1. **State the problem:** We need to evaluate the indefinite integral $$\int \frac{7}{\sqrt{x+7} + \sqrt{x}} \, dx.$$\n\n2. **Rewrite the integral:** The integrand is $$\frac{7}{\sqrt{x+7} + \sqrt{x}}.$$ To simplify, multiply numerator and denominator by the conjugate $$\sqrt{x+7} - \sqrt{x}$$ to rationalize the denominator.\n\n3. **Multiply numerator and denominator by the conjugate:**\n$$\int \frac{7}{\sqrt{x+7} + \sqrt{x}} \, dx = \int \frac{7(\sqrt{x+7} - \sqrt{x})}{(\sqrt{x+7} + \sqrt{x})(\sqrt{x+7} - \sqrt{x})} \, dx.$$\n\n4. **Simplify the denominator using difference of squares:**\n$$(\sqrt{x+7})^2 - (\sqrt{x})^2 = (x+7) - x = 7.$$\n\n5. **Substitute back:**\n$$\int \frac{7(\sqrt{x+7} - \sqrt{x})}{7} \, dx = \int (\sqrt{x+7} - \sqrt{x}) \, dx.$$\n\n6. **Cancel the 7's:**\n$$\int \cancel{7} \frac{(\sqrt{x+7} - \sqrt{x})}{\cancel{7}} \, dx = \int (\sqrt{x+7} - \sqrt{x}) \, dx.$$\n\n7. **Split the integral:**\n$$\int \sqrt{x+7} \, dx - \int \sqrt{x} \, dx.$$\n\n8. **Use the power rule for integration:** For any $n \neq -1$, $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.$$\n\n9. **Rewrite the square roots as powers:**\n$$\sqrt{x+7} = (x+7)^{1/2}, \quad \sqrt{x} = x^{1/2}.$$\n\n10. **Integrate each term:**\n$$\int (x+7)^{1/2} \, dx = \frac{2}{3} (x+7)^{3/2} + C_1,$$\n$$\int x^{1/2} \, dx = \frac{2}{3} x^{3/2} + C_2.$$\n\n11. **Combine results:**\n$$\int \frac{7}{\sqrt{x+7} + \sqrt{x}} \, dx = \frac{2}{3} (x+7)^{3/2} - \frac{2}{3} x^{3/2} + C,$$ where $C = C_1 - C_2$ is the constant of integration.