How to evaluate one particular integral from physics problem?

Question

Long story short, there was a problem about oscillatory motion in 1D due to potential V=P₄(x)- polynomial of 4^th degree. And I got to the integral:
$$\int_0^a \frac{dx}{\sqrt{-x(x-a)(x-b)(x-c)}}$$ and $$\int_b^c \frac{dx}{\sqrt{-x(x-a)(x-b)(x-c)}}$$ which are essentially the same with some change of variables. Restriction on a, b, c is that a<b<c. The gist of the problem is to prove that both yield the same result.
So, trying to convert the integral above into elliptic integral of the first kind I got to(ignoring constants) $$\int_0^a\frac{dx}{\sqrt{-(1-\frac{4}{a^2}(x-\frac{a}{2})^2})\sqrt{1-\frac{4}{(b-c)^2}(x-\frac{(b+c)}{2})^2}}$$ And that is the place I'm currently at, any shifts of the origin and changes of the variables have been tried, I didn't expand it into series since it seems to be ineffective here. Also, I'd like to know more about elliptic integrals, since my uni skipped that topic for whatever reason, so any additional material is welcome

Just to check whether both integrals give the same result I substituted some numbers and enquired WolframAlpha and it did give the same number, but obviously it is general expresion I'm after hence this question/

Answer 1

For those, who might stumble upon the similliar question try these substitutions