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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
                                         1/2 + x                              
                                 1 + ---------------                  / 2    \
                                        ____________                  |x    x|
   ____________                        /  2                 (1/2 + x)*|-- - -|
  /  2          /  1   x\            \/  x  + x + 1                   \4    4/
\/  x  + x + 1 *|- - + -| + ----------------------------- + ------------------
                \  4   2/     /             ____________\       ____________  
                              |            /  2         |      /  2           
                            4*\x + 1/2 + \/  x  + x + 1 /    \/  x  + x + 1
(x+12)(x24−x4)(x2+x)+1+((x2−14)(x2+x)+1+x+12(x2+x)+1+14((x+12)+(x2+x)+1))\frac{\left(x + \frac{1}{2}\right) \left(\frac{x^{2}}{4} - \frac{x}{4}\right)}{\sqrt{\left(x^{2} + x\right) + 1}} + \left(\left(\frac{x}{2} - \frac{1}{4}\right) \sqrt{\left(x^{2} + x\right) + 1} + \frac{\frac{x + \frac{1}{2}}{\sqrt{\left(x^{2} + x\right) + 1}} + 1}{4 \left(\left(x + \frac{1}{2}\right) + \sqrt{\left(x^{2} + x\right) + 1}\right)}\right) 
sqrt(x^2 + x + 1)*(-1/4 + x/2) + (1 + (1/2 + x)/sqrt(x^2 + x + 1))/((4*(x + 1/2 + sqrt(x^2 + x + 1)))) + ((1/2 + x)*(x^2/4 - x/4))/sqrt(x^2 + x + 1)
Simplificación general [src] 
   /           2\
 x*\1 + x + 6*x /
-----------------
     ____________
    /          2 
8*\/  1 + x + x
x(6x2+x+1)8x2+x+1\frac{x \left(6 x^{2} + x + 1\right)}{8 \sqrt{x^{2} + x + 1}} 
x*(1 + x + 6*x^2)/(8*sqrt(1 + x + x^2))
Respuesta numérica [src] 
(1.0 + x + x^2)^0.5*(-0.25 + 0.5*x) + (1.0 + (1.0 + x + x^2)^(-0.5)*(0.5 + x))/(2.0 + 4.0*x + 4.0*(1.0 + x + x^2)^0.5) + (1.0 + x + x^2)^(-0.5)*(0.5 + x)*(0.25*x^2 - 0.25*x)
(1.0 + x + x^2)^0.5*(-0.25 + 0.5*x) + (1.0 + (1.0 + x + x^2)^(-0.5)*(0.5 + x))/(2.0 + 4.0*x + 4.0*(1.0 + x + x^2)^0.5) + (1.0 + x + x^2)^(-0.5)*(0.5 + x)*(0.25*x^2 - 0.25*x)
Compilar la expresión [src] 
                                        1/2 + x                               
                                1 + ---------------                 /       2\
                                       ____________                 |  x   x |
   ____________                       /          2        (1/2 + x)*|- - + --|
  /          2  /  1   x\           \/  1 + x + x                   \  4   4 /
\/  1 + x + x  *|- - + -| + --------------------------- + --------------------
                \  4   2/                  ____________        ____________   
                                          /          2        /          2    
                            2 + 4*x + 4*\/  1 + x + x       \/  1 + x + x
(x2−14)x2+x+1+(x+12)(x24−x4)x2+x+1+x+12x2+x+1+14x+4x2+x+1+2\left(\frac{x}{2} - \frac{1}{4}\right) \sqrt{x^{2} + x + 1} + \frac{\left(x + \frac{1}{2}\right) \left(\frac{x^{2}}{4} - \frac{x}{4}\right)}{\sqrt{x^{2} + x + 1}} + \frac{\frac{x + \frac{1}{2}}{\sqrt{x^{2} + x + 1}} + 1}{4 x + 4 \sqrt{x^{2} + x + 1} + 2} 
sqrt(1 + x + x^2)*(-1/4 + x/2) + (1 + (1/2 + x)/sqrt(1 + x + x^2))/(2 + 4*x + 4*sqrt(1 + x + x^2)) + (1/2 + x)*(-x/4 + x^2/4)/sqrt(1 + x + x^2)
Parte trigonométrica [src] 
                                        1/2 + x                               
                                1 + ---------------                 /       2\
                                       ____________                 |  x   x |
   ____________                       /          2        (1/2 + x)*|- - + --|
  /          2  /  1   x\           \/  1 + x + x                   \  4   4 /
\/  1 + x + x  *|- - + -| + --------------------------- + --------------------
                \  4   2/                  ____________        ____________   
                                          /          2        /          2    
                            2 + 4*x + 4*\/  1 + x + x       \/  1 + x + x
(x2−14)x2+x+1+(x+12)(x24−x4)x2+x+1+x+12x2+x+1+14x+4x2+x+1+2\left(\frac{x}{2} - \frac{1}{4}\right) \sqrt{x^{2} + x + 1} + \frac{\left(x + \frac{1}{2}\right) \left(\frac{x^{2}}{4} - \frac{x}{4}\right)}{\sqrt{x^{2} + x + 1}} + \frac{\frac{x + \frac{1}{2}}{\sqrt{x^{2} + x + 1}} + 1}{4 x + 4 \sqrt{x^{2} + x + 1} + 2} 
sqrt(1 + x + x^2)*(-1/4 + x/2) + (1 + (1/2 + x)/sqrt(1 + x + x^2))/(2 + 4*x + 4*sqrt(1 + x + x^2)) + (1/2 + x)*(-x/4 + x^2/4)/sqrt(1 + x + x^2)
Denominador común [src] 
                                            ____________        ____________           ____________        
           2                       2       /          2        /          2       2   /          2         
1   x   3*x           2 + 2*x + 2*x  + 4*\/  1 + x + x   - x*\/  1 + x + x   + 9*x *\/  1 + x + x          
- - - + ---- - --------------------------------------------------------------------------------------------
4   4    4                       ____________                          ____________            ____________
                       3        /          2               2          /          2        2   /          2 
               8 + 16*x  + 16*\/  1 + x + x   + 24*x + 24*x  + 16*x*\/  1 + x + x   + 16*x *\/  1 + x + x
3x24−x4−9x2x2+x+1+2x2−xx2+x+1+2x+4x2+x+1+216x3+16x2x2+x+1+24x2+16xx2+x+1+24x+16x2+x+1+8+14\frac{3 x^{2}}{4} - \frac{x}{4} - \frac{9 x^{2} \sqrt{x^{2} + x + 1} + 2 x^{2} - x \sqrt{x^{2} + x + 1} + 2 x + 4 \sqrt{x^{2} + x + 1} + 2}{16 x^{3} + 16 x^{2} \sqrt{x^{2} + x + 1} + 24 x^{2} + 16 x \sqrt{x^{2} + x + 1} + 24 x + 16 \sqrt{x^{2} + x + 1} + 8} + \frac{1}{4} 
1/4 - x/4 + 3*x^2/4 - (2 + 2*x + 2*x^2 + 4*sqrt(1 + x + x^2) - x*sqrt(1 + x + x^2) + 9*x^2*sqrt(1 + x + x^2))/(8 + 16*x^3 + 16*sqrt(1 + x + x^2) + 24*x + 24*x^2 + 16*x*sqrt(1 + x + x^2) + 16*x^2*sqrt(1 + x + x^2))
Unión de expresiones racionales [src] 
2 + 2*(1 + x*(1 + x))*(-1 + 2*x) + x*(1 + 2*x)*(-1 + x)
-------------------------------------------------------
                      _______________                  
                  8*\/ 1 + x*(1 + x)
x(x−1)(2x+1)+2(2x−1)(x(x+1)+1)+28x(x+1)+1\frac{x \left(x - 1\right) \left(2 x + 1\right) + 2 \left(2 x - 1\right) \left(x \left(x + 1\right) + 1\right) + 2}{8 \sqrt{x \left(x + 1\right) + 1}} 
(2 + 2*(1 + x*(1 + x))*(-1 + 2*x) + x*(1 + 2*x)*(-1 + x))/(8*sqrt(1 + x*(1 + x)))
Combinatoria [src] 
   /           2\
 x*\1 + x + 6*x /
-----------------
     ____________
    /          2 
8*\/  1 + x + x
x(6x2+x+1)8x2+x+1\frac{x \left(6 x^{2} + x + 1\right)}{8 \sqrt{x^{2} + x + 1}} 
x*(1 + x + 6*x^2)/(8*sqrt(1 + x + x^2))
Potencias [src] 
                                        1/2 + x                               
                                1 + ---------------                 /       2\
                                       ____________                 |  x   x |
   ____________                       /          2        (1/2 + x)*|- - + --|
  /          2  /  1   x\           \/  1 + x + x                   \  4   4 /
\/  1 + x + x  *|- - + -| + --------------------------- + --------------------
                \  4   2/                  ____________        ____________   
                                          /          2        /          2    
                            2 + 4*x + 4*\/  1 + x + x       \/  1 + x + x
(x2−14)x2+x+1+(x+12)(x24−x4)x2+x+1+x+12x2+x+1+14x+4x2+x+1+2\left(\frac{x}{2} - \frac{1}{4}\right) \sqrt{x^{2} + x + 1} + \frac{\left(x + \frac{1}{2}\right) \left(\frac{x^{2}}{4} - \frac{x}{4}\right)}{\sqrt{x^{2} + x + 1}} + \frac{\frac{x + \frac{1}{2}}{\sqrt{x^{2} + x + 1}} + 1}{4 x + 4 \sqrt{x^{2} + x + 1} + 2} 
sqrt(1 + x + x^2)*(-1/4 + x/2) + (1 + (1/2 + x)/sqrt(1 + x + x^2))/(2 + 4*x + 4*sqrt(1 + x + x^2)) + (1/2 + x)*(-x/4 + x^2/4)/sqrt(1 + x + x^2)
Denominador racional [src] 
             ____________              ____________              ____________              ____________            ____________                      3/2                       3/2                       3/2
        4   /          2          5   /          2          3   /          2          2   /          2            /          2           /         2\            2 /         2\            3 /         2\   
- 7168*x *\/  1 + x + x   - 6144*x *\/  1 + x + x   - 3584*x *\/  1 + x + x   - 1280*x *\/  1 + x + x   - 256*x*\/  1 + x + x   + 1024*x*\1 + x + x /    + 1024*x *\1 + x + x /    + 6144*x *\1 + x + x /   
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                2                                                                                           
                                                                                          6144 + 6144*x + 6144*x
−6144x5x2+x+1−7168x4x2+x+1+6144x3(x2+x+1)32−3584x3x2+x+1+1024x2(x2+x+1)32−1280x2x2+x+1+1024x(x2+x+1)32−256xx2+x+16144x2+6144x+6144\frac{- 6144 x^{5} \sqrt{x^{2} + x + 1} - 7168 x^{4} \sqrt{x^{2} + x + 1} + 6144 x^{3} \left(x^{2} + x + 1\right)^{\frac{3}{2}} - 3584 x^{3} \sqrt{x^{2} + x + 1} + 1024 x^{2} \left(x^{2} + x + 1\right)^{\frac{3}{2}} - 1280 x^{2} \sqrt{x^{2} + x + 1} + 1024 x \left(x^{2} + x + 1\right)^{\frac{3}{2}} - 256 x \sqrt{x^{2} + x + 1}}{6144 x^{2} + 6144 x + 6144} 
(-7168*x^4*sqrt(1 + x + x^2) - 6144*x^5*sqrt(1 + x + x^2) - 3584*x^3*sqrt(1 + x + x^2) - 1280*x^2*sqrt(1 + x + x^2) - 256*x*sqrt(1 + x + x^2) + 1024*x*(1 + x + x^2)^(3/2) + 1024*x^2*(1 + x + x^2)^(3/2) + 6144*x^3*(1 + x + x^2)^(3/2))/(6144 + 6144*x + 6144*x^2)
Abrimos la expresión [src] 
                                                              1/2 + x         
                                      / 2    \        1 + ---------------     
                                      |x    x|               ____________     
   ____________             (1/2 + x)*|-- - -|              /  2              
  /  2          /  1   x\             \4    4/            \/  x  + x + 1      
\/  x  + x + 1 *|- - + -| + ------------------ + -----------------------------
                \  4   2/       ____________       /             ____________\
                               /  2                |            /  2         |
                             \/  x  + x + 1      4*\x + 1/2 + \/  x  + x + 1 /
(x+12)(x24−x4)(x2+x)+1+(x2−14)(x2+x)+1+x+12(x2+x)+1+14((x+12)+(x2+x)+1)\frac{\left(x + \frac{1}{2}\right) \left(\frac{x^{2}}{4} - \frac{x}{4}\right)}{\sqrt{\left(x^{2} + x\right) + 1}} + \left(\frac{x}{2} - \frac{1}{4}\right) \sqrt{\left(x^{2} + x\right) + 1} + \frac{\frac{x + \frac{1}{2}}{\sqrt{\left(x^{2} + x\right) + 1}} + 1}{4 \left(\left(x + \frac{1}{2}\right) + \sqrt{\left(x^{2} + x\right) + 1}\right)} 
sqrt(x^2 + x + 1)*(-1/4 + x/2) + ((1/2 + x)*(x^2/4 - x/4))/sqrt(x^2 + x + 1) + (1 + (1/2 + x)/sqrt(x^2 + x + 1))/(4*(x + 1/2 + sqrt(x^2 + x + 1)))
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