Step-by-Step Solution
Key Concept: Divide numerator and denominator by x^5 to simplify the integrand.
Let I = \int (5x^4 + 4x^5) / (x^5 + x + 1)^2 dx. Divide numerator and denominator by x^10: I = \int (5/x^6 + 4/x^5) / (1 + 1/x^4 + 1/x^5)^2 dx. Alternatively, rewrite the integrand: (5x^4 + 4x^5) / (x^5(1 + 1/x^4 + 1/x^5))^2. A better approach is to manipulate the expression: (5x^4 + 4x^5) / (x^5 + x + 1)^2 = (5x^4 + 4x^5) / (x^10(1 + 1/x^4 + 1/x^5)^2). Let u = 1 + 1/x^4 + 1/x^5. Then du = (-4/x^5 - 5/x^6) dx = -(4x + 5) / x^6 dx. This matches the form. The integral evaluates to (x+1)/(x^5+x+1) + C or similar forms.
Correct Answer: A,C