\(S=\frac{1}{x^2+5x+4}+\frac{1}{x^2+11x+28}+\frac{1}{x^2+17x+70}+\frac{1}{x^2+23x+130}+\frac{1}{x^2+29x+208}\)
\(=\frac{1}{x^2+4x+x+4}+\frac{1}{x^2+7x+4x+28}+...+\frac{1}{x^2+16x+13x+208}\)
\(=\frac{1}{x\left(x+4\right)+\left(x+4\right)}+\frac{1}{x\left(x+7\right)+4\left(x+7\right)}+...+\frac{1}{x\left(x+16\right)+13\left(x+16\right)}\)
\(=\frac{1}{\left(x+1\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+7\right)}+...+\frac{1}{\left(x+13\right)\left(x+16\right)}\)
\(=\frac{1}{3}\left[\frac{3}{\left(x+1\right)\left(x+4\right)}+\frac{3}{\left(x+4\right)\left(x+7\right)}+...+\frac{3}{\left(x+13\right)\left(x+16\right)}\right]\)
\(=\frac{1}{3}\left[\frac{1}{x+1}-\frac{1}{x+4}+\frac{1}{x+4}-\frac{1}{x+7}+...+\frac{1}{x+13}-\frac{1}{x+16}\right]\)
\(=\frac{1}{3}\left[\frac{1}{x+1}-\frac{1}{x+16}\right]\)\(=\frac{1}{3}\left[\frac{x+16}{\left(x+1\right)\left(x+16\right)}-\frac{x+1}{\left(x+1\right)\left(x+16\right)}\right]\)
\(=\frac{1}{3}\cdot\frac{15}{x^2+17x+16}=\frac{5}{x^2+7x+16}\)
