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Simplificación de expresiones/ Descomposición en fracciones simples/ (1+(1+(1+x)/(1-3x))/(1-3\(1+x)/(1-3x)))/(-3\(1+(1+x)/(1-3x))/(1-3\(1+x)/(1-3x)))

¿Cómo vas a descomponer esta (1+(1+(1+x)/(1-3x))/(1-3\(1+x)/(1-3x)))/(-3\(1+(1+x)/(1-3x))/(1-3\(1+x)/(1-3x))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
         1 + x 
    1 + -------
        1 - 3*x
1 + -----------
        /  3  \
        |-----|
        \1 + x/
    1 - -------
        1 - 3*x
---------------
//    -3     \\
||-----------||
||     1 + x ||
||1 + -------||
|\    1 - 3*x/|
|-------------|
|     /  3  \ |
|     |-----| |
|     \1 + x/ |
| 1 - ------- |
\     1 - 3*x /
1+1+x+11−3x1−31x+11−3x11−31x+11−3x(−31+x+11−3x)\frac{1 + \frac{1 + \frac{x + 1}{1 - 3 x}}{1 - \frac{3 \frac{1}{x + 1}}{1 - 3 x}}}{\frac{1}{1 - \frac{3 \frac{1}{x + 1}}{1 - 3 x}} \left(- \frac{3}{1 + \frac{x + 1}{1 - 3 x}}\right)} 
(1 + (1 + (1 + x)/(1 - 3*x))/(1 - 3/(1 + x)/(1 - 3*x)))/(((-3/(1 + (1 + x)/(1 - 3*x)))/(1 - 3/(1 + x)/(1 - 3*x))))
Descomposición de una fracción [src] 
-10/27 + 1/(4*(1 + x)) + 11/(27*(-1 + 3*x)^2) + 31/(108*(-1 + 3*x))
−1027+31108(3x−1)+1127(3x−1)2+14(x+1)- \frac{10}{27} + \frac{31}{108 \left(3 x - 1\right)} + \frac{11}{27 \left(3 x - 1\right)^{2}} + \frac{1}{4 \left(x + 1\right)} 
  10       1             11               31      
- -- + --------- + -------------- + --------------
  27   4*(1 + x)                2   108*(-1 + 3*x)
                   27*(-1 + 3*x)
Simplificación general [src] 
      /       2      \ 
  2*x*\2 - 5*x  + 3*x/ 
-----------------------
              2       3
3 - 15*x + 9*x  + 27*x
2x(−5x2+3x+2)27x3+9x2−15x+3\frac{2 x \left(- 5 x^{2} + 3 x + 2\right)}{27 x^{3} + 9 x^{2} - 15 x + 3} 
2*x*(2 - 5*x^2 + 3*x)/(3 - 15*x + 9*x^2 + 27*x^3)
Respuesta numérica [src] 
-0.333333333333333*(1.0 + (1.0 + x)/(1.0 - 3.0*x))*(1.0 + (1.0 + (1.0 + x)/(1.0 - 3.0*x))/(1.0 - 3.0/((1.0 + x)*(1.0 - 3.0*x))))*(1.0 - 3.0/((1.0 + x)*(1.0 - 3.0*x)))
-0.333333333333333*(1.0 + (1.0 + x)/(1.0 - 3.0*x))*(1.0 + (1.0 + (1.0 + x)/(1.0 - 3.0*x))/(1.0 - 3.0/((1.0 + x)*(1.0 - 3.0*x))))*(1.0 - 3.0/((1.0 + x)*(1.0 - 3.0*x)))
Compilar la expresión [src] 
               /              1 + x      \                         
               |         1 + -------     |                         
 /     1 + x \ |             1 - 3*x     | /            3        \ 
-|1 + -------|*|1 + ---------------------|*|1 - -----------------| 
 \    1 - 3*x/ |                3        | \    (1 + x)*(1 - 3*x)/ 
               |    1 - -----------------|                         
               \        (1 + x)*(1 - 3*x)/                         
-------------------------------------------------------------------
                                 3
−(1−3(1−3x)(x+1))(1+x+11−3x)(1+1+x+11−3x1−3(1−3x)(x+1))3- \frac{\left(1 - \frac{3}{\left(1 - 3 x\right) \left(x + 1\right)}\right) \left(1 + \frac{x + 1}{1 - 3 x}\right) \left(1 + \frac{1 + \frac{x + 1}{1 - 3 x}}{1 - \frac{3}{\left(1 - 3 x\right) \left(x + 1\right)}}\right)}{3} 
-(1 + (1 + x)/(1 - 3*x))*(1 + (1 + (1 + x)/(1 - 3*x))/(1 - 3/((1 + x)*(1 - 3*x))))*(1 - 3/((1 + x)*(1 - 3*x)))/3
Parte trigonométrica [src] 
               /              1 + x      \                         
               |         1 + -------     |                         
 /     1 + x \ |             1 - 3*x     | /            3        \ 
-|1 + -------|*|1 + ---------------------|*|1 - -----------------| 
 \    1 - 3*x/ |                3        | \    (1 + x)*(1 - 3*x)/ 
               |    1 - -----------------|                         
               \        (1 + x)*(1 - 3*x)/                         
-------------------------------------------------------------------
                                 3
−(1−3(1−3x)(x+1))(1+x+11−3x)(1+1+x+11−3x1−3(1−3x)(x+1))3- \frac{\left(1 - \frac{3}{\left(1 - 3 x\right) \left(x + 1\right)}\right) \left(1 + \frac{x + 1}{1 - 3 x}\right) \left(1 + \frac{1 + \frac{x + 1}{1 - 3 x}}{1 - \frac{3}{\left(1 - 3 x\right) \left(x + 1\right)}}\right)}{3} 
-(1 + (1 + x)/(1 - 3*x))*(1 + (1 + (1 + x)/(1 - 3*x))/(1 - 3/((1 + x)*(1 - 3*x))))*(1 - 3/((1 + x)*(1 - 3*x)))/3
Abrimos la expresión [src] 
/         1 + x \ /    /  3  \\                    
|    1 + -------| |    |-----||                    
|        1 - 3*x| |    \1 + x/| /  1      1 + x   \
|1 + -----------|*|1 - -------|*|- - - -----------|
|        /  3  \| \    1 - 3*x/ \  3   3*(1 - 3*x)/
|        |-----||                                  
|        \1 + x/|                                  
|    1 - -------|                                  
\        1 - 3*x/
(−13−x+13(1−3x))(1−31x+11−3x)(1+1+x+11−3x1−31x+11−3x)\left(- \frac{1}{3} - \frac{x + 1}{3 \left(1 - 3 x\right)}\right) \left(1 - \frac{3 \frac{1}{x + 1}}{1 - 3 x}\right) \left(1 + \frac{1 + \frac{x + 1}{1 - 3 x}}{1 - \frac{3 \frac{1}{x + 1}}{1 - 3 x}}\right) 
(1 + (1 + (1 + x)/(1 - 3*x))/(1 - 3/(1 + x)/(1 - 3*x)))*(1 - 3/(1 + x)/(1 - 3*x))*(-1/3 - (1 + x)/(3*(1 - 3*x)))
Denominador racional [src] 
(2 - 2*x)*((1 - 3*x)*(-3 + (1 + x)*(1 - 3*x)) + (1 + x)*(1 - 3*x)*(2 - 2*x))
----------------------------------------------------------------------------
                                        2                                   
                       (1 + x)*(1 - 3*x) *(-3 + 9*x)
(2−2x)((1−3x)(2−2x)(x+1)+(1−3x)((1−3x)(x+1)−3))(1−3x)2(x+1)(9x−3)\frac{\left(2 - 2 x\right) \left(\left(1 - 3 x\right) \left(2 - 2 x\right) \left(x + 1\right) + \left(1 - 3 x\right) \left(\left(1 - 3 x\right) \left(x + 1\right) - 3\right)\right)}{\left(1 - 3 x\right)^{2} \left(x + 1\right) \left(9 x - 3\right)} 
(2 - 2*x)*((1 - 3*x)*(-3 + (1 + x)*(1 - 3*x)) + (1 + x)*(1 - 3*x)*(2 - 2*x))/((1 + x)*(1 - 3*x)^2*(-3 + 9*x))
Combinatoria [src] 
-2*x*(-1 + x)*(2 + 5*x)
-----------------------
                     2 
 3*(1 + x)*(-1 + 3*x)
−2x(x−1)(5x+2)3(x+1)(3x−1)2- \frac{2 x \left(x - 1\right) \left(5 x + 2\right)}{3 \left(x + 1\right) \left(3 x - 1\right)^{2}} 
-2*x*(-1 + x)*(2 + 5*x)/(3*(1 + x)*(-1 + 3*x)^2)
Unión de expresiones racionales [src] 
-2*(1 - x)*(-3 + (1 + x)*(1 - 3*x) + 2*(1 + x)*(1 - x))
-------------------------------------------------------
                                     2                 
                  3*(1 + x)*(1 - 3*x)
−2(1−x)((1−3x)(x+1)+2(1−x)(x+1)−3)3(1−3x)2(x+1)- \frac{2 \left(1 - x\right) \left(\left(1 - 3 x\right) \left(x + 1\right) + 2 \left(1 - x\right) \left(x + 1\right) - 3\right)}{3 \left(1 - 3 x\right)^{2} \left(x + 1\right)} 
-2*(1 - x)*(-3 + (1 + x)*(1 - 3*x) + 2*(1 + x)*(1 - x))/(3*(1 + x)*(1 - 3*x)^2)
Denominador común [src] 
                            2     
  10        10 - 14*x + 84*x      
- -- + ---------------------------
  27                    2        3
       27 - 135*x + 81*x  + 243*x
84x2−14x+10243x3+81x2−135x+27−1027\frac{84 x^{2} - 14 x + 10}{243 x^{3} + 81 x^{2} - 135 x + 27} - \frac{10}{27} 
-10/27 + (10 - 14*x + 84*x^2)/(27 - 135*x + 81*x^2 + 243*x^3)
Potencias [src] 
               /              1 + x      \                         
               |         1 + -------     |                         
 /     1 + x \ |             1 - 3*x     | /            3        \ 
-|1 + -------|*|1 + ---------------------|*|1 - -----------------| 
 \    1 - 3*x/ |                3        | \    (1 + x)*(1 - 3*x)/ 
               |    1 - -----------------|                         
               \        (1 + x)*(1 - 3*x)/                         
-------------------------------------------------------------------
                                 3
−(1−3(1−3x)(x+1))(1+x+11−3x)(1+1+x+11−3x1−3(1−3x)(x+1))3- \frac{\left(1 - \frac{3}{\left(1 - 3 x\right) \left(x + 1\right)}\right) \left(1 + \frac{x + 1}{1 - 3 x}\right) \left(1 + \frac{1 + \frac{x + 1}{1 - 3 x}}{1 - \frac{3}{\left(1 - 3 x\right) \left(x + 1\right)}}\right)}{3} 
-(1 + (1 + x)/(1 - 3*x))*(1 + (1 + (1 + x)/(1 - 3*x))/(1 - 3/((1 + x)*(1 - 3*x))))*(1 - 3/((1 + x)*(1 - 3*x)))/3
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