Clear[ll, ii, n, l, rho, a, np, ii2, ii3]
(*
Mathematica:xy''+(a+1-x)y'+ny=0
whereas, Desai's is:
x y'' + (2 (l+ 1) - x) y' + n' y = 0
where,
n' = n - l -1
so:
a + 1 == 2 ( l + 1)
*)

a[l_] = 2(l + 1)-1
np[n_, l_] = n - l -1
ll[n_, l_, rho_] := LaguerreL[np[n, l], a[l], rho]
ii[n_, l_] := Integrate[rho^(2 l + 2) E^(-rho) (ll[n, l, rho])^2, {rho, 0, Infinity}]
ii2[n_, l_] = 2 n ((n + l)!)^3/((n-l-1)!)
(*ii3[n_, l_] = ii2[np[n, l], l]*)
ii[1, 0] 
ii2[1, 0]
ii[2, 0] 
ii2[2, 0]
ii[2, 1] 
ii2[2, 1]

np[n, n-1]

864/24

36 (* 6^2 *)

32/8