I am evaluating an integral with constants that are not specified, but I am not sure why it takes so long for it to give an output, so I just decided to cancel the running. The integral is given by,

$\int_0^1 dy \frac{ y^2 (1 - b^3 y^3)^{1/2} }{ (1 - a^4 c^2 y^4)^{1/2} }$

 d=2; z=1; b=a/z; Integrate[ ( y^d (1 - (b y)^(d+1))^1/2 )/(1 - c^2 (a y)^2d)^1/2 , {y,0,1}, Assumptions -> {y>0, c>0} ] 

d indicates dimensions so in this case I set for example, d=2, while a,b are constants (leave it open so I can put values later). In the end, I want to get an expression for "c" in terms of "a" (since "a" also gives "b") through evaluation of the integral.