Define the set $D$ to be the subset of $\mathbb{R}^{5}$ specified as follows:
$$D:=\{\left(a,b,c,d,z\right)\in\mathbb{R}^{5}\mid c>0\land z\ge a\land z>d\}.$$
For each $\left(a,b,c,d,z\right)\in D$, the improper integral $\int_{z}^{\infty}\mathrm{d}x\,\frac{1}{\left(x-d\right)\sqrt{\left(x-a\right)\left[\left(x-b\right)^{2}+c^{2}\right]}}$ can be shown to converge using the standard comparison test. Thus, the function $\mathcal{E}:D\rightarrow\mathbb{R}$ is well-defined via the elliptic integral representation
$$\begin{align} \mathcal{E}{\left(a,b,c,d,z\right)} &:=\int_{z}^{\infty}\mathrm{d}x\,\frac{1}{\left(x-d\right)\sqrt{\left(x-a\right)\left[\left(x-b\right)^{2}+c^{2}\right]}}.\\ \end{align}$$
(Note: I shall adopt the same convention as DLMF for defining the elliptic integrals of the 1st, 2nd and 3rd kind.)
My initial goal was to obtain a general expression for the elliptic integral $\mathcal{E}$ reduced to Legendre normal form using the substitution
$$\frac{2\sqrt{A\left(x-a\right)}}{x-a+A}=t;~~~\small{A:=\sqrt{\left(a-b\right)^{2}+c^{2}}\in\mathbb{R}_{>0}},$$
but proceeding this way seems to require a good bit of casework. Some of the casework can be eliminated by adding the assumptions $a<d\land a+A<z$, which is what I'm primarily interested in. This seems to result in elliptic integrals of the third kind having characteristic greater than unity.
Question: Is there an alternative way to reduce $\mathcal{E}$ to Legendre's three elliptic integrals such all elliptic integral variables are in their normal ranges, i.e., $0<\theta<\frac{\pi}{2}\land0<\kappa<1\land0<n<1$?