How to integrate $\int_{0}^{\infty} t^{-\frac{3}{2}} e^{-at - \frac{b}{t}}dt$ without gamma function?

Question

How to solve this type of integral \begin{equation} \int_{0}^{\infty} t^{-\frac{3}{2}} e^{-at - \frac{b}{t}}dt \end{equation} where $a$ and $b$ are positive.

Please help me out with the integration without using the gamma function.

Answer 1

Substitute $t=\sqrt{\frac ba }\frac1{x^2 }$

\begin{align}\int_{0}^{\infty} t^{-\frac{3}{2}} e^{-at - \frac{b}{t}}dt &=\sqrt[4]{\frac ab } e^{-2\sqrt{ab}}\int_{-\infty}^\infty e^{-\sqrt{ab }\>(x-\frac1{x})^2}dx\\ &= \sqrt[4]{\frac ab } e^{-2\sqrt{ab}}\int_{-\infty}^\infty e^{-\sqrt{ab }\>x^2}dx\\ &=\sqrt[4]{\frac ab } e^{-2\sqrt{ab}}\cdot\frac{\sqrt\pi}{\sqrt[4]{ab}}= \sqrt{\frac\pi b} e^{-2\sqrt{ab}} \end{align} where $\int_{-\infty}^\infty e^{-cx^2}dx=\sqrt{\frac \pi c}$ and $ \int_{-\infty}^{\infty} f\left(x-\frac 1x \right) dx =\int_{-\infty}^{\infty} f(x) dx $ are used.