The easiest way to check this is probably to calculate the derivative of the solution that WolframAlpha gives:
D[2 Log[2 - x] + 3 Log[x] - Log[2 + x], x]
(*-(2/(2 - x)) + 3/x - 1/(2 + x)*)
This is identical to the integrand:
Simplify[3/x + 2/(x - 2) - 1/(x + 2)==-(2/(2 - x)) + 3/x - 1/(2 + x)]
(*True*)
Cheers,
M.
