The reduction formula for $\int\frac{x^ndx}{Ax + B}$ is given by $$ \int\frac{x^ndx}{Ax+B} = x\int\frac{x^{n-1}dx}{Ax+B} - \int\left(\int\frac{x^{n-1}dx}{Ax+B}\right)dx. $$
Is there any special notation for $\int\left(\int\frac{x^{n-1}dx}{Ax+B}\right)dx$ that I should know about? Something like $\iint\frac{x^{n-1}dx^2}{Ax+B}$ maybe?
That notation is completely understandable.
It is rare to see a double integration of the same variable, because usually they are simplified beforehand. However, you do not 'need' to change the form of the integral if you are doing multiple integration of multiple variables, so I do not see why you would need to here.
Although, sometimes you do get stuff like this,
$$ \int_\mathbf{D} f(X) dX = \int \cdots \int_\mathbf{D}f(x_1,x_2,\ldots,x_n) {dx_1\cdots dx_n}.$$
Your last suggestion might be confused as $d(x^2) = 2x dx$, but I'm not sure.