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Simplificación de expresiones/ Descomposición en fracciones simples/ exp(x*t)*(-2)/(x/5+1)^2+10*t*exp(x*t)/(x/5+1)

¿Cómo vas a descomponer esta exp(x*t)*(-2)/(x/5+1)^2+10*t*exp(x*t)/(x/5+1) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
 x*t              x*t
e   *(-2)   10*t*e   
--------- + ---------
        2     x      
 /x    \      - + 1  
 |- + 1|      5      
 \5    /
(−2)etx(x5+1)2+10tetxx5+1\frac{\left(-2\right) e^{t x}}{\left(\frac{x}{5} + 1\right)^{2}} + \frac{10 t e^{t x}}{\frac{x}{5} + 1} 
(exp(x*t)*(-2))/(x/5 + 1)^2 + ((10*t)*exp(x*t))/(x/5 + 1)
Simplificación general [src] 
   /                  2\  t*x
50*\-5 - x + t*(5 + x) /*e   
-----------------------------
                  3          
           (5 + x)
50(t(x+5)2−x−5)etx(x+5)3\frac{50 \left(t \left(x + 5\right)^{2} - x - 5\right) e^{t x}}{\left(x + 5\right)^{3}} 
50*(-5 - x + t*(5 + x)^2)*exp(t*x)/(5 + x)^3
Respuesta numérica [src] 
-2.0*exp(x*t)/(1.0 + 0.2*x)^2 + 10.0*t*exp(x*t)/(1.0 + 0.2*x)
-2.0*exp(x*t)/(1.0 + 0.2*x)^2 + 10.0*t*exp(x*t)/(1.0 + 0.2*x)
Compilar la expresión [src] 
      x*t          x*t
   2*e       10*t*e   
- -------- + ---------
         2         x  
  /    x\      1 + -  
  |1 + -|          5  
  \    5/
10tetxx5+1−2etx(x5+1)2\frac{10 t e^{t x}}{\frac{x}{5} + 1} - \frac{2 e^{t x}}{\left(\frac{x}{5} + 1\right)^{2}} 
-2*exp(x*t)/(1 + x/5)^2 + 10*t*exp(x*t)/(1 + x/5)
Unión de expresiones racionales [src] 
                     t*x
50*(-1 + t*(5 + x))*e   
------------------------
               2        
        (5 + x)
50(t(x+5)−1)etx(x+5)2\frac{50 \left(t \left(x + 5\right) - 1\right) e^{t x}}{\left(x + 5\right)^{2}} 
50*(-1 + t*(5 + x))*exp(t*x)/(5 + x)^2
Potencias [src] 
      t*x          t*x
   2*e       10*t*e   
- -------- + ---------
         2         x  
  /    x\      1 + -  
  |1 + -|          5  
  \    5/
10tetxx5+1−2etx(x5+1)2\frac{10 t e^{t x}}{\frac{x}{5} + 1} - \frac{2 e^{t x}}{\left(\frac{x}{5} + 1\right)^{2}} 
-2*exp(t*x)/(1 + x/5)^2 + 10*t*exp(t*x)/(1 + x/5)
Denominador racional [src] 
       t*x         t*x               2  t*x
- 250*e    - 50*x*e    + 50*t*(5 + x) *e   
-------------------------------------------
                         3                 
                  (5 + x)
50t(x+5)2etx−50xetx−250etx(x+5)3\frac{50 t \left(x + 5\right)^{2} e^{t x} - 50 x e^{t x} - 250 e^{t x}}{\left(x + 5\right)^{3}} 
(-250*exp(t*x) - 50*x*exp(t*x) + 50*t*(5 + x)^2*exp(t*x))/(5 + x)^3
Combinatoria [src] 
                     t*x
50*(-1 + 5*t + t*x)*e   
------------------------
               2        
        (5 + x)
50(tx+5t−1)etx(x+5)2\frac{50 \left(t x + 5 t - 1\right) e^{t x}}{\left(x + 5\right)^{2}} 
50*(-1 + 5*t + t*x)*exp(t*x)/(5 + x)^2
Denominador común [src] 
      t*x          t*x           t*x
- 50*e    + 250*t*e    + 50*t*x*e   
------------------------------------
                 2                  
           25 + x  + 10*x
50txetx+250tetx−50etxx2+10x+25\frac{50 t x e^{t x} + 250 t e^{t x} - 50 e^{t x}}{x^{2} + 10 x + 25} 
(-50*exp(t*x) + 250*t*exp(t*x) + 50*t*x*exp(t*x))/(25 + x^2 + 10*x)
Parte trigonométrica [src] 
  2*(cosh(t*x) + sinh(t*x))   10*t*(cosh(t*x) + sinh(t*x))
- ------------------------- + ----------------------------
                  2                          x            
           /    x\                       1 + -            
           |1 + -|                           5            
           \    5/
10t(sinh⁡(tx)+cosh⁡(tx))x5+1−2(sinh⁡(tx)+cosh⁡(tx))(x5+1)2\frac{10 t \left(\sinh{\left(t x \right)} + \cosh{\left(t x \right)}\right)}{\frac{x}{5} + 1} - \frac{2 \left(\sinh{\left(t x \right)} + \cosh{\left(t x \right)}\right)}{\left(\frac{x}{5} + 1\right)^{2}} 
      t*x          t*x
   2*e       10*t*e   
- -------- + ---------
         2         x  
  /    x\      1 + -  
  |1 + -|          5  
  \    5/
10tetxx5+1−2etx(x5+1)2\frac{10 t e^{t x}}{\frac{x}{5} + 1} - \frac{2 e^{t x}}{\left(\frac{x}{5} + 1\right)^{2}} 
-2*cosh(t*x) - 2*sinh(t*x)   10*t*(cosh(t*x) + sinh(t*x))
-------------------------- + ----------------------------
                2                           x            
         /    x\                        1 + -            
         |1 + -|                            5            
         \    5/
10t(sinh⁡(tx)+cosh⁡(tx))x5+1+−2sinh⁡(tx)−2cosh⁡(tx)(x5+1)2\frac{10 t \left(\sinh{\left(t x \right)} + \cosh{\left(t x \right)}\right)}{\frac{x}{5} + 1} + \frac{- 2 \sinh{\left(t x \right)} - 2 \cosh{\left(t x \right)}}{\left(\frac{x}{5} + 1\right)^{2}} 
  2*cosh(t*x) + 2*sinh(t*x)   10*t*(cosh(t*x) + sinh(t*x))
- ------------------------- + ----------------------------
                  2                          x            
           /    x\                       1 + -            
           |1 + -|                           5            
           \    5/
10t(sinh⁡(tx)+cosh⁡(tx))x5+1−2sinh⁡(tx)+2cosh⁡(tx)(x5+1)2\frac{10 t \left(\sinh{\left(t x \right)} + \cosh{\left(t x \right)}\right)}{\frac{x}{5} + 1} - \frac{2 \sinh{\left(t x \right)} + 2 \cosh{\left(t x \right)}}{\left(\frac{x}{5} + 1\right)^{2}} 
-(2*cosh(t*x) + 2*sinh(t*x))/(1 + x/5)^2 + 10*t*(cosh(t*x) + sinh(t*x))/(1 + x/5)
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