Factonimator.

We note that $\begin{aligned} \frac{10}{625n^4 + 1250n^3 + 873n^2 + 250n + 24} & = \; \frac{10}{(5n+1)(5n+2)(5n+3)(5n+4)} \\ & = \; \frac{5}{3}\left[\frac{1}{5n+1} - \frac{3}{5n+2} + \frac{3}{5n+3} - \frac{1}{5n+4}\right] \; = \; \frac13\left[\frac{1}{n+\frac15} - \frac{3}{n+\frac25} + \frac{3}{n+\frac35} - \frac{1}{n + \frac45}\right] \end{aligned}$ so that $\begin{aligned} \sum_{n=0}^\infty \frac{10}{625n^4 + 1250n^3 + 873n^2 + 250n + 24} & = \; \frac13\left[\psi(\tfrac45) - 3\psi(\tfrac35) + 3\psi(\tfrac25) - \psi(\tfrac15)\right] \\ & = \; \frac{\pi}{3}\left[\sqrt{1 + \frac{2}{\sqrt{5}}} - 3\sqrt{1 - \frac{2}{\sqrt{5}}}\right] \end{aligned}$ using Gauss' Digamma Theorem. Thus $(a+b+c+d)^2 = (3+2+1+5)^2 = \boxed{121}$ .