From a physics problem I'm interested by a closed form of this integral : $$\int_{-\infty}^{+\infty} \frac{1}{\sqrt{P(x)}}e^{-ax^2 - bx - c} dx$$
where $P(x) = \lambda_6 x^6 + ... + \lambda_0$
I just know that $\int_{-\infty}^{+\infty} e^{-ax^2 - bx} dx = \sqrt{\frac{\pi}{a}}e^{\frac{b^2}{4a}}$ but here I have no ideas. Thanks for any suggestions.
Edit : more details for $P(x)$ : $P(x) = \Big(1+(\frac{x+c_1}{c_2})^2\Big)\Big(1+(\frac{x-c_1}{c_2})^2\Big)\Big(\lambda_1[1+(\frac{x-c_3}{c_4})^2] + \lambda_2[1+(\frac{x+c_3}{c_4})^2]\Big)$ with $c_1,c_2,c_3,c_4,\lambda_1, \lambda_2 \in \mathbb R$