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

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

Expresión a simplificar:

Solución

Ha introducido [src] 
           /  ___\ /    /  ___\                  \
       atan\\/ x / |atan\\/ x /      log(x + 1)  |
(x + 1)           *|----------- + ---------------|
                   |   x + 1          ___        |
                   \              2*\/ x *(1 + x)/
$$\left(x + 1\right)^{\operatorname{atan}{\left(\sqrt{x} \right)}} \left(\frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{x + 1} + \frac{\log{\left(x + 1 \right)}}{2 \sqrt{x} \left(x + 1\right)}\right)$$ 
(x + 1)^atan(sqrt(x))*(atan(sqrt(x))/(x + 1) + log(x + 1)/(((2*sqrt(x))*(1 + x))))
Simplificación general [src] 
                /  ___\                                 
       -1 + atan\\/ x / /log(1 + x)     ___     /  ___\\
(1 + x)                *|---------- + \/ x *atan\\/ x /|
                        \    2                         /
--------------------------------------------------------
                           ___                          
                         \/ x
$$\frac{\left(x + 1\right)^{\operatorname{atan}{\left(\sqrt{x} \right)} - 1} \left(\sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} + \frac{\log{\left(x + 1 \right)}}{2}\right)}{\sqrt{x}}$$ 
(1 + x)^(-1 + atan(sqrt(x)))*(log(1 + x)/2 + sqrt(x)*atan(sqrt(x)))/sqrt(x)
Respuesta numérica [src] 
(1.0 + x)^atan(sqrt(x))*(atan(sqrt(x))/(1.0 + x) + 0.5*x^(-0.5)*log(x + 1)/(1.0 + x))
(1.0 + x)^atan(sqrt(x))*(atan(sqrt(x))/(1.0 + x) + 0.5*x^(-0.5)*log(x + 1)/(1.0 + x))
Denominador común [src] 
           /  ___\                                 /  ___\            
       atan\\/ x /                  ___        atan\\/ x /     /  ___\
(1 + x)           *log(1 + x) + 2*\/ x *(1 + x)           *atan\\/ x /
----------------------------------------------------------------------
                               ___      3/2                           
                           2*\/ x  + 2*x
$$\frac{2 \sqrt{x} \left(x + 1\right)^{\operatorname{atan}{\left(\sqrt{x} \right)}} \operatorname{atan}{\left(\sqrt{x} \right)} + \left(x + 1\right)^{\operatorname{atan}{\left(\sqrt{x} \right)}} \log{\left(x + 1 \right)}}{2 x^{\frac{3}{2}} + 2 \sqrt{x}}$$ 
((1 + x)^atan(sqrt(x))*log(1 + x) + 2*sqrt(x)*(1 + x)^atan(sqrt(x))*atan(sqrt(x)))/(2*sqrt(x) + 2*x^(3/2))
Unión de expresiones racionales [src] 
           /  ___\                                   
       atan\\/ x / /    ___     /  ___\             \
(1 + x)           *\2*\/ x *atan\\/ x / + log(1 + x)/
-----------------------------------------------------
                       ___                           
                   2*\/ x *(1 + x)
$$\frac{\left(x + 1\right)^{\operatorname{atan}{\left(\sqrt{x} \right)}} \left(2 \sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} + \log{\left(x + 1 \right)}\right)}{2 \sqrt{x} \left(x + 1\right)}$$ 
(1 + x)^atan(sqrt(x))*(2*sqrt(x)*atan(sqrt(x)) + log(1 + x))/(2*sqrt(x)*(1 + x))
Combinatoria [src] 
           /  ___\                                   
       atan\\/ x / /    ___     /  ___\             \
(1 + x)           *\2*\/ x *atan\\/ x / + log(1 + x)/
-----------------------------------------------------
                       ___                           
                   2*\/ x *(1 + x)
$$\frac{\left(x + 1\right)^{\operatorname{atan}{\left(\sqrt{x} \right)}} \left(2 \sqrt{x} \operatorname{atan}{\left(\sqrt{x} \right)} + \log{\left(x + 1 \right)}\right)}{2 \sqrt{x} \left(x + 1\right)}$$ 
(1 + x)^atan(sqrt(x))*(2*sqrt(x)*atan(sqrt(x)) + log(1 + x))/(2*sqrt(x)*(1 + x))
Potencias [src] 
           /  ___\ /    /  ___\                  \
       atan\\/ x / |atan\\/ x /      log(1 + x)  |
(1 + x)           *|----------- + ---------------|
                   |   1 + x        ___          |
                   \              \/ x *(2 + 2*x)/
$$\left(x + 1\right)^{\operatorname{atan}{\left(\sqrt{x} \right)}} \left(\frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{x + 1} + \frac{\log{\left(x + 1 \right)}}{\sqrt{x} \left(2 x + 2\right)}\right)$$ 
(1 + x)^atan(sqrt(x))*(atan(sqrt(x))/(1 + x) + log(1 + x)/(sqrt(x)*(2 + 2*x)))
Denominador racional [src] 
                /  ___\                                /  ___\                                      /  ___\                                      /  ___\            
       -2 + atan\\/ x /                       -2 + atan\\/ x /                  ___        -2 + atan\\/ x /     /  ___\      3/2        -2 + atan\\/ x /     /  ___\
(1 + x)                *log(1 + x) + x*(1 + x)                *log(1 + x) + 2*\/ x *(1 + x)                *atan\\/ x / + 2*x   *(1 + x)                *atan\\/ x /
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                                                                              2*\/ x
$$\frac{2 x^{\frac{3}{2}} \left(x + 1\right)^{\operatorname{atan}{\left(\sqrt{x} \right)} - 2} \operatorname{atan}{\left(\sqrt{x} \right)} + 2 \sqrt{x} \left(x + 1\right)^{\operatorname{atan}{\left(\sqrt{x} \right)} - 2} \operatorname{atan}{\left(\sqrt{x} \right)} + x \left(x + 1\right)^{\operatorname{atan}{\left(\sqrt{x} \right)} - 2} \log{\left(x + 1 \right)} + \left(x + 1\right)^{\operatorname{atan}{\left(\sqrt{x} \right)} - 2} \log{\left(x + 1 \right)}}{2 \sqrt{x}}$$ 
((1 + x)^(-2 + atan(sqrt(x)))*log(1 + x) + x*(1 + x)^(-2 + atan(sqrt(x)))*log(1 + x) + 2*sqrt(x)*(1 + x)^(-2 + atan(sqrt(x)))*atan(sqrt(x)) + 2*x^(3/2)*(1 + x)^(-2 + atan(sqrt(x)))*atan(sqrt(x)))/(2*sqrt(x))
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