We know that $$\int_{t=0}^\infty t \cdot \exp(-p^2t^2) \text{I}_{\nu}(at) \operatorname{I}_\nu (bt) \, \mathrm{d}t = \frac{1}{2p^2}\exp\left(\frac{a^2+b^2}{4p^2}\right) \operatorname{I}_\nu \left(\frac{ab}{2p^2}\right),$$ where $\text{I}_\nu(at)$ is the modified Bessel function of the first kind. What do we know about following integral? $$\int_{t=0}^\infty t^2 \cdot \exp(-p^2t^2) \operatorname{I}_\nu(at) \operatorname{I}_\nu(bt) \, \mathrm{d}t$$ I know following general answers, but I am wondering if they could be simplified for the case of $t^2$ or not.
The First one is from the book "A Treatise on the Theory of Bessel Functions" by Watson:
The second one is from the book "Table of Integrals":
