Why won't Eliminate reduce the more complex form of this equation to the simpler form?

Question

It's been asserted to me that there are two forms of representing a certain variable q. One is more complicated than the other, but I've been told that they should deliver the same result.

The more complex form is:

q == (H (1 - H)/(a + c)) ((F (s - 1) s)/(F s - F - H s)^2)^2 + (F (1 - F)/(b + d)) (-(H (s - 1) s)/(F (s - 1) - H s)^2)^2

, where H == (a/(a + c)), F == (b/(b + d)), s == (a + c)/(a + b + c + d).

The simpler form is:

q == z^4((1-H)/a+(1-F)/b)a^2/b^2

, where in addition the variables I've previously defined we also have z == (b/(a + b)).

I've tested various input values on the two forms, and they seem to always produce the same result. I wrote some code before that has worked in similar contexts. However, here Mathematica just runs for hours without reaching an answer.

Eliminate[{q == (H (1 - H)/(a + c)) ((F (s - 1) s)/(F s - F - H s)^2)^2 + (F (1 - F)/(b + d)) (-(H (s - 1) s)/(F (s - 1) - H s)^2)^2, H == (a/(a + c)), F == (b/(b + d)), s == (a + c)/(a + b + c + d), z == (b/(a + b))}, {s, c, d}]

Just in case it matters, I'll mention that a, b, c, and d all have to be positive integers. However, I haven't told Mathematica that.

Answer 1

This simplifies to

Solve[{q == (H (1 - H)/(a + c)) ((F (s - 1) s)/(F s - F - H s)^2)^2 + (F (1 - F)/(b + d)) (-(H (s - 1) s)/(F (s - 1) - H s)^2)^2, 
       H == (a/(a + c)), 
       F == (b/(b + d)), 
       s == (a + c)/(a + b + c + d)}, q, {s, c, d}]

(* q -> -((a b (-a - b + a F + b H))/(a + b)^4) *)

which is equal to your simple form:

FullSimplify[-((a b (-a - b + a F + b H))/(a + b)^4) == (b/(a + b))^4 ((1 - H)/a + (1 - F)/b) a^2/b^2]
(* True *)