m = 1
d = 2
a = 1
b = 2
p = 0.5
F = function(x) (1 - ((1+x^m)^-d) * exp(-b*x^a))  /
    (1 - (p*(1+x^m)^-d) * exp(-b*x^a))

InvF1 = function(a) uniroot(function(x) F(x) - a, c(0,10))$root
InvF1(0.5)
[1] 0.1038906
F(InvF1(0.5))
[1] 0.4999983

x = c(seq(0,3, 0.001), seq(3.1,10,0.1))
y = F(x)
InvF2 = approxfun(y, x)

InvF2(0.5)
[1] 0.1038916
F(InvF2(0.5))
[1] 0.5000011