The SageMath result (output of your code in SageMath 9.2) has coefficient a $2/5$ instead of the $2/15$ you claim:
sage: fi

Then it agrees with your hand calculation, e.g. because for $z = \sqrt{x+1}$ we have $$\frac{2}{5} z^5 - \frac{2}{3} z^3= \frac{2}{3}(z^2-1)z^3 - \frac{4}{15}z^5.$$