calculus and analysis - Integrating the Associated Legendre Polynomials

I know the following identity:

$\qquad \int_{-1}^1 P_l^m(t)^2dt=\frac{2(m+n)!}{(2n+1)(n-m)!}$

I would like to verify this result using Mathematica. This is what I entered:

Integrate[LegendreP[l, m, t]^2, {t, -1, 1}, 

  Assumptions -> (m | l) ∈ Integers && l >= 0 && -l <= m <= l]

However it did not give the result I wanted. Any tips or advice would be appreciated!