Series RL AC circuit Laplace analysis
Take partial fractions: -

\$\dfrac{1}{(s^2+\omega^2)(s+b)}\$ becomes \$\dfrac{b-s}{(\omega^2+b^2)(s^2+\omega^2)}+\dfrac{1}{(\omega^2+b^2)(s+b)}\$

The reverse laplace of this is moderate to easy. I don't think you need me to help you with the \$\omega\$ on top or the reduction of (sL + R) to (s + b).