Consider the polynomial \begin{align*} p(x) &= \frac{a^3 (x - b)(x - c)(x - d)(x - e)}{(a - b)(a - c)(a - d)(a - e)} + \frac{b^3 (x - a)(x - c)(x - d)(x - e)}{(b - a)(b - c)(b - d)(b - e)} \\ &\quad + \frac{c^3 (x - a)(x - b)(x - d)(x - e)}{(c - a)(c - b)(c - d)(c - e)} + \frac{d^3 (x - a)(x - b)(x - c)(x - e)}{(d - a)(d - b)(d - c)(d - e)} \\ &\quad + \frac{e^3 (x - a)(x - b)(x - c)(x - d)}{(e - a)(e - b)(e - c)(e - d)}.\end{align*}Note that $ p(x) $ is a polynomial of degree at most 4. Also, $ p(a) = a^3, $ $ p(b) = b^3, $ $ p(c) = c^3, $ $ p(d) = d^3, $ and $ p(e) = e^3 $. Since the polynomial $ p(x) $ and $ x^3 $ agree at five different values, by the Identity Theorem, they are the same polynomial. The expression given in the problem is the coefficient of $ x^4 $ in $ p(x), $ which is then $ \boxed{0} $.