Laplace Transform of Kelvin functions

Question

What is the value of the Laplace transform, in terms of the G-function, \begin{align} \int_{0}^{\infty} e^{-st} \, t^{m} \, \left(ber_{\nu}^{2}(t) + bei_{\nu}^{2}(t)\right) \, dt \hspace{5mm} ? \end{align} As a limited case can it be shown that \begin{align} \int_{0}^{\infty} e^{-4 t} \, t \, \left( ber_{0}^{2}(t) + bei_{0}^{2}(t) \right) \, dt = - \frac{1}{8 \pi^{5/2}} \, G_{1,5}^{5,1}\left(\left. \begin{array}{cc} 1 \\ \frac{1}{4}, \frac{3}{4}, \frac{1}{2}, \frac{3}{4}, \frac{5}{4} \end{array} \right| \, - \frac{1}{4} \right) \hspace{5mm} ? \end{align}