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}]

Thanks a lot .I have just started using mathematica. So It will take sometime to get used to with the notation.

I have modified my integral a bit. But it is again not working.I have also tried to plot the function but it is showing an error.

enter code here
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[h[x], {x, 0, 1}, Assumptions -> {x > 0, 0 < a < 1, b > 0, d > 2}]

it is giving the following error {{{{Integrate[x^(-1+d)/Sqrt[(1-a x^d) (-x^(-2+2 d)+(1+a b x^d)/(1+a b))],{x,0,1},Assumptions->{x>0,0<a<1,b>0,d>2}]}}}}

Thanks a lot Gianluca Gorni . I was looking for some analytic results. But it seems there is none for this Integral.