Try:
$Version
(*"12.3.0 for Microsoft Windows (64-bit) (May 10, 2021)"*)
f = (k x^3 + b x^4 + c y x^3) - Sqrt[x^2 + y^2]*(k a x^3 + a b x^4 + a c y x^3)
Integrate[f, {y, 0, R}, {x, 0, Sqrt[R^2 - y^2]}, Assumptions -> {R > 0}]
(*-(1/45) k R^5 (-6 + 5 a R) - 1/672 (4 c + 3 b \[Pi]) R^6 (-7 + 6 a R)*)
