1. **State the problem:** Evaluate the integral $$\int \frac{x+1}{x + 2\sqrt{x} - 3} \, dx.$$\n\n2. **Substitution:** Let $$y = \sqrt{x}$$ so that $$x = y^2$$ and $$dx = 2y \, dy.$$\n\n3. **Rewrite the integral in terms of $$y$$:**\n$$\int \frac{y^2 + 1}{y^2 + 2y - 3} \, dx = \int \frac{y^2 + 1}{y^2 + 2y - 3} \cdot 2y \, dy = 2 \int \frac{y^3 + y}{y^2 + 2y - 3} \, dy.$$\n\n4. **Factor the denominator:**\n$$y^2 + 2y - 3 = (y + 3)(y - 1).$$\n\n5. **Set up polynomial division or partial fractions:** Since numerator degree (3) is greater than denominator degree (2), perform polynomial division:\nDivide $$y^3 + y$$ by $$y^2 + 2y - 3$$:\n- Leading term: $$y^3 / y^2 = y$$\n- Multiply divisor by $$y$$: $$y^3 + 2y^2 - 3y$$\n- Subtract: $$(y^3 + y) - (y^3 + 2y^2 - 3y) = -2y^2 + 4y$$\n\n6. **Rewrite integral:**\n$$2 \int \left(y + \frac{-2y^2 + 4y}{y^2 + 2y - 3}\right) dy = 2 \int y \, dy + 2 \int \frac{-2y^2 + 4y}{(y + 3)(y - 1)} \, dy.$$\n\n7. **Integrate the first part:**\n$$2 \int y \, dy = 2 \cdot \frac{y^2}{2} = y^2.$$\n\n8. **Partial fraction decomposition for the second integral:**\nSet $$\frac{-2y^2 + 4y}{(y + 3)(y - 1)} = \frac{A}{y + 3} + \frac{B}{y - 1}.$$\nMultiply both sides by denominator:\n$$-2y^2 + 4y = A(y - 1) + B(y + 3).$$\n\n9. **Solve for A and B:**\n- Let $$y = 1$$: $$-2(1)^2 + 4(1) = A(0) + B(4) \Rightarrow -2 + 4 = 4B \Rightarrow 2 = 4B \Rightarrow B = \frac{1}{2}.$$\n- Let $$y = -3$$: $$-2(9) + 4(-3) = A(-4) + B(0) \Rightarrow -18 - 12 = -4A \Rightarrow -30 = -4A \Rightarrow A = \frac{30}{4} = \frac{15}{2}.$$\n\n10. **Rewrite integral:**\n$$2 \int \left( \frac{15/2}{y + 3} + \frac{1/2}{y - 1} \right) dy = 2 \left( \frac{15}{2} \int \frac{1}{y + 3} dy + \frac{1}{2} \int \frac{1}{y - 1} dy \right) = 15 \int \frac{1}{y + 3} dy + \int \frac{1}{y - 1} dy.$$\n\n11. **Integrate:**\n$$15 \ln|y + 3| + \ln|y - 1| + C.$$\n\n12. **Combine all parts:**\n$$y^2 + 15 \ln|y + 3| + \ln|y - 1| + C.$$\n\n13. **Back-substitute $$y = \sqrt{x}$$:**\n$$\boxed{\sqrt{x}^2 + 15 \ln|\sqrt{x} + 3| + \ln|\sqrt{x} - 1| + C = x + 15 \ln|\sqrt{x} + 3| + \ln|\sqrt{x} - 1| + C}.$$