I am evaluating an integral with constants that are to be specified after the evaluation. The integral is given by,
$$\int_0^1 dy \frac{ y^2 (1 - b^3 y^3)^{1/2} }{ (1 - c^2 a^4 y^4)^{1/2} }$$
where $b=a/z$, $z=10~$ (actually I can change this value but for a specific case and to remove constraints I placed a value), and $a$ is a constant which I want to leave open until after evaluation of the integral (I want to tune this later); also $c=c(a)$ is an unknown function of $a$.
To make things clear, I want to find the minimal value of this integral as a function of $a,c$ then find a relation between $a,c$ after extremizing. Can anyone point me in the direction on how to do this?
I hope everything is clear.