Is this series diverging? If not, what's the sum?
Heuristically, the summand is the ratio of a linear polynomial to a quadratic polynomial, and so it grows similarly to the series $\sum_{i=1}^\infty \frac 1 i$, which diverges. This tells us that the original sum diverges as well. To show this, note that for sufficiently large $i$ ($i>80$, to be exact) we have $$\frac{4i+1}{(4i+3)^2+(4i+7)^2} = \frac{4i+1}{32i^2+80i+58} \geq \frac{4i}{33i^2} = \frac{4}{33}\frac 1 i$$ Now you can use the comparison test.