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Long story short, there was a problem about oscillatory motion in 1D due to potential V=P4(x)- polynomial of 4th 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/

bruhndough D1.0
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    This brief set of notes on the reduction of elliptic integrals is so useful, it almost feels like cheating to offer it to students: notes. Skip to the last couple of pages for a complete case-by-case list of the substitutions you're looking for. – David H Jan 23 '23 at 05:52
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    @David H: Thanks for mentioning these notes. I wasn't aware of them when I gathered some references here for an extensive 2-part MSE answer, so I've added a comment there to remind myself to include a reference to these notes if I later make any changes/additions there. FYI, I know next to nothing about elliptic integrals, and that 2-part answer was the result of a 2-weeks long diversion during my spare time in looking into this topic (at the level of what a fairly strong U.S. 2nd semester calculus student could follow). – Dave L. Renfro Jan 23 '23 at 07:05
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    Words cannot describe my gratitude for those notes – bruhndough D1.0 Jan 23 '23 at 11:44
  • @bruhndoughD1.0 =D – David H Jan 23 '23 at 13:21

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

bruhndough D1.0
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