I am interested in integrals of the following form:
$\int\limits_{0}^{M} \left( -x^3 +bx^2 -\omega \right)^{-\frac{1}{2}} \,dx \, ,$
where $b, \omega>0$, and $M$ is one of the roots of the polynomial ($x=0$ is not a root). I used the trigonometric method to find all three real roots of this polynomial, but have not found any good way/closed form expression for this integral.