Do you have any reason to think there is a closed form? Most integrals don't have one. Maybe the best you can do is numerical methods.
In this case only way is NIntegrate:
H[z_, h_, Ωm_, ωd_, Ωr_] := 100 h Sqrt[Ωr (1 + z)^4 + Ωm (1 + z)^3 + (1 - Ωm - Ωr) (1 + z)^(3 (1 + ωd))]
NIntegrate[1/H[z, 5, 1/2, 1/3, 1/4], {z, 0, 3}]
(*0.00167544*)