Given the context of your previous problems, it's possible there might be a typo and the denominator was intended to be 3x2+4x+2. If that were the case, the integral would be solvable using standar

This integral involves a cubic polynomial in the denominator that does not have easily found rational roots. Integrals of this form are generally not solvable using elementary functions (functions expressible in terms of polynomials, exponentials, logarithms, and trigonometric functions) and typically require advanced mathematical techniques or numerical methods.

Given the context of your previous problems, it's possible there might be a typo and the denominator was intended to be $3x^2+4x+2$. If that were the case, the integral would be solvable using standard calculus techniques.

Could you please confirm if the exponent of $x$ in the denominator is indeed 3, or if it should be 2?