I have a problem with the notation.
rational function of n variables
If I go by this notation, the square root term can be taken as y and t as x. For the numerator and denominator to be polynomials only even powers will be taken of square root term.
Is it correct?
For the numerator and denominator to be polynomials only even powers will be taken of square root term.
Note there is a difference between being a polynomial in $t$ and being a polynomial in $t$ and $y.$ ($x$ is the upper limit, so it has no bearing on this description of the integrand.)
The excerpt you gave specifically says that the integrand is a rational function of $t$ and the radical. Thus, letting $y$ denote the radical, the integrand could be any of the following. Incidentally, unless we're told that the coefficients have to be restricted to be integers, or restricted to be rational numbers (this actually winds up giving you nothing additional to being restricted to integers), or restricted to be radical-expressed numbers, etc., then something like the third example below is also possible.
$$R(t,y) \;\; = \;\; t^2y \;\; = \;\; t^2\sqrt{a_4t^4 + a_3t^3 + a_2t^2 + a_1t^1 + a_0}$$
or
$$R(t,y) \;\; = \;\; \frac{t^3}{y} \;\; = \;\; \frac{t^3}{\sqrt{a_4t^4 + a_3t^3 + a_2t^2 + a_1t^1 + a_0}}$$
or
$$R(t,y) \;\; = \;\; \frac{\pi t^2 + 3ty + 2y}{y^8 + \sqrt[3]{5}t^3}$$
$$ = \;\;\; \frac{\pi t^2 \;\; + \;\; 3t\sqrt{a_4t^4 + a_3t^3 + a_2t^2 + a_1t^1 + a_0} \;\; + \;\; 2\sqrt{a_4t^4 + a_3t^3 + a_2t^2 + a_1t^1 + a_0}}{(a_4t^4 + a_3t^3 + a_2t^2 + a_1t^1 + a_0)^4 \; + \; \sqrt[3]{5}t^3}$$
or
$$R(t,y) \;\; = \;\; \frac{2t^3y^3 \; + \; 5t^2y^4 \; - \; 6t^2y \; + \;3t^4}{8t^4y^7 \; + \; 2ty^5 \; + \; 2ty^5 \; - \; 5t^2y^3 \; + \; y^5}$$
$$ = \;\; \text{a messy expression when only} \; t \; \text{is used}$$
If you're interested in seeing how to actually evaluate, or at least reduce to standard forms, a fairly simple looking such integral, see How to integrate $ \int \frac{x}{\sqrt{x^4+10x^2-96x-71}}dx$. Note that in your notation this integrand corresponds to $R(t,y) = \frac{t}{y}$ for $a_4 = 1$ and $a_3 = 0$ and $a_2 = 10$ and $a_1 = -96$ and $a_0 = -71.$
