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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
            /        _____                                            \
            |      \/ q*x           2*q                               |
            |      ------- + -----------------                        |
            |         x        _____     _____             _____      |
  _________ |  1             \/ q*x  + \/ r*s          2*\/ q*x       |
\/ q*r*s*x *|- - + --------------------------- - ---------------------|
            |  x          _____     _____          /  _____     _____\|
            \           \/ q*x  + \/ r*s         x*\\/ q*x  + \/ r*s //
-----------------------------------------------------------------------
                            /  _____     _____\                        
                        4*x*\\/ q*x  + \/ r*s /
xsqr(−2qxx(qx+rs)+(2qqx+rs+qxxqx+rs−1x))4x(qx+rs)\frac{\sqrt{x s q r} \left(- \frac{2 \sqrt{q x}}{x \left(\sqrt{q x} + \sqrt{r s}\right)} + \left(\frac{\frac{2 q}{\sqrt{q x} + \sqrt{r s}} + \frac{\sqrt{q x}}{x}}{\sqrt{q x} + \sqrt{r s}} - \frac{1}{x}\right)\right)}{4 x \left(\sqrt{q x} + \sqrt{r s}\right)} 
(sqrt(((q*r)*s)*x)*(-1/x + (sqrt(q*x)/x + (2*q)/(sqrt(q*x) + sqrt(r*s)))/(sqrt(q*x) + sqrt(r*s)) - 2*sqrt(q*x)/(x*(sqrt(q*x) + sqrt(r*s)))))/(((4*x)*(sqrt(q*x) + sqrt(r*s))))
Simplificación general [src] 
            _________ /          _____   _____\           
         -\/ q*r*s*x *\r*s + 3*\/ q*x *\/ r*s /           
----------------------------------------------------------
   2 /     3/2        3/2           _____           _____\
4*x *\(q*x)    + (r*s)    + 3*q*x*\/ r*s  + 3*r*s*\/ q*x /
−qrsx(rs+3qxrs)4x2(3qxrs+3rsqx+(qx)32+(rs)32)- \frac{\sqrt{q r s x} \left(r s + 3 \sqrt{q x} \sqrt{r s}\right)}{4 x^{2} \left(3 q x \sqrt{r s} + 3 r s \sqrt{q x} + \left(q x\right)^{\frac{3}{2}} + \left(r s\right)^{\frac{3}{2}}\right)} 
-sqrt(q*r*s*x)*(r*s + 3*sqrt(q*x)*sqrt(r*s))/(4*x^2*((q*x)^(3/2) + (r*s)^(3/2) + 3*q*x*sqrt(r*s) + 3*r*s*sqrt(q*x)))
Respuesta numérica [src] 
0.25*(q*r*s*x)^0.5*(-1/x + ((q*x)^0.5/x + 2.0*q/((q*x)^0.5 + (r*s)^0.5))/((q*x)^0.5 + (r*s)^0.5) - 2.0*(q*x)^0.5/(x*((q*x)^0.5 + (r*s)^0.5)))/(x*((q*x)^0.5 + (r*s)^0.5))
0.25*(q*r*s*x)^0.5*(-1/x + ((q*x)^0.5/x + 2.0*q/((q*x)^0.5 + (r*s)^0.5))/((q*x)^0.5 + (r*s)^0.5) - 2.0*(q*x)^0.5/(x*((q*x)^0.5 + (r*s)^0.5)))/(x*((q*x)^0.5 + (r*s)^0.5))
Combinatoria [src] 
   _________ /          _____   _____\ 
-\/ q*r*s*x *\r*s + 3*\/ q*x *\/ r*s / 
---------------------------------------
                               3       
          2 /  _____     _____\        
       4*x *\\/ q*x  + \/ r*s /
−qrsx(rs+3qxrs)4x2(qx+rs)3- \frac{\sqrt{q r s x} \left(r s + 3 \sqrt{q x} \sqrt{r s}\right)}{4 x^{2} \left(\sqrt{q x} + \sqrt{r s}\right)^{3}} 
-sqrt(q*r*s*x)*(r*s + 3*sqrt(q*x)*sqrt(r*s))/(4*x^2*(sqrt(q*x) + sqrt(r*s))^3)
Abrimos la expresión [src] 
            /        _____                                            \
            |      \/ q*x           2*q                               |
            |      ------- + -----------------                        |
            |         x        _____     _____             _____      |
  _________ |  1             \/ q*x  + \/ r*s          2*\/ q*x       |
\/ q*r*s*x *|- - + --------------------------- - ---------------------|
            |  x          _____     _____          /  _____     _____\|
            \           \/ q*x  + \/ r*s         x*\\/ q*x  + \/ r*s //
-----------------------------------------------------------------------
                            /  _____     _____\                        
                        4*x*\\/ q*x  + \/ r*s /
qrsx(2qqx+rs+qxxqx+rs−2qxx(qx+rs)−1x)4x(qx+rs)\frac{\sqrt{q r s x} \left(\frac{\frac{2 q}{\sqrt{q x} + \sqrt{r s}} + \frac{\sqrt{q x}}{x}}{\sqrt{q x} + \sqrt{r s}} - \frac{2 \sqrt{q x}}{x \left(\sqrt{q x} + \sqrt{r s}\right)} - \frac{1}{x}\right)}{4 x \left(\sqrt{q x} + \sqrt{r s}\right)} 
          /        ___                                                                \
          |      \/ q               2*q                                               |
          |      ----- + -------------------------                                    |
          |        ___     ___   ___     ___   ___                    ___             |
  _______ |  1   \/ x    \/ q *\/ x  + \/ r *\/ s                 2*\/ q              |
\/ q*r*s *|- - + --------------------------------- - ---------------------------------|
          |  x         ___   ___     ___   ___         ___ /  ___   ___     ___   ___\|
          \          \/ q *\/ x  + \/ r *\/ s        \/ x *\\/ q *\/ x  + \/ r *\/ s //
---------------------------------------------------------------------------------------
                              ___ /  ___   ___     ___   ___\                          
                          4*\/ x *\\/ q *\/ x  + \/ r *\/ s /
qrs(−2qx(qx+rs)+qx+2qqx+rsqx+rs−1x)4x(qx+rs)\frac{\sqrt{q r s} \left(- \frac{2 \sqrt{q}}{\sqrt{x} \left(\sqrt{q} \sqrt{x} + \sqrt{r} \sqrt{s}\right)} + \frac{\frac{\sqrt{q}}{\sqrt{x}} + \frac{2 q}{\sqrt{q} \sqrt{x} + \sqrt{r} \sqrt{s}}}{\sqrt{q} \sqrt{x} + \sqrt{r} \sqrt{s}} - \frac{1}{x}\right)}{4 \sqrt{x} \left(\sqrt{q} \sqrt{x} + \sqrt{r} \sqrt{s}\right)} 
sqrt(q*r*s)*(-1/x + (sqrt(q)/sqrt(x) + 2*q/(sqrt(q)*sqrt(x) + sqrt(r)*sqrt(s)))/(sqrt(q)*sqrt(x) + sqrt(r)*sqrt(s)) - 2*sqrt(q)/(sqrt(x)*(sqrt(q)*sqrt(x) + sqrt(r)*sqrt(s))))/(4*sqrt(x)*(sqrt(q)*sqrt(x) + sqrt(r)*sqrt(s)))
Parte trigonométrica [src] 
            /        _____                                            \
            |      \/ q*x           2*q                               |
            |      ------- + -----------------                        |
            |         x        _____     _____             _____      |
  _________ |  1             \/ q*x  + \/ r*s          2*\/ q*x       |
\/ q*r*s*x *|- - + --------------------------- - ---------------------|
            |  x          _____     _____          /  _____     _____\|
            \           \/ q*x  + \/ r*s         x*\\/ q*x  + \/ r*s //
-----------------------------------------------------------------------
                            /  _____     _____\                        
                        4*x*\\/ q*x  + \/ r*s /
qrsx(2qqx+rs+qxxqx+rs−2qxx(qx+rs)−1x)4x(qx+rs)\frac{\sqrt{q r s x} \left(\frac{\frac{2 q}{\sqrt{q x} + \sqrt{r s}} + \frac{\sqrt{q x}}{x}}{\sqrt{q x} + \sqrt{r s}} - \frac{2 \sqrt{q x}}{x \left(\sqrt{q x} + \sqrt{r s}\right)} - \frac{1}{x}\right)}{4 x \left(\sqrt{q x} + \sqrt{r s}\right)} 
sqrt(q*r*s*x)*(-1/x + (sqrt(q*x)/x + 2*q/(sqrt(q*x) + sqrt(r*s)))/(sqrt(q*x) + sqrt(r*s)) - 2*sqrt(q*x)/(x*(sqrt(q*x) + sqrt(r*s))))/(4*x*(sqrt(q*x) + sqrt(r*s)))
Denominador racional [src] 
                                            2                    4                                               2                    4                                                 4                                                 4                                                 4
   2   _____   _________ /  _____     _____\  /  _____     _____\       2   _____   _________ /  _____     _____\  /  _____     _____\         3   _____   _________ /  _____     _____\         3   _____   _________ /  _____     _____\         2   _____   _________ /  _____     _____\ 
- x *\/ r*s *\/ q*r*s*x *\\/ q*x  + \/ r*s / *\\/ q*x  - \/ r*s /  - 3*x *\/ q*x *\/ q*r*s*x *\\/ q*x  + \/ r*s / *\\/ q*x  - \/ r*s /  + 3*q*x *\/ q*x *\/ q*r*s*x *\\/ q*x  - \/ r*s /  + 4*q*x *\/ r*s *\/ q*r*s*x *\\/ q*x  - \/ r*s /  + r*s*x *\/ q*x *\/ q*r*s*x *\\/ q*x  - \/ r*s / 
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                         4            4                                                                                                                                      
                                                                                                                                      4*x *(q*x - r*s)
3qx3qxqrsx(qx−rs)4+4qx3rsqrsx(qx−rs)4+rsx2qxqrsx(qx−rs)4−3x2qxqrsx(qx−rs)4(qx+rs)2−x2rsqrsx(qx−rs)4(qx+rs)24x4(qx−rs)4\frac{3 q x^{3} \sqrt{q x} \sqrt{q r s x} \left(\sqrt{q x} - \sqrt{r s}\right)^{4} + 4 q x^{3} \sqrt{r s} \sqrt{q r s x} \left(\sqrt{q x} - \sqrt{r s}\right)^{4} + r s x^{2} \sqrt{q x} \sqrt{q r s x} \left(\sqrt{q x} - \sqrt{r s}\right)^{4} - 3 x^{2} \sqrt{q x} \sqrt{q r s x} \left(\sqrt{q x} - \sqrt{r s}\right)^{4} \left(\sqrt{q x} + \sqrt{r s}\right)^{2} - x^{2} \sqrt{r s} \sqrt{q r s x} \left(\sqrt{q x} - \sqrt{r s}\right)^{4} \left(\sqrt{q x} + \sqrt{r s}\right)^{2}}{4 x^{4} \left(q x - r s\right)^{4}} 
(-x^2*sqrt(r*s)*sqrt(q*r*s*x)*(sqrt(q*x) + sqrt(r*s))^2*(sqrt(q*x) - sqrt(r*s))^4 - 3*x^2*sqrt(q*x)*sqrt(q*r*s*x)*(sqrt(q*x) + sqrt(r*s))^2*(sqrt(q*x) - sqrt(r*s))^4 + 3*q*x^3*sqrt(q*x)*sqrt(q*r*s*x)*(sqrt(q*x) - sqrt(r*s))^4 + 4*q*x^3*sqrt(r*s)*sqrt(q*r*s*x)*(sqrt(q*x) - sqrt(r*s))^4 + r*s*x^2*sqrt(q*x)*sqrt(q*r*s*x)*(sqrt(q*x) - sqrt(r*s))^4)/(4*x^4*(q*x - r*s)^4)
Unión de expresiones racionales [src] 
            /                     2                                      \
  _________ |  /  _____     _____\      _____ /  _____     _____\        |
\/ q*r*s*x *\- \\/ q*x  + \/ r*s /  - \/ q*x *\\/ q*x  + \/ r*s / + 2*q*x/
--------------------------------------------------------------------------
                                                3                         
                           2 /  _____     _____\                          
                        4*x *\\/ q*x  + \/ r*s /
qrsx(2qx−qx(qx+rs)−(qx+rs)2)4x2(qx+rs)3\frac{\sqrt{q r s x} \left(2 q x - \sqrt{q x} \left(\sqrt{q x} + \sqrt{r s}\right) - \left(\sqrt{q x} + \sqrt{r s}\right)^{2}\right)}{4 x^{2} \left(\sqrt{q x} + \sqrt{r s}\right)^{3}} 
sqrt(q*r*s*x)*(-(sqrt(q*x) + sqrt(r*s))^2 - sqrt(q*x)*(sqrt(q*x) + sqrt(r*s)) + 2*q*x)/(4*x^2*(sqrt(q*x) + sqrt(r*s))^3)
Compilar la expresión [src] 
            /             _____         _____                    \
            |         2*\/ q*x        \/ q*x           2*q       |
            |-1 - -----------------   ------- + -----------------|
            |       _____     _____      x        _____     _____|
  _________ |     \/ q*x  + \/ r*s              \/ q*x  + \/ r*s |
\/ q*r*s*x *|---------------------- + ---------------------------|
            |          x                     _____     _____     |
            \                              \/ q*x  + \/ r*s      /
------------------------------------------------------------------
                         /  _____     _____\                      
                     4*x*\\/ q*x  + \/ r*s /
qrsx(2qqx+rs+qxxqx+rs+−2qxqx+rs−1x)4x(qx+rs)\frac{\sqrt{q r s x} \left(\frac{\frac{2 q}{\sqrt{q x} + \sqrt{r s}} + \frac{\sqrt{q x}}{x}}{\sqrt{q x} + \sqrt{r s}} + \frac{- \frac{2 \sqrt{q x}}{\sqrt{q x} + \sqrt{r s}} - 1}{x}\right)}{4 x \left(\sqrt{q x} + \sqrt{r s}\right)} 
            /        _____                                            \
            |      \/ q*x           2*q                               |
            |      ------- + -----------------                        |
            |         x        _____     _____             _____      |
  _________ |  1             \/ q*x  + \/ r*s          2*\/ q*x       |
\/ q*r*s*x *|- - + --------------------------- - ---------------------|
            |  x          _____     _____          /  _____     _____\|
            \           \/ q*x  + \/ r*s         x*\\/ q*x  + \/ r*s //
-----------------------------------------------------------------------
                            /  _____     _____\                        
                        4*x*\\/ q*x  + \/ r*s /
qrsx(2qqx+rs+qxxqx+rs−2qxx(qx+rs)−1x)4x(qx+rs)\frac{\sqrt{q r s x} \left(\frac{\frac{2 q}{\sqrt{q x} + \sqrt{r s}} + \frac{\sqrt{q x}}{x}}{\sqrt{q x} + \sqrt{r s}} - \frac{2 \sqrt{q x}}{x \left(\sqrt{q x} + \sqrt{r s}\right)} - \frac{1}{x}\right)}{4 x \left(\sqrt{q x} + \sqrt{r s}\right)} 
sqrt(q*r*s*x)*(-1/x + (sqrt(q*x)/x + 2*q/(sqrt(q*x) + sqrt(r*s)))/(sqrt(q*x) + sqrt(r*s)) - 2*sqrt(q*x)/(x*(sqrt(q*x) + sqrt(r*s))))/(4*x*(sqrt(q*x) + sqrt(r*s)))
Potencias [src] 
            /          _____                                              \
            |        \/ q*x           2*q                                 |
            |        ------- + -----------------                          |
            |           x        _____     _____             _____        |
  _________ |   1              \/ q*x  + \/ r*s            \/ q*x         |
\/ q*r*s*x *|- --- + --------------------------- - -----------------------|
            |  4*x        /  _____     _____\          /  _____     _____\|
            \           4*\\/ q*x  + \/ r*s /      2*x*\\/ q*x  + \/ r*s //
---------------------------------------------------------------------------
                             /  _____     _____\                           
                           x*\\/ q*x  + \/ r*s /
qrsx(2qqx+rs+qxx4(qx+rs)−qx2x(qx+rs)−14x)x(qx+rs)\frac{\sqrt{q r s x} \left(\frac{\frac{2 q}{\sqrt{q x} + \sqrt{r s}} + \frac{\sqrt{q x}}{x}}{4 \left(\sqrt{q x} + \sqrt{r s}\right)} - \frac{\sqrt{q x}}{2 x \left(\sqrt{q x} + \sqrt{r s}\right)} - \frac{1}{4 x}\right)}{x \left(\sqrt{q x} + \sqrt{r s}\right)} 
            /        _____                                            \
            |      \/ q*x           2*q                               |
            |      ------- + -----------------                        |
            |         x        _____     _____             _____      |
  _________ |  1             \/ q*x  + \/ r*s          2*\/ q*x       |
\/ q*r*s*x *|- - + --------------------------- - ---------------------|
            |  x          _____     _____          /  _____     _____\|
            \           \/ q*x  + \/ r*s         x*\\/ q*x  + \/ r*s //
-----------------------------------------------------------------------
                            /  _____     _____\                        
                        4*x*\\/ q*x  + \/ r*s /
qrsx(2qqx+rs+qxxqx+rs−2qxx(qx+rs)−1x)4x(qx+rs)\frac{\sqrt{q r s x} \left(\frac{\frac{2 q}{\sqrt{q x} + \sqrt{r s}} + \frac{\sqrt{q x}}{x}}{\sqrt{q x} + \sqrt{r s}} - \frac{2 \sqrt{q x}}{x \left(\sqrt{q x} + \sqrt{r s}\right)} - \frac{1}{x}\right)}{4 x \left(\sqrt{q x} + \sqrt{r s}\right)} 
sqrt(q*r*s*x)*(-1/x + (sqrt(q*x)/x + 2*q/(sqrt(q*x) + sqrt(r*s)))/(sqrt(q*x) + sqrt(r*s)) - 2*sqrt(q*x)/(x*(sqrt(q*x) + sqrt(r*s))))/(4*x*(sqrt(q*x) + sqrt(r*s)))
Denominador común [src] 
       3/2   _________        3/2   _________         _____   _________           _____   _________           _____   _________
- (q*x)   *\/ q*r*s*x  - (r*s)   *\/ q*r*s*x  + q*x*\/ q*x *\/ q*r*s*x  - 4*r*s*\/ q*x *\/ q*r*s*x  - 3*q*x*\/ r*s *\/ q*r*s*x 
-------------------------------------------------------------------------------------------------------------------------------
                         2  4      2  2  2       2   _____      3/2       2      3/2   _____             3                     
                      4*q *x  + 4*r *s *x  + 16*x *\/ q*x *(r*s)    + 16*x *(q*x)   *\/ r*s  + 24*q*r*s*x
qxqxqrsx−3qxrsqrsx−4rsqxqrsx−(qx)32qrsx−(rs)32qrsx4q2x4+24qrsx3+4r2s2x2+16x2(qx)32rs+16x2qx(rs)32\frac{q x \sqrt{q x} \sqrt{q r s x} - 3 q x \sqrt{r s} \sqrt{q r s x} - 4 r s \sqrt{q x} \sqrt{q r s x} - \left(q x\right)^{\frac{3}{2}} \sqrt{q r s x} - \left(r s\right)^{\frac{3}{2}} \sqrt{q r s x}}{4 q^{2} x^{4} + 24 q r s x^{3} + 4 r^{2} s^{2} x^{2} + 16 x^{2} \left(q x\right)^{\frac{3}{2}} \sqrt{r s} + 16 x^{2} \sqrt{q x} \left(r s\right)^{\frac{3}{2}}} 
(-(q*x)^(3/2)*sqrt(q*r*s*x) - (r*s)^(3/2)*sqrt(q*r*s*x) + q*x*sqrt(q*x)*sqrt(q*r*s*x) - 4*r*s*sqrt(q*x)*sqrt(q*r*s*x) - 3*q*x*sqrt(r*s)*sqrt(q*r*s*x))/(4*q^2*x^4 + 4*r^2*s^2*x^2 + 16*x^2*sqrt(q*x)*(r*s)^(3/2) + 16*x^2*(q*x)^(3/2)*sqrt(r*s) + 24*q*r*s*x^3)
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