Solving $ \frac{x^2}{x-1} + \sqrt{x-1} +\frac{\sqrt{x-1}}{x^2} = \frac{x-1}{x^2} + \frac{1}{\sqrt{x-1}} + \frac{x^2}{\sqrt{x-1}}$ 
Find all real values of $ x>1$ which satisfy$$ \frac{x^2}{x-1} + \sqrt{x-1} +\frac{\sqrt{x-1}}{x^2} = \frac{x-1}{x^2} + \frac{1}{\sqrt{x-1}} + \frac{x^2}{\sqrt{x-1}}$$

Well I first observe that each factor has its own inverse involve in the equation and rewriting 
gives $$\left( \frac{x^2}{x-1}-\frac{x-1}{x^2} \right) + \left(\sqrt{x-1}  -\frac{1}{\sqrt{x-1}} \right)+\left(\frac{\sqrt{x-1}}{x^2} -\frac{x^2}{\sqrt{x-1}}\right) =0  $$
However after expanding all the terms it does not look wise to do so.  Does anyone has an idea  

Let $$A=\frac{x^2}{x-1},\qquad B=\sqrt{x-1},\qquad C=\frac{\sqrt{x-1}}{x^2}$$
Then, we have
$$A+B+C=\frac 1A+\frac 1B+\frac 1C\qquad\text{and}\qquad ABC=1$$
from which we have
$$A+B+C=AB+BC+CA\qquad\text{and}\qquad ABC=1$$
Let $s:=A+B+C=AB+BC+CA$.
It follows from these that $A,B,C$ are the solutions of
$$t^3-st^2+st-1=0,$$
i.e.
$$(t-1)(t^2+t+1-st)=0$$
So, we see that either $A,B$ or $C$ has to be equal to $1$.
$A=\frac{x^2}{x-1}=1\implies x^2-x+1=0$ has no real solutions.
$B=\sqrt{x-1}=1\implies x=2$ which is sufficient.
$C=\frac{\sqrt{x-1}}{x^2}=1\implies x^4-x+1=0$.
Let $f(x):=x^4-x+1$. Then, $f'(x)=4x^3-1=0$. So, $f'(x)\gt 0$ for $x\gt 1$. 
Since $f(1)=1\gt 0$, we see that $f(x)=0$ has no real solutions.
Therefore, $\color{red}{x=2}$ is the only solution.
