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Simplificación de expresiones/ Descomposición en fracciones simples/ log(x)/log(5)/5*((x+exp(-x)*x)^(1/5))+(x+5)/(2*sin(x)+x)

¿Cómo vas a descomponer esta log(x)/log(5)/5*((x+exp(-x)*x)^(1/5))+(x+5)/(2*sin(x)+x) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
/log(x)\                              
|------|    ___________               
\log(5)/ 5 /      -x         x + 5    
--------*\/  x + e  *x  + ------------
   5                      2*sin(x) + x
$$\frac{\frac{1}{\log{\left(5 \right)}} \log{\left(x \right)}}{5} \sqrt[5]{x + x e^{- x}} + \frac{x + 5}{x + 2 \sin{\left(x \right)}}$$ 
((log(x)/log(5))/5)*(x + exp(-x)*x)^(1/5) + (x + 5)/(2*sin(x) + x)
Simplificación general [src] 
                       ________________                      
                    5 /   /     x\  -x                       
(5 + x)*log(3125) + \/  x*\1 + e /*e   *(x + 2*sin(x))*log(x)
-------------------------------------------------------------
                   5*(x + 2*sin(x))*log(5)
$$\frac{\sqrt[5]{x \left(e^{x} + 1\right) e^{- x}} \left(x + 2 \sin{\left(x \right)}\right) \log{\left(x \right)} + \left(x + 5\right) \log{\left(3125 \right)}}{5 \left(x + 2 \sin{\left(x \right)}\right) \log{\left(5 \right)}}$$ 
((5 + x)*log(3125) + (x*(1 + exp(x))*exp(-x))^(1/5)*(x + 2*sin(x))*log(x))/(5*(x + 2*sin(x))*log(5))
Unión de expresiones racionales [src] 
                      ________________                      
                   5 /   /     x\  -x                       
5*(5 + x)*log(5) + \/  x*\1 + e /*e   *(x + 2*sin(x))*log(x)
------------------------------------------------------------
                  5*(x + 2*sin(x))*log(5)
$$\frac{\sqrt[5]{x \left(e^{x} + 1\right) e^{- x}} \left(x + 2 \sin{\left(x \right)}\right) \log{\left(x \right)} + 5 \left(x + 5\right) \log{\left(5 \right)}}{5 \left(x + 2 \sin{\left(x \right)}\right) \log{\left(5 \right)}}$$ 
(5*(5 + x)*log(5) + (x*(1 + exp(x))*exp(-x))^(1/5)*(x + 2*sin(x))*log(x))/(5*(x + 2*sin(x))*log(5))
Denominador racional [src] 
                              ___________               ___________              
                           5 /        -x             5 /        -x               
25*log(5) + 5*x*log(5) + x*\/  x + x*e   *log(x) + 2*\/  x + x*e   *log(x)*sin(x)
---------------------------------------------------------------------------------
                             5*(x + 2*sin(x))*log(5)
$$\frac{x \sqrt[5]{x + x e^{- x}} \log{\left(x \right)} + 5 x \log{\left(5 \right)} + 2 \sqrt[5]{x + x e^{- x}} \log{\left(x \right)} \sin{\left(x \right)} + 25 \log{\left(5 \right)}}{5 \left(x + 2 \sin{\left(x \right)}\right) \log{\left(5 \right)}}$$ 
(25*log(5) + 5*x*log(5) + x*(x + x*exp(-x))^(1/5)*log(x) + 2*(x + x*exp(-x))^(1/5)*log(x)*sin(x))/(5*(x + 2*sin(x))*log(5))
Respuesta numérica [src] 
(5.0 + x)/(x + 2.0*sin(x)) + 0.124266986911922*(x + x*exp(-x))^0.2*log(x)
(5.0 + x)/(x + 2.0*sin(x)) + 0.124266986911922*(x + x*exp(-x))^0.2*log(x)
Denominador común [src] 
                                        ___________               ___________              
                                     5 /        -x             5 /        -x               
    25*log(5) - 10*log(5)*sin(x) + x*\/  x + x*e   *log(x) + 2*\/  x + x*e   *log(x)*sin(x)
1 + ---------------------------------------------------------------------------------------
                                 5*x*log(5) + 10*log(5)*sin(x)
$$1 + \frac{x \sqrt[5]{x + x e^{- x}} \log{\left(x \right)} + 2 \sqrt[5]{x + x e^{- x}} \log{\left(x \right)} \sin{\left(x \right)} - 10 \log{\left(5 \right)} \sin{\left(x \right)} + 25 \log{\left(5 \right)}}{5 x \log{\left(5 \right)} + 10 \log{\left(5 \right)} \sin{\left(x \right)}}$$ 
1 + (25*log(5) - 10*log(5)*sin(x) + x*(x + x*exp(-x))^(1/5)*log(x) + 2*(x + x*exp(-x))^(1/5)*log(x)*sin(x))/(5*x*log(5) + 10*log(5)*sin(x))
Potencias [src] 
                  ___________       
               5 /        -x        
   5 + x       \/  x + x*e   *log(x)
------------ + ---------------------
x + 2*sin(x)          5*log(5)
$$\frac{x + 5}{x + 2 \sin{\left(x \right)}} + \frac{\sqrt[5]{x + x e^{- x}} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
                            ___________       
                         5 /        -x        
        5 + x            \/  x + x*e   *log(x)
---------------------- + ---------------------
      /   -I*x    I*x\          5*log(5)      
x - I*\- e     + e   /
$$\frac{x + 5}{x - i \left(e^{i x} - e^{- i x}\right)} + \frac{\sqrt[5]{x + x e^{- x}} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
(5 + x)/(x - i*(-exp(-i*x) + exp(i*x))) + (x + x*exp(-x))^(1/5)*log(x)/(5*log(5))
Compilar la expresión [src] 
                  ___________       
               5 /        -x        
   5 + x       \/  x + x*e   *log(x)
------------ + ---------------------
x + 2*sin(x)          5*log(5)
$$\frac{x + 5}{x + 2 \sin{\left(x \right)}} + \frac{\sqrt[5]{x + x e^{- x}} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
(5 + x)/(x + 2*sin(x)) + (x + x*exp(-x))^(1/5)*log(x)/(5*log(5))
Abrimos la expresión [src] 
                                 ___________       
                              5 /      -x          
     5              x         \/  x + e  *x *log(x)
------------ + ------------ + ---------------------
2*sin(x) + x   2*sin(x) + x          5*log(5)
$$\frac{x}{x + 2 \sin{\left(x \right)}} + \frac{\sqrt[5]{x + x e^{- x}} \log{\left(x \right)}}{5 \log{\left(5 \right)}} + \frac{5}{x + 2 \sin{\left(x \right)}}$$ 
5/(2*sin(x) + x) + x/(2*sin(x) + x) + (x + exp(-x)*x)^(1/5)*log(x)/(5*log(5))
Combinatoria [src] 
                              ___________               ___________              
                           5 /        -x             5 /        -x               
25*log(5) + 5*x*log(5) + x*\/  x + x*e   *log(x) + 2*\/  x + x*e   *log(x)*sin(x)
---------------------------------------------------------------------------------
                             5*(x + 2*sin(x))*log(5)
$$\frac{x \sqrt[5]{x + x e^{- x}} \log{\left(x \right)} + 5 x \log{\left(5 \right)} + 2 \sqrt[5]{x + x e^{- x}} \log{\left(x \right)} \sin{\left(x \right)} + 25 \log{\left(5 \right)}}{5 \left(x + 2 \sin{\left(x \right)}\right) \log{\left(5 \right)}}$$ 
(25*log(5) + 5*x*log(5) + x*(x + x*exp(-x))^(1/5)*log(x) + 2*(x + x*exp(-x))^(1/5)*log(x)*sin(x))/(5*(x + 2*sin(x))*log(5))
Parte trigonométrica [src] 
               5 ____________________________       
   5 + x       \/ x + x*(-sinh(x) + cosh(x)) *log(x)
------------ + -------------------------------------
x + 2*sin(x)                  5*log(5)
$$\frac{x + 5}{x + 2 \sin{\left(x \right)}} + \frac{\sqrt[5]{x \left(- \sinh{\left(x \right)} + \cosh{\left(x \right)}\right) + x} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
                  5 ____________________________       
     5 + x        \/ x - x*(-cosh(x) + sinh(x)) *log(x)
--------------- + -------------------------------------
           /x\                   5*log(5)              
      4*tan|-|                                         
           \2/                                         
x + -----------                                        
           2/x\                                        
    1 + tan |-|                                        
            \2/
$$\frac{x + 5}{x + \frac{4 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} + \frac{\sqrt[5]{- x \left(\sinh{\left(x \right)} - \cosh{\left(x \right)}\right) + x} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
                ___________       
             5 /        -x        
  5 + x      \/  x + x*e   *log(x)
---------- + ---------------------
      2             5*log(5)      
x + ------                        
    csc(x)
$$\frac{x + 5}{x + \frac{2}{\csc{\left(x \right)}}} + \frac{\sqrt[5]{x + x e^{- x}} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
             5 ____________________________       
  5 + x      \/ x - x*(-cosh(x) + sinh(x)) *log(x)
---------- + -------------------------------------
      2                     5*log(5)              
x + ------                                        
    csc(x)
$$\frac{x + 5}{x + \frac{2}{\csc{\left(x \right)}}} + \frac{\sqrt[5]{- x \left(\sinh{\left(x \right)} - \cosh{\left(x \right)}\right) + x} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
                     ___________       
                  5 /        -x        
     5 + x        \/  x + x*e   *log(x)
--------------- + ---------------------
         2               5*log(5)      
x + -----------                        
       /    pi\                        
    sec|x - --|                        
       \    2 /
$$\frac{x + 5}{x + \frac{2}{\sec{\left(x - \frac{\pi}{2} \right)}}} + \frac{\sqrt[5]{x + x e^{- x}} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
                    5 ____________________________       
      5 + x         \/ x + x*(-sinh(x) + cosh(x)) *log(x)
----------------- + -------------------------------------
         /    pi\                  5*log(5)              
x + 2*cos|x - --|                                        
         \    2 /
$$\frac{x + 5}{x + 2 \cos{\left(x - \frac{\pi}{2} \right)}} + \frac{\sqrt[5]{x \left(- \sinh{\left(x \right)} + \cosh{\left(x \right)}\right) + x} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
                     ___________       
                  5 /        -x        
     5 + x        \/  x + x*e   *log(x)
--------------- + ---------------------
           /x\           5*log(5)      
      4*tan|-|                         
           \2/                         
x + -----------                        
           2/x\                        
    1 + tan |-|                        
            \2/
$$\frac{x + 5}{x + \frac{4 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}} + \frac{\sqrt[5]{x + x e^{- x}} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
               5 ____________________________       
   5 + x       \/ x - x*(-cosh(x) + sinh(x)) *log(x)
------------ + -------------------------------------
x + 2*sin(x)                  5*log(5)
$$\frac{x + 5}{x + 2 \sin{\left(x \right)}} + \frac{\sqrt[5]{- x \left(\sinh{\left(x \right)} - \cosh{\left(x \right)}\right) + x} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
                  5 ____________________________       
     5 + x        \/ x - x*(-cosh(x) + sinh(x)) *log(x)
--------------- + -------------------------------------
         2                       5*log(5)              
x + -----------                                        
       /    pi\                                        
    sec|x - --|                                        
       \    2 /
$$\frac{x + 5}{x + \frac{2}{\sec{\left(x - \frac{\pi}{2} \right)}}} + \frac{\sqrt[5]{- x \left(\sinh{\left(x \right)} - \cosh{\left(x \right)}\right) + x} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
                     ___________       
                  5 /        -x        
     5 + x        \/  x + x*e   *log(x)
--------------- + ---------------------
           /x\           5*log(5)      
      4*cot|-|                         
           \2/                         
x + -----------                        
           2/x\                        
    1 + cot |-|                        
            \2/
$$\frac{x + 5}{x + \frac{4 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} + \frac{\sqrt[5]{x + x e^{- x}} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
                  ___________       
               5 /        -x        
   5 + x       \/  x + x*e   *log(x)
------------ + ---------------------
x + 2*sin(x)          5*log(5)
$$\frac{x + 5}{x + 2 \sin{\left(x \right)}} + \frac{\sqrt[5]{x + x e^{- x}} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
                    5 ____________________________       
      5 + x         \/ x - x*(-cosh(x) + sinh(x)) *log(x)
----------------- + -------------------------------------
         /    pi\                  5*log(5)              
x + 2*cos|x - --|                                        
         \    2 /
$$\frac{x + 5}{x + 2 \cos{\left(x - \frac{\pi}{2} \right)}} + \frac{\sqrt[5]{- x \left(\sinh{\left(x \right)} - \cosh{\left(x \right)}\right) + x} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
                  5 ____________________________       
     5 + x        \/ x - x*(-cosh(x) + sinh(x)) *log(x)
--------------- + -------------------------------------
           /x\                   5*log(5)              
      4*cot|-|                                         
           \2/                                         
x + -----------                                        
           2/x\                                        
    1 + cot |-|                                        
            \2/
$$\frac{x + 5}{x + \frac{4 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}} + \frac{\sqrt[5]{- x \left(\sinh{\left(x \right)} - \cosh{\left(x \right)}\right) + x} \log{\left(x \right)}}{5 \log{\left(5 \right)}}$$ 
(5 + x)/(x + 4*cot(x/2)/(1 + cot(x/2)^2)) + (x - x*(-cosh(x) + sinh(x)))^(1/5)*log(x)/(5*log(5))
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