Summations can be easy!

Calculus Level 5

\[\sum_{n=0}^{\infty}\left( \dfrac{n}{2+\int_{0}^{n} \left(18x^2+26x+9\right)\, dx}\right)= \ln\left(\dfrac{a}{b}\right)-\dfrac{c \pi }{\sqrt{d}} \]

\(\text{Find:}\) \(a+b+c+d+5\).

Details and Assumptions:

\(\bullet\) All the constants in the RHS are positive integers. All fractions are in reduced form (co-prime). \(d\) is a square free integer.

This question is part of the set Best of Me.
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