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