Ta có

$\int_{0}^{1}{\frac{3}{{{x}^{2}}-4x-5}dx}\int_{0}^{1}{\frac{3}{\left( x-5 \right)\left( x+1 \right)}dx}$

$=\int_{0}^{1}{\left( \frac{\frac{1}{2}}{x-5}-\frac{\frac{1}{2}}{x+1} \right)dx}=\left. \frac{1}{2}\left( \ln \left| x-5 \right|-\ln \left| x+1 \right| \right) \right|_{0}^{1}$

Vậy $I=\left. \frac{1}{2}\ln \left| \frac{x-5}{x+1} \right| \right|_{0}^{1}=\frac{1}{2}\ln \frac{2}{5}$.

Ta chọn phương án D.