The term  independent of x in the binomial expansion  ${(\frac{x+1}{\sqrt[3]{x^2}-\sqrt[3]{x}+1}-\frac{x-1}{x-\sqrt{x}})}^{10}$ is 

Rationalize 

$={\left[\left(x^{1/3}+1\right)-\frac{(\sqrt{x}+1)}{\sqrt{x}}\right]}^{10}={\left(x^{1/3}-x^{-1/2}\right)}^{10}$

$\begin{array}{r}{T}_{r+1}{=}^{10}{C}_r{\left(x^{1/3}\right)}^{10-r}{\left(-x^{-1/2}\right)}^r\\ {=}^{10}{C}_r(-1{)}^r{x}^{\frac{10-r}{3}-\frac{r}{2}}\end{array}$

For term independent of x

$\begin{array}{l}\frac{10-r}{3}-\frac{r}{2}=0\\ \Rightarrow 20-2r-3r=0\\ \Rightarrow 20=5r\Rightarrow r=4\\ \therefore T_5{=}^{10}{C}_4=\frac{10\times 9\times 8\times 7}{4\times 3\times 2\times 1}=210\end{array}$