In Mathematica syntax the curly braces {} are for lists. To indicate the order of algebraic operations you should use parentheses () instead.
In general your integral seems too difficult to calculate symbolically:
With[{a = 1/2, b = 1, d = 3},
f[x_] := 1 - a*x^d;
k[x_] := 1 + b*a*x^d;
h[x_] := x^(d - 1)/Sqrt[f[x]*((k[x]/k[1]) - x^(2*d - 2))];
Integrate[Simplify@h[x], {x, 0, 1}]]
However you can calculate it numerically:
Manipulate[
f[x_] := 1 - a*x^d;
k[x_] := 1 + b*a*x^d;
h[x_] := x^(d - 1)/Sqrt[f[x]*((k[x]/k[1]) - x^(2*d - 2))];
NIntegrate[h[x], {x, 0, 1}],
{a, 0, 1}, {b, 0, 2}, {d, 2, 4}]
You can also plot the function:
Manipulate[
f[x_] := 1 - a*x^d;
k[x] := 1 + bax^d;
h[x] := x^(d - 1)/Sqrt[f[x]((k[x]/k[1]) - x^(2d - 2))];
Plot[h[x], {x, 0, 1}],
{a, 0, 1}, {b, 0, 2}, {d, 2, 4}]