I am working on a model for my research, and I need to solve the expectation of a random variable for it. Specifically, the part where I am stuck is of the form $\int_{\underline{x}}^{\infty} \dfrac{a}{bx+c} g(x)dx$, where $g(.)$ is the pdf of the random variable $x$, with cdf $G(.)$.
I assume for now that the support of $x$ is $[0,\infty]$, but the upper bound can be specified if necessary. $\underline{x}$ is some threshold value of $x$, determined by other parameters in my model. $G(.)$ is a general distribution, so it is okay if any transformation is applied to the distribution for solving the integral. Additionally, $G(0)=0, G(\infty)=1$.
I have tried a few things to simplify this integral and they haven't worked: substituting the denominator $bx + c$, or $\dfrac{a}{bx+c}$. The problem in integration by parts due to the variable being in the denominator is that the power of the denominator keeps increasing (by differentiation rule of the second part). If I treat this fraction as the first part of integration, I do not end up with a simplification that is meaningful enough to analyze/interpret further.
Edit: Using the substitution approach highlighted here, I tried substituting $G(x)=y$, i.e. $G^{-1}(y)=x$. That led me to the expression $\int_{\underline{y}}^{1}\dfrac{1}{bG^{-1}(y)+c}dy$. I wonder if this could be simplified further, as the present form is not easy to analyze/interpret. I am also not sure if this attempt of directly plugging in the inverse (quantile) function is mathematically correct? My expectation is over $\frac{1}{x}$, but the distribution is defined over $x$.