I ran into the following integral in my research that I believe has no closed-form solution:
$$ I = \int_{s_0}^{s_1} \frac{(\alpha_x s + \beta_x)^{\lambda_x}}{(\alpha_y s + \beta_y)^{\lambda_y}} ds $$
The values $s_0$, $s_1$, $\alpha_x$, $\alpha_y$, $\beta_x$, $\beta_y$, $\lambda_x$, and $\lambda_y$ are all real constants. $s_0$ and $s_1$ are always contained within $[0,0.5]$, the polynomials underneath $\lambda_x$ and $\lambda_y$ respectively are strictly positive, and $\lambda_x$ and $\lambda_y$ are both strictly positive (you can assume they're both positive irrational reals). There are no relationships between the constants with subscripts $x$ and those with subscript $y$ (they're independent).
My questions are:
Is there a name for the class of functions (of $s$) matching the form of the integrand, perhaps so I can look for existing research on this? I don't come from a pure math background, so I'm not aware of any names for it.
Assuming this integral has no closed-form solution, does there exist an approximation that's reasonably efficient to compute (preferably outside the realm of Riemann sums)? Or, can I approximate (with a reasonable degree of accuracy) the integrand with some other function that's easier to integrate?