Đáp án:

Vì \(f'(x) \geq 0\), \(f(x) > 0\), \(\forall x \in (0, +\infty)\):  

\(\Rightarrow f'(x) = \sqrt{(x+1)f(x)}.\)

\(\Rightarrow\frac{f'(x)}{\sqrt{f(x)}} = \sqrt{x+1} \quad \Rightarrow \quad \int \frac{f'(x)}{\sqrt{f(x)}} dx = \int \sqrt{x+1} dx.\)

\(\Rightarrow2\sqrt{f(x)} = \frac{2}{3} \sqrt{(x+1)}^{3} + C.\)

\(\Rightarrow \sqrt{f(x)} = \frac{1}{3}\sqrt{(x+1)}^{3} + C.\)

Với \(f(2) = 3 \Rightarrow \quad C = 0.\)

\(\Rightarrow f(x) = \frac{1}{9}(x+1)^{3}.\)

 \( \Rightarrow f(8) = 81 \Rightarrow \boxed{81}.\)