1. **State the problem:** We want to find the integral $$\int \frac{ax^3 + bx^2 + cx + d}{ex^2 + fx + g} \, dx$$ where $a,b,c,d,e,f,g$ are constants. 2. **Understand the approach:** Since the numerator is a cubic polynomial and the denominator is quadratic, perform polynomial division first to simplify the integrand. 3. **Polynomial division:** Divide $ax^3 + bx^2 + cx + d$ by $ex^2 + fx + g$. Let quotient be $Q(x) = A x + B$ and remainder $R(x) = C x + D$ such that: $$ax^3 + bx^2 + cx + d = (ex^2 + fx + g)(A x + B) + C x + D$$ 4. **Find coefficients $A,B,C,D$ by equating coefficients:** Expand right side: $$(ex^2)(A x) + (fx)(A x) + g (A x) + (ex^2)(B) + (fx)(B) + g B + C x + D$$ $$= A e x^3 + A f x^2 + A g x + B e x^2 + B f x + B g + C x + D$$ Group by powers: - $x^3$: $A e$ - $x^2$: $A f + B e$ - $x^1$: $A g + B f + C$ - constant: $B g + D$ Set equal to numerator coefficients: $$a = A e$$ $$b = A f + B e$$ $$c = A g + B f + C$$ $$d = B g + D$$ 5. **Solve for $A$ and $B$:** $$A = \frac{a}{e}$$ $$b = \frac{a}{e} f + B e \implies B = \frac{b - \frac{a}{e} f}{e} = \frac{b e - a f}{e^2}$$ 6. **Express $C$ and $D$ in terms of $A,B$:** $$C = c - A g - B f$$ $$D = d - B g$$ 7. **Rewrite integral:** $$\int \frac{ax^3 + bx^2 + cx + d}{ex^2 + fx + g} dx = \int (A x + B) dx + \int \frac{C x + D}{ex^2 + fx + g} dx$$ 8. **Integrate polynomial part:** $$\int (A x + B) dx = \frac{A}{2} x^2 + B x + C_1$$ 9. **Integrate rational part:** Use substitution or complete the square for denominator: $$ex^2 + fx + g = e \left(x^2 + \frac{f}{e} x + \frac{g}{e}\right)$$ Complete the square: $$x^2 + \frac{f}{e} x + \left(\frac{f}{2e}\right)^2 - \left(\frac{f}{2e}\right)^2 + \frac{g}{e} = \left(x + \frac{f}{2e}\right)^2 + \left(\frac{4 e g - f^2}{4 e^2}\right)$$ Let $$u = x + \frac{f}{2e}$$ and $$\Delta = 4 e g - f^2$$. 10. **Integral of form:** $$\int \frac{C x + D}{e (u^2 + \frac{\Delta}{4 e^2})} dx$$ Rewrite numerator in terms of $u$ and integrate using standard formulas for rational functions with quadratic denominators. 11. **Final answer:** $$\int \frac{ax^3 + bx^2 + cx + d}{ex^2 + fx + g} dx = \frac{A}{2} x^2 + B x + \frac{1}{e} \int \frac{C x + D}{u^2 + \frac{\Delta}{4 e^2}} dx + C_2$$ where $A,B,C,D$ are as above and the last integral can be solved by splitting numerator and using arctangent or logarithm formulas depending on $\Delta$. This completes the integration process.