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Simplificación de expresiones/ Descomposición en fracciones simples/ atan(y/x)/(2*x^3)+y/(2*x^4+2*x^2*y^2)

¿Cómo vas a descomponer esta atan(y/x)/(2*x^3)+y/(2*x^4+2*x^2*y^2) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    /y\                 
atan|-|                 
    \x/         y       
------- + --------------
     3       4      2  2
  2*x     2*x  + 2*x *y
$$\frac{y}{2 x^{4} + 2 x^{2} y^{2}} + \frac{\operatorname{atan}{\left(\frac{y}{x} \right)}}{2 x^{3}}$$ 
atan(y/x)/((2*x^3)) + y/(2*x^4 + (2*x^2)*y^2)
Simplificación general [src] 
      / 2    2\     /y\
x*y + \x  + y /*atan|-|
                    \x/
-----------------------
        3 / 2    2\    
     2*x *\x  + y /
$$\frac{x y + \left(x^{2} + y^{2}\right) \operatorname{atan}{\left(\frac{y}{x} \right)}}{2 x^{3} \left(x^{2} + y^{2}\right)}$$ 
(x*y + (x^2 + y^2)*atan(y/x))/(2*x^3*(x^2 + y^2))
Respuesta numérica [src] 
y/(2.0*x^4 + 2.0*x^2*y^2) + 0.5*atan(y/x)/x^3
y/(2.0*x^4 + 2.0*x^2*y^2) + 0.5*atan(y/x)/x^3
Denominador común [src] 
       2     /y\    2     /y\
x*y + x *atan|-| + y *atan|-|
             \x/          \x/
-----------------------------
           5      3  2       
        2*x  + 2*x *y
$$\frac{x^{2} \operatorname{atan}{\left(\frac{y}{x} \right)} + x y + y^{2} \operatorname{atan}{\left(\frac{y}{x} \right)}}{2 x^{5} + 2 x^{3} y^{2}}$$ 
(x*y + x^2*atan(y/x) + y^2*atan(y/x))/(2*x^5 + 2*x^3*y^2)
Denominador racional [src] 
/   4      2  2\     /y\        3
\2*x  + 2*x *y /*atan|-| + 2*y*x 
                     \x/         
---------------------------------
         3 /   4      2  2\      
      2*x *\2*x  + 2*x *y /
$$\frac{2 x^{3} y + \left(2 x^{4} + 2 x^{2} y^{2}\right) \operatorname{atan}{\left(\frac{y}{x} \right)}}{2 x^{3} \left(2 x^{4} + 2 x^{2} y^{2}\right)}$$ 
((2*x^4 + 2*x^2*y^2)*atan(y/x) + 2*y*x^3)/(2*x^3*(2*x^4 + 2*x^2*y^2))
Unión de expresiones racionales [src] 
      / 2    2\     /y\
x*y + \x  + y /*atan|-|
                    \x/
-----------------------
        3 / 2    2\    
     2*x *\x  + y /
$$\frac{x y + \left(x^{2} + y^{2}\right) \operatorname{atan}{\left(\frac{y}{x} \right)}}{2 x^{3} \left(x^{2} + y^{2}\right)}$$ 
(x*y + (x^2 + y^2)*atan(y/x))/(2*x^3*(x^2 + y^2))
Combinatoria [src] 
       2     /y\    2     /y\
x*y + x *atan|-| + y *atan|-|
             \x/          \x/
-----------------------------
           3 / 2    2\       
        2*x *\x  + y /
$$\frac{x^{2} \operatorname{atan}{\left(\frac{y}{x} \right)} + x y + y^{2} \operatorname{atan}{\left(\frac{y}{x} \right)}}{2 x^{3} \left(x^{2} + y^{2}\right)}$$ 
(x*y + x^2*atan(y/x) + y^2*atan(y/x))/(2*x^3*(x^2 + y^2))
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