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I am interested in finding the following definite integral: $$\int_{\text{x0}}^{\infty } x^s \exp \left(a x^2+b x\right) \, dx$$ with assumptions $\text{xo}>0\land s<0\land a\in \mathbb{R}\land b\in \mathbb{R}$

I found this very similar integral in this book:

$$\int_0^{\infty } y^{\nu } \exp \left(-\left(a y^{\delta }+b y^{-\rho }\right)\right) \, dy$$

with assumptions $a>0\land b>0\land \delta >0\land \rho >0$

The answer for this is given in terms of Fox's H function. However I am not able to subtract the finite integral part from that ($0$ to $x0$).

Any help is appreciated. Two related posts are this and this.

mattTheMathLearner
  • 715
  • I believe you want to assume $a<0.$ – Idividedbyzero Apr 24 '25 at 11:22
  • For $b=0$ and $a<0$ you get with the simple change of variable, $u= ax^2:$ $$ \int_{x_0}^{\infty } x^s \exp \left(a x^2\right) , dx = \frac{1}{2} (-a)^{-\frac{s}{2}-\frac{1}{2}} \Gamma \left(\frac{s+1}{2},-a x_0^2\right) $$ with the incomplete Gamma function $\Gamma.$ – Idividedbyzero Apr 24 '25 at 11:48

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