special functions - Relation between full elliptic integrals of the first and third kind

I am working on a calculation involving the Ronkin function of a hyperplane in 3-space.
I get a horrible matrix with full elliptic integrals as entries. A priori I know that the matrix is symmetrical and that give me a relation between full elliptic integrals of the first and third kind.
I can not find transformations in the literature that explain the relation and I think I need one in order to simplify my matrix.

The relation

With the notation

$operatorname{K}(k) = int_0^{frac{pi}{2}}frac{dvarphi}{sqrt{1-k^2sin^2varphi}},$
$qquad$ $Pi(alpha^2,k)=int_0^{frac{pi}{2}}frac{dvarphi}{(1-alpha^2sin^2varphi)sqrt{1-k^2sin^2varphi}}$

$k^2 = frac{(1+a+b-c)(1+a-b+c)(1-a+b+c)(-1+a+b+c)}{16abc},quad a,b,c > 0$

the following is true:

$2frac{(1+a+b-c)(1-a-b+c)(a-b)}{(a-c)(b-c)}operatorname{K}(k)+$

$(1-a-b+c)(1+a-b-c)Pileft( frac{(1+a-b+c)(-1+a+b+c)}{4ac},kright) +$

$frac{(a+c)(1+b)(1-a-b+c)(-1-a+b+c)}{(a-c)(-1+b)}Pileft( frac{(1+a-b+c)(-1+a+b+c)(a-c)^2}{4ac(1-b)^2},kright)+$

$(1-a-b+c)(-1+a+b+c)Pileft( frac{(1-a+b+c)(-1+a+b+c)}{4ac},kright)+$

$frac{(1+a)(b+c)(-1-a+b+c)(-1+a+b-c)}{(1-a)(c-b)}Pileft( frac{(1-a+b+c)(-1+a+b+c)(b-c)^2}{4ac(1-a)^2},kright)$

$==0$.

Is there some addition formula or transformation between elliptic integrals of the first and third kind that will explain this?