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Simplificación de expresiones/ Descomposición en fracciones simples/ atan(sqrt(3)*x)/(8*3^(3/2))+(3*x^3-x)/(24+144*x^2+216*x^4)

¿Cómo vas a descomponer esta atan(sqrt(3)*x)/(8*3^(3/2))+(3*x^3-x)/(24+144*x^2+216*x^4) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
    /  ___  \            3          
atan\\/ 3 *x/         3*x  - x      
------------- + --------------------
       3/2                2        4
    8*3         24 + 144*x  + 216*x
$$\frac{3 x^{3} - x}{216 x^{4} + \left(144 x^{2} + 24\right)} + \frac{\operatorname{atan}{\left(\sqrt{3} x \right)}}{8 \cdot 3^{\frac{3}{2}}}$$ 
atan(sqrt(3)*x)/((8*3^(3/2))) + (3*x^3 - x)/(24 + 144*x^2 + 216*x^4)
Simplificación general [src] 
          3     ___ /       2      4\     /    ___\
-3*x + 9*x  + \/ 3 *\1 + 6*x  + 9*x /*atan\x*\/ 3 /
---------------------------------------------------
                   /       2      4\               
                72*\1 + 6*x  + 9*x /
$$\frac{9 x^{3} - 3 x + \sqrt{3} \left(9 x^{4} + 6 x^{2} + 1\right) \operatorname{atan}{\left(\sqrt{3} x \right)}}{72 \left(9 x^{4} + 6 x^{2} + 1\right)}$$ 
(-3*x + 9*x^3 + sqrt(3)*(1 + 6*x^2 + 9*x^4)*atan(x*sqrt(3)))/(72*(1 + 6*x^2 + 9*x^4))
Abrimos la expresión [src] 
         3               ___     /  ___  \
      3*x  - x         \/ 3 *atan\\/ 3 *x/
-------------------- + -------------------
          2        4            72        
24 + 144*x  + 216*x
$$\frac{3 x^{3} - x}{216 x^{4} + \left(144 x^{2} + 24\right)} + \frac{\sqrt{3} \operatorname{atan}{\left(\sqrt{3} x \right)}}{72}$$ 
(3*x^3 - x)/(24 + 144*x^2 + 216*x^4) + sqrt(3)*atan(sqrt(3)*x)/72
Potencias [src] 
             3           ___     /    ___\
     -x + 3*x          \/ 3 *atan\x*\/ 3 /
-------------------- + -------------------
          2        4            72        
24 + 144*x  + 216*x
$$\frac{3 x^{3} - x}{216 x^{4} + 144 x^{2} + 24} + \frac{\sqrt{3} \operatorname{atan}{\left(\sqrt{3} x \right)}}{72}$$ 
(-x + 3*x^3)/(24 + 144*x^2 + 216*x^4) + sqrt(3)*atan(x*sqrt(3))/72
Respuesta numérica [src] 
0.0240562612162344*atan(sqrt(3)*x) + (-x + 3.0*x^3)/(24.0 + 144.0*x^2 + 216.0*x^4)
0.0240562612162344*atan(sqrt(3)*x) + (-x + 3.0*x^3)/(24.0 + 144.0*x^2 + 216.0*x^4)
Unión de expresiones racionales [src] 
    /        2\     ___ /       2      4\     /    ___\
3*x*\-1 + 3*x / + \/ 3 *\1 + 6*x  + 9*x /*atan\x*\/ 3 /
-------------------------------------------------------
                     /       2      4\                 
                  72*\1 + 6*x  + 9*x /
$$\frac{3 x \left(3 x^{2} - 1\right) + \sqrt{3} \left(9 x^{4} + 6 x^{2} + 1\right) \operatorname{atan}{\left(\sqrt{3} x \right)}}{72 \left(9 x^{4} + 6 x^{2} + 1\right)}$$ 
(3*x*(-1 + 3*x^2) + sqrt(3)*(1 + 6*x^2 + 9*x^4)*atan(x*sqrt(3)))/(72*(1 + 6*x^2 + 9*x^4))
Combinatoria [src] 
          3     ___     /    ___\       ___  2     /    ___\       ___  4     /    ___\
-3*x + 9*x  + \/ 3 *atan\x*\/ 3 / + 6*\/ 3 *x *atan\x*\/ 3 / + 9*\/ 3 *x *atan\x*\/ 3 /
---------------------------------------------------------------------------------------
                                                  2                                    
                                        /       2\                                     
                                     72*\1 + 3*x /
$$\frac{9 \sqrt{3} x^{4} \operatorname{atan}{\left(\sqrt{3} x \right)} + 9 x^{3} + 6 \sqrt{3} x^{2} \operatorname{atan}{\left(\sqrt{3} x \right)} - 3 x + \sqrt{3} \operatorname{atan}{\left(\sqrt{3} x \right)}}{72 \left(3 x^{2} + 1\right)^{2}}$$ 
(-3*x + 9*x^3 + sqrt(3)*atan(x*sqrt(3)) + 6*sqrt(3)*x^2*atan(x*sqrt(3)) + 9*sqrt(3)*x^4*atan(x*sqrt(3)))/(72*(1 + 3*x^2)^2)
Compilar la expresión [src] 
             3           ___     /  ___  \
     -x + 3*x          \/ 3 *atan\\/ 3 *x/
-------------------- + -------------------
          2        4            72        
24 + 144*x  + 216*x
$$\frac{3 x^{3} - x}{216 x^{4} + 144 x^{2} + 24} + \frac{\sqrt{3} \operatorname{atan}{\left(\sqrt{3} x \right)}}{72}$$ 
(-x + 3*x^3)/(24 + 144*x^2 + 216*x^4) + sqrt(3)*atan(sqrt(3)*x)/72
Denominador común [src] 
             3           ___     /    ___\
     -x + 3*x          \/ 3 *atan\x*\/ 3 /
-------------------- + -------------------
          2        4            72        
24 + 144*x  + 216*x
$$\frac{3 x^{3} - x}{216 x^{4} + 144 x^{2} + 24} + \frac{\sqrt{3} \operatorname{atan}{\left(\sqrt{3} x \right)}}{72}$$ 
(-x + 3*x^3)/(24 + 144*x^2 + 216*x^4) + sqrt(3)*atan(x*sqrt(3))/72
Denominador racional [src] 
          3     ___     /    ___\       ___  2     /    ___\       ___  4     /    ___\
-3*x + 9*x  + \/ 3 *atan\x*\/ 3 / + 6*\/ 3 *x *atan\x*\/ 3 / + 9*\/ 3 *x *atan\x*\/ 3 /
---------------------------------------------------------------------------------------
                                            2        4                                 
                                  72 + 432*x  + 648*x
$$\frac{9 \sqrt{3} x^{4} \operatorname{atan}{\left(\sqrt{3} x \right)} + 9 x^{3} + 6 \sqrt{3} x^{2} \operatorname{atan}{\left(\sqrt{3} x \right)} - 3 x + \sqrt{3} \operatorname{atan}{\left(\sqrt{3} x \right)}}{648 x^{4} + 432 x^{2} + 72}$$ 
(-3*x + 9*x^3 + sqrt(3)*atan(x*sqrt(3)) + 6*sqrt(3)*x^2*atan(x*sqrt(3)) + 9*sqrt(3)*x^4*atan(x*sqrt(3)))/(72 + 432*x^2 + 648*x^4)
Parte trigonométrica [src] 
             3           ___     /    ___\
     -x + 3*x          \/ 3 *atan\x*\/ 3 /
-------------------- + -------------------
          2        4            72        
24 + 144*x  + 216*x
$$\frac{3 x^{3} - x}{216 x^{4} + 144 x^{2} + 24} + \frac{\sqrt{3} \operatorname{atan}{\left(\sqrt{3} x \right)}}{72}$$ 
(-x + 3*x^3)/(24 + 144*x^2 + 216*x^4) + sqrt(3)*atan(x*sqrt(3))/72
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