Đáp án:

\(\int \frac{f(\sqrt{x + 1})}{\sqrt{x+1}} = 2 \int f(\sqrt{x + 1}) \, d(\sqrt{x + 1}) = \frac{2(\sqrt{x + 1} + 3)}{(\sqrt{x + 1})^2 + 4}. \)

\( \Rightarrow \int f(x) \, dx = \frac{x + 3}{x^2 + 4} + C. \)

\( \int f(2x) \, dx = \frac{1}{2} \int f(x) \, d(2x) = \frac{1}{2} \left( \frac{2x + 3}{4x^2 + 4} \right) + C. \)

\( = \frac{2x + 3}{8(x^2 + 1)} + C \quad \Rightarrow \boxed{D}. \)