Is this correct:? $$\frac{A}{C+B+sD} = A \frac{1} {s \frac{C+B}{D}}$$ using $$1 = \frac{1}{s}$$ so: $$f(t) = \frac{A}{\frac{C+B}{D}}$$?
it's time to teach yourself about partial fractions.
$\dfrac{A}{s(B+C+sD)} = \dfrac{\alpha}{s} + \dfrac{\beta}{B+C+sD}$
and there is a method for determining $\alpha$ and $\beta$
Once separated into partial fractions it's much easier to compute the inverse transforms of the terms.
that would work if the two terms were being added but the way you have it they are being multiplied.
What the method of partial fractions gets you is
$\dfrac{A}{s (B+C+D s)}=\dfrac{A}{s (B+C)}-\dfrac{A D}{(B+C) (B+C+D s)}$
Now you can mangle these terms to match the common transform templates and invert them as usual.
I'm not checking that what you've done to reach above is correct. But assuming it is what you do is break the equation up into powers of $s$
$\alpha(B+C) = A$ (equation for constant terms)
$(\alpha D + \beta) = 0$ (equation for terms of $s$ )
solve this system for $\alpha$ and $\beta$
IMPORTANT NOTE FOR THE FUTURE
The rules for the partial fraction decomposition get more complicated than this as powers of $s$ go up.
Please take a moment to read about it in your textbook or on the web somewhere.