Suppose $a,b,c,$ and $d$ are constants such that the following holds for all real numbers $x$ such that all denominators are nonzero: $\begin{aligned} & \frac{10}{x(x+10)}+\frac{10}{(x+5)(x+15)}+\frac{10}{(x+10)(x+20)} \\ &+ \frac{10}{(x+15)(x+25)}+\frac{10}{(x+20)(x+30)} \\ &= \frac{a(x^2+30x+75)}{x(x+b)(x+c)(x+d)}. \end{aligned}$ What is the value of $a+b+c+d?$