This is my very first question on stack exchange, nice to meet you all! Let me apologise in advance in case I did not respect some conventions for a first question.
I came across this integral: $$I:=\int_0^\infty \frac{\exp\left(-\frac{1}{4}\frac{b_1^2}{s+a_1}-\frac{1}{4}\frac{b_2^2}{s+a_2}-\frac{1}{4}\frac{b_3^2}{s+a_3}\right)}{\sqrt{s+a_1}\sqrt{s+a_2}\sqrt{s+a_3}}\,\text{d}s\,,$$ where $(a_1, a_2, a_3, b_1, b_2, b_3)\in\mathbb C^6$.
Let me say that in the case where $a_1=a_2=a_3=:a$, the solution is given by $$I=\frac{2\sqrt\pi\,\text{erf}\left(\frac{\sqrt{b_1^2+b_2^2+b_3^2}}{2\sqrt a}\right)}{\sqrt{b_1^2+b_2^2+b_3^2}}\,.$$
This can be obtained by performing the change of variable $$u^2:=\frac{1}{4}\frac{b_1^2+b_2^2+b_3^2}{s+a}\,,$$ leading immediately to the result.
However, for generic $(a_1, a_2, a_3)\in\mathbb C^3$, I cannot seem to find an expression for that integral. Would you know how to proceed?
Thank you very much!