$$ {(s^2 -3)} \over (s+2)(s-3)(s^2+2s+5)$$
I was trying to get the correct result of this inverse of Laplace but I still get the wrong result because I get these results as coefficients:
$$A = \frac 15$$ $$B = 0$$ $$C = -\frac15$$ $$D = 1$$ But applying the inverse of Laplace does not agree with what it says in the book as the result, which is the following:
$$\frac 3{50}e^{3t} -\frac 1{25}e^{-2t}- \frac 1{50}e^{-t}\cos(2t) +\frac9{25}e^t \sec(2t)$$
your coefficients are not correct.
There are different methods to find the coefficient.
Use a different method and make sure that you do not make any arithmetic error.
You may find coefficients of $1/(s-3)$ and $ 1/(s+2)$ by Heaviside's method and subtract the result from the left side to find the other two coefficients.