How to arrange the expression in ascending powers of the variable

Not sure this is what you want, but here is a way to bring it into a simple form:

a = (24 (15 + 4 Sqrt[3] x - 4 x^2 + x^4))/((3 + Sqrt[3] x)^4 (29 - 
       2 x^2 + x^4)) + (64 (58 + 165 x^2 + 28 x^4 + x^6))/(-87 + 
      35 x^2 - 5 x^4 + x^6)^2

b = Denominator[a[[2]]]
c = Numerator[a[[2]]]
d = a[[1]] + c/Factor[b]
e = Together[d]
f = Numerator[e]
g = Denominator[e]
h = Factor[f, Extension -> Sqrt[3]]/g // Simplify

$$frac{8 left(x^8-4 sqrt{3} x^7+42 x^6-8 sqrt{3} x^5+676 x^4-84 sqrt{3} x^3+4510 x^2-464 sqrt{3} x+1827right)}{3 left(x^2-3right)^2 left(x^4-2 x^2+29right)^2}$$