It seems that the spaces between the letters bax went lost in the text. Write b a x with explicit spaces.

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}]

I would also remove the curly braces around a*x^d:

f[x_] := 1 - a x^d;
h[x_] := Sqrt[f[x]];
Integrate[h[x], {x, 0, 1}, Assumptions -> {a > 0, d > 1}]

Capitalize sqrt