How to simplify an expression with no constraints?

Question

Simplifying expressions on Mathematica is a recurrent topic. However I found myself stuck with this one:

Solving an rational equation on Mathematica gives me the following expression:

expression := -((2 x)/3) + (2^(1/3) (3 - x^2))/(
  3 (9 x - 2 x^3 + Sqrt[4 (3 - x^2)^3 + (9 x - 2 x^3)^2])^(
   1/3)) - (9 x - 2 x^3 + Sqrt[4 (3 - x^2)^3 + (9 x - 2 x^3)^2])^(
  1/3)/(3 2^(1/3))

I know that the expression should simply to $-x$. If I plot the expression I get indeed a plot of $y=-x$.

Plot[expression, {x, -1, 1}]

I tried using the function "Simplify", "FullSimplify" even "Expand" and then "Simplify" but got no luck as I do not have restriction on $x$. The expression is very messy and I do not manage to simplify it by hand. How could I use mathematica to simplify "expression" to $-x$? (other than using a graphical plot)

Any idea are much appreciated! Thanks

Answer 1

This isn't the best or most general solution, but you can experiment a bit by Taylor expanding about $x=0$ to some sufficiently high order.

Series[expression, {x, 0, 20}] // Simplify

This results in -x+O[x]^21 which means there are no corrections to $-x$ upto order 21 in the Taylor expansion about $x=0$. If this remains stable to even higher orders you have a probable guess as to what the result is.