I cannot evaluate this integral how either: $$ \int_{0}^{\infty}\left(8xe^{-x}(I_{0}(x)-I_{1}(x))-\frac{1}{x^{3/2}(x+1)^{1/2}}\right)dx, $$ where $I_{n}$ is the Bessel function of the first kind.
I expect that this value is zero because I think positive part and negative part of the integrand are cancelled out. So, I have tried to find the value by its direction but I couldn't find well. Moreover, I don't know even whether this integral converges or not. (Wolfram Alpha says it diverges but I'm not sure whether it's true.)
My question : Does this integral converge? or diverge? If it converges, is its value zero?
I'm glad if you give me even only hints of argument direction.
Thank you in advance.
HINT 1:
The series representation for $I_n$ is
$$I_n(x)=\sum_{k=0}^{\infty}\frac{1}{k!(k+n)!}\left(\frac{x}{2}\right)^{2k+n}$$
HINT 2:
The large-argument asymptotic expansion for $I_n$ is
$$I_n(x)\sim \frac{e^x}{\sqrt{2\pi x}}\left(1-\frac{4n^2-1}{8x}+\frac{(4n^2-1)(4n^2-9)}{2!(8x)^2}-\frac{(4n^2-1)(4n^2-9)(4n^2-25)}{3!(8x)^3}+\cdots\right)$$
HINT 3:
Write out the first few terms of these expansions for $I_0$ and $I_1$ and determine the expansions for $I_0-I_1$. Then, follow through with the analysis and determine whether there are convergence issues at both $0$ and $\infty$.