A beautiful question

$\huge{\frac{x + 1}{x^{\frac{2}{3}} - x^{\frac{1}{3}} + 1} = \frac{(x^{1/3} + 1)(x^{2/3} - x^{1/3} + 1)}{(x^{2/3} - x^{1/3} + 1)} = x^{1/3} + 1}$

$\huge{\frac{x - 1}{x^{1/2}(x^{1/2} - 1)} = \frac{1}{x^{1/2}} + 1}$

Thus it becomes

$[ x^{1/3} - \frac{1}{x^{1/2}}]^{10}$

$T_{r + 1} = \binom {n}{r} a^{r} b^{n - r}$

$T_{r + 1} = \binom {10}{r} x^{r/3} . x^{(-10 + r)/2}$

For constant term variable term power should be zero

Therefore $\frac{r}{3} + \frac{r}{2} - \frac{10}{2} = 0$

$r = 6$

$T_{6 + 1} = \binom{10}{6} = 210$