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Simplificación de expresiones/ Descomposición en fracciones simples/ 3*(1/(8*x^(5/2))-16*x^(7/2)/(1+x^2)^3+12*x^(3/2)/(1+x^2)^2-1/(2*sqrt(x)*(1+x^2)))/(1+x^2)

¿Cómo vas a descomponer esta 3*(1/(8*x^(5/2))-16*x^(7/2)/(1+x^2)^3+12*x^(3/2)/(1+x^2)^2-1/(2*sqrt(x)*(1+x^2)))/(1+x^2) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
  /              7/2         3/2                    \
  |  1       16*x        12*x              1        |
3*|------ - --------- + --------- - ----------------|
  |   5/2           3           2       ___ /     2\|
  |8*x      /     2\    /     2\    2*\/ x *\1 + x /|
  \         \1 + x /    \1 + x /                    /
-----------------------------------------------------
                             2                       
                        1 + x
3((12x32(x2+1)2+(−16x72(x2+1)3+18x52))−12x(x2+1))x2+1\frac{3 \left(\left(\frac{12 x^{\frac{3}{2}}}{\left(x^{2} + 1\right)^{2}} + \left(- \frac{16 x^{\frac{7}{2}}}{\left(x^{2} + 1\right)^{3}} + \frac{1}{8 x^{\frac{5}{2}}}\right)\right) - \frac{1}{2 \sqrt{x} \left(x^{2} + 1\right)}\right)}{x^{2} + 1} 
(3*(1/(8*x^(5/2)) - 16*x^(7/2)/(1 + x^2)^3 + (12*x^(3/2))/(1 + x^2)^2 - 1/((2*sqrt(x))*(1 + x^2))))/(1 + x^2)
Simplificación general [src] 
   /          4      2        6\    
  -\-3 - 273*x  + 3*x  + 105*x /    
------------------------------------
   5/2 /     8      2      6      4\
8*x   *\1 + x  + 4*x  + 4*x  + 6*x /
−105x6−273x4+3x2−38x52(x8+4x6+6x4+4x2+1)- \frac{105 x^{6} - 273 x^{4} + 3 x^{2} - 3}{8 x^{\frac{5}{2}} \left(x^{8} + 4 x^{6} + 6 x^{4} + 4 x^{2} + 1\right)} 
-(-3 - 273*x^4 + 3*x^2 + 105*x^6)/(8*x^(5/2)*(1 + x^8 + 4*x^2 + 4*x^6 + 6*x^4))
Abrimos la expresión [src] 
               7/2         3/2                   
  3      3*16*x      3*12*x             3        
------ - --------- + --------- - ----------------
   5/2           3           2       ___ /     2\
8*x      /     2\    /     2\    2*\/ x *\1 + x /
         \1 + x /    \1 + x /                    
-------------------------------------------------
                           2                     
                      1 + x
3⋅12x32(x2+1)2−3⋅16x72(x2+1)3−32x(x2+1)+38x52x2+1\frac{\frac{3 \cdot 12 x^{\frac{3}{2}}}{\left(x^{2} + 1\right)^{2}} - \frac{3 \cdot 16 x^{\frac{7}{2}}}{\left(x^{2} + 1\right)^{3}} - \frac{3}{2 \sqrt{x} \left(x^{2} + 1\right)} + \frac{3}{8 x^{\frac{5}{2}}}}{x^{2} + 1} 
(3/(8*x^(5/2)) - 3*16*x^(7/2)/(1 + x^2)^3 + 3*(12*x^(3/2))/(1 + x^2)^2 - 3/(2*sqrt(x)*(1 + x^2)))/(1 + x^2)
Denominador racional [src] 
                  5                    /        2 /        3         \                 3\
      5/2 /     2\        ___ /     2\ |/     2\  |/     2\         6|       4 /     2\ |
- 24*x   *\1 + x /  + 6*\/ x *\1 + x /*\\1 + x / *\\1 + x /  - 128*x / + 96*x *\1 + x / /
-----------------------------------------------------------------------------------------
                                                   7                                     
                                         3 /     2\                                      
                                     16*x *\1 + x /
−24x52(x2+1)5+6x(x2+1)(96x4(x2+1)3+(x2+1)2(−128x6+(x2+1)3))16x3(x2+1)7\frac{- 24 x^{\frac{5}{2}} \left(x^{2} + 1\right)^{5} + 6 \sqrt{x} \left(x^{2} + 1\right) \left(96 x^{4} \left(x^{2} + 1\right)^{3} + \left(x^{2} + 1\right)^{2} \left(- 128 x^{6} + \left(x^{2} + 1\right)^{3}\right)\right)}{16 x^{3} \left(x^{2} + 1\right)^{7}} 
(-24*x^(5/2)*(1 + x^2)^5 + 6*sqrt(x)*(1 + x^2)*((1 + x^2)^2*((1 + x^2)^3 - 128*x^6) + 96*x^4*(1 + x^2)^3))/(16*x^3*(1 + x^2)^7)
Respuesta numérica [src] 
(0.375*x^(-2.5) + 36.0*x^1.5/(1.0 + x^2)^2 - 1.5*x^(-0.5)/(1.0 + x^2) - 48.0*x^3.5/(1.0 + x^2)^3)/(1.0 + x^2)
(0.375*x^(-2.5) + 36.0*x^1.5/(1.0 + x^2)^2 - 1.5*x^(-0.5)/(1.0 + x^2) - 48.0*x^3.5/(1.0 + x^2)^3)/(1.0 + x^2)
Denominador común [src] 
              /          6        8        4      2       10        12\              
             -\-3 - 708*x  - 501*x  - 273*x  - 6*x  + 42*x   + 105*x  /              
-------------------------------------------------------------------------------------
   5/2      33/2       9/2       29/2        13/2        25/2        17/2        21/2
8*x    + 8*x     + 56*x    + 56*x     + 168*x     + 168*x     + 280*x     + 280*x
−105x12+42x10−501x8−708x6−273x4−6x2−38x332+56x292+168x252+280x212+280x172+168x132+56x92+8x52- \frac{105 x^{12} + 42 x^{10} - 501 x^{8} - 708 x^{6} - 273 x^{4} - 6 x^{2} - 3}{8 x^{\frac{33}{2}} + 56 x^{\frac{29}{2}} + 168 x^{\frac{25}{2}} + 280 x^{\frac{21}{2}} + 280 x^{\frac{17}{2}} + 168 x^{\frac{13}{2}} + 56 x^{\frac{9}{2}} + 8 x^{\frac{5}{2}}} 
-(-3 - 708*x^6 - 501*x^8 - 273*x^4 - 6*x^2 + 42*x^10 + 105*x^12)/(8*x^(5/2) + 8*x^(33/2) + 56*x^(9/2) + 56*x^(29/2) + 168*x^(13/2) + 168*x^(25/2) + 280*x^(17/2) + 280*x^(21/2))
Compilar la expresión [src] 
              7/2         3/2                    
  3       48*x        36*x              3        
------ - --------- + --------- - ----------------
   5/2           3           2       ___ /     2\
8*x      /     2\    /     2\    2*\/ x *\1 + x /
         \1 + x /    \1 + x /                    
-------------------------------------------------
                           2                     
                      1 + x
−48x72(x2+1)3+36x32(x2+1)2−32x(x2+1)+38x52x2+1\frac{- \frac{48 x^{\frac{7}{2}}}{\left(x^{2} + 1\right)^{3}} + \frac{36 x^{\frac{3}{2}}}{\left(x^{2} + 1\right)^{2}} - \frac{3}{2 \sqrt{x} \left(x^{2} + 1\right)} + \frac{3}{8 x^{\frac{5}{2}}}}{x^{2} + 1} 
(3/(8*x^(5/2)) - 48*x^(7/2)/(1 + x^2)^3 + 36*x^(3/2)/(1 + x^2)^2 - 3/(2*sqrt(x)*(1 + x^2)))/(1 + x^2)
Combinatoria [src] 
   /      2       4       6\
-3*\-1 + x  - 91*x  + 35*x /
----------------------------
                     4      
         5/2 /     2\       
      8*x   *\1 + x /
−3(35x6−91x4+x2−1)8x52(x2+1)4- \frac{3 \left(35 x^{6} - 91 x^{4} + x^{2} - 1\right)}{8 x^{\frac{5}{2}} \left(x^{2} + 1\right)^{4}} 
-3*(-1 + x^2 - 91*x^4 + 35*x^6)/(8*x^(5/2)*(1 + x^2)^4)
Unión de expresiones racionales [src] 
  /        3                         2                 \
  |/     2\         6      2 /     2\        4 /     2\|
3*\\1 + x /  - 128*x  - 4*x *\1 + x /  + 96*x *\1 + x //
--------------------------------------------------------
                                   4                    
                       5/2 /     2\                     
                    8*x   *\1 + x /
3(−128x6+96x4(x2+1)−4x2(x2+1)2+(x2+1)3)8x52(x2+1)4\frac{3 \left(- 128 x^{6} + 96 x^{4} \left(x^{2} + 1\right) - 4 x^{2} \left(x^{2} + 1\right)^{2} + \left(x^{2} + 1\right)^{3}\right)}{8 x^{\frac{5}{2}} \left(x^{2} + 1\right)^{4}} 
3*((1 + x^2)^3 - 128*x^6 - 4*x^2*(1 + x^2)^2 + 96*x^4*(1 + x^2))/(8*x^(5/2)*(1 + x^2)^4)
Potencias [src] 
              7/2         3/2                    
  3       48*x        36*x              3        
------ - --------- + --------- - ----------------
   5/2           3           2       ___ /     2\
8*x      /     2\    /     2\    2*\/ x *\1 + x /
         \1 + x /    \1 + x /                    
-------------------------------------------------
                           2                     
                      1 + x
−48x72(x2+1)3+36x32(x2+1)2−32x(x2+1)+38x52x2+1\frac{- \frac{48 x^{\frac{7}{2}}}{\left(x^{2} + 1\right)^{3}} + \frac{36 x^{\frac{3}{2}}}{\left(x^{2} + 1\right)^{2}} - \frac{3}{2 \sqrt{x} \left(x^{2} + 1\right)} + \frac{3}{8 x^{\frac{5}{2}}}}{x^{2} + 1} 
              7/2                            3/2 
  3       48*x              3            36*x    
------ - --------- - ---------------- + ---------
   5/2           3     ___ /       2\           2
8*x      /     2\    \/ x *\2 + 2*x /   /     2\ 
         \1 + x /                       \1 + x / 
-------------------------------------------------
                           2                     
                      1 + x
−48x72(x2+1)3+36x32(x2+1)2−3x(2x2+2)+38x52x2+1\frac{- \frac{48 x^{\frac{7}{2}}}{\left(x^{2} + 1\right)^{3}} + \frac{36 x^{\frac{3}{2}}}{\left(x^{2} + 1\right)^{2}} - \frac{3}{\sqrt{x} \left(2 x^{2} + 2\right)} + \frac{3}{8 x^{\frac{5}{2}}}}{x^{2} + 1} 
(3/(8*x^(5/2)) - 48*x^(7/2)/(1 + x^2)^3 - 3/(sqrt(x)*(2 + 2*x^2)) + 36*x^(3/2)/(1 + x^2)^2)/(1 + x^2)
Parte trigonométrica [src] 
              7/2         3/2                    
  3       48*x        36*x              3        
------ - --------- + --------- - ----------------
   5/2           3           2       ___ /     2\
8*x      /     2\    /     2\    2*\/ x *\1 + x /
         \1 + x /    \1 + x /                    
-------------------------------------------------
                           2                     
                      1 + x
−48x72(x2+1)3+36x32(x2+1)2−32x(x2+1)+38x52x2+1\frac{- \frac{48 x^{\frac{7}{2}}}{\left(x^{2} + 1\right)^{3}} + \frac{36 x^{\frac{3}{2}}}{\left(x^{2} + 1\right)^{2}} - \frac{3}{2 \sqrt{x} \left(x^{2} + 1\right)} + \frac{3}{8 x^{\frac{5}{2}}}}{x^{2} + 1} 
(3/(8*x^(5/2)) - 48*x^(7/2)/(1 + x^2)^3 + 36*x^(3/2)/(1 + x^2)^2 - 3/(2*sqrt(x)*(1 + x^2)))/(1 + x^2)
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