x}{x + 4}} \left(x + 4\right) \left(- \frac{x}{2 \left(x + 4\right)^{2}} + \frac{1}{2 x + 8}\right)}{x \sqrt{- \frac{x}{x + 4} + 1}} 
sqrt(x/(4 + x))*(4 + x)*(1/(8 + 2*x) - x/(2*(4 + x)^2))/(x*sqrt(1 - x/(4 + x)))
Unión de expresiones racionales [src] 
         _______     
        /   x        
       /  -----      
     \/   4 + x      
---------------------
      _______        
     /   1           
x*  /  ----- *(4 + x)
  \/   4 + x
xx+4x(x+4)1x+4\frac{\sqrt{\frac{x}{x + 4}}}{x \left(x + 4\right) \sqrt{\frac{1}{x + 4}}} 
sqrt(x/(4 + x))/(x*sqrt(1/(4 + x))*(4 + x))
Denominador racional [src] 
        _______     ___________                     _______     ___________           _______     ___________
       /   x       /       x           2      2    /   x       /       x             /   x       /       x   
- 2*  /  ----- *  /  1 - ----- *(4 + x)  + 2*x *  /  ----- *  /  1 - -----  + 8*x*  /  ----- *  /  1 - ----- 
    \/   x + 4  \/       4 + x                  \/   x + 4  \/       4 + x        \/   x + 4  \/       4 + x 
-------------------------------------------------------------------------------------------------------------
                                          /       x  \                                                       
                                      2*x*|-1 + -----|*(4 + x)*(8 + 2*x)                                     
                                          \     4 + x/
2x2xx+4−xx+4+1+8xxx+4−xx+4+1−2xx+4(x+4)2−xx+4+12x(x+4)(2x+8)(xx+4−1)\frac{2 x^{2} \sqrt{\frac{x}{x + 4}} \sqrt{- \frac{x}{x + 4} + 1} + 8 x \sqrt{\frac{x}{x + 4}} \sqrt{- \frac{x}{x + 4} + 1} - 2 \sqrt{\frac{x}{x + 4}} \left(x + 4\right)^{2} \sqrt{- \frac{x}{x + 4} + 1}}{2 x \left(x + 4\right) \left(2 x + 8\right) \left(\frac{x}{x + 4} - 1\right)} 
(-2*sqrt(x/(x + 4))*sqrt(1 - x/(4 + x))*(4 + x)^2 + 2*x^2*sqrt(x/(x + 4))*sqrt(1 - x/(4 + x)) + 8*x*sqrt(x/(x + 4))*sqrt(1 - x/(4 + x)))/(2*x*(-1 + x/(4 + x))*(4 + x)*(8 + 2*x))
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