It seems to slimplify nicely for me:
syms A B C x
g(x) = (A*(x + 1)^3 + C*(x + 1)^3 + B*(x - 1)*(x + 1)^2)/((x - 1)*(x + 1)^3);
g = simplify(g, 'Steps', 20)
g(x) =
A/(x - 1) + B/(x + 1) + C/(x - 1)
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