Let I (x)=∫ (3 d x)/((4 x+6)(√(4 x^2+8 x+3))) and I (0)=(√ 3)/4+20. If I (1/2)=(a √ 2)/b+ c, where a, b, c ∈ N, gcd (a, b)=1, then a+b+c is equal to

Let $\displaystyle \mathrm{I}(x)=\int \frac{3 d x}{(4 x+6)\left(\sqrt{4 x^2+8 x+3}\right)}$ and $\displaystyle \mathrm{I}(0)=\frac{\sqrt{3}}{4}+20$. If $\displaystyle \mathrm{I}\left(\frac{1}{2}\right)=\frac{a \sqrt{2}}{b}+\mathrm{c}$, where $\displaystyle a, b, \mathrm{c} \in \mathrm{N}, \operatorname{gcd}(a, b)=1$, then $\displaystyle a+b+c$ is equal to