1. **Stating the problem:** We need to find the integral $$\int x \sqrt{ax+b} \, dx$$ where $a$ and $b$ are constants. 2. **Formula and substitution:** To solve this integral, we use substitution. Let $$u = ax + b$$ so that $$du = a \, dx$$ or $$dx = \frac{du}{a}$$. 3. **Rewrite the integral:** Express $x$ in terms of $u$: $$x = \frac{u - b}{a}$$. Substitute into the integral: $$\int x \sqrt{ax+b} \, dx = \int \frac{u - b}{a} \sqrt{u} \frac{du}{a} = \int \frac{u - b}{a^2} u^{1/2} \, du = \frac{1}{a^2} \int (u - b) u^{1/2} \, du$$ 4. **Simplify the integrand:** $$ (u - b) u^{1/2} = u^{3/2} - b u^{1/2} $$ So the integral becomes: $$ \frac{1}{a^2} \int (u^{3/2} - b u^{1/2}) \, du = \frac{1}{a^2} \left( \int u^{3/2} \, du - b \int u^{1/2} \, du \right) $$ 5. **Integrate each term:** $$ \int u^{3/2} \, du = \frac{u^{5/2}}{\frac{5}{2}} = \frac{2}{5} u^{5/2} $$ $$ \int u^{1/2} \, du = \frac{u^{3/2}}{\frac{3}{2}} = \frac{2}{3} u^{3/2} $$ 6. **Substitute back:** $$ \frac{1}{a^2} \left( \frac{2}{5} u^{5/2} - b \frac{2}{3} u^{3/2} \right) + C = \frac{2}{a^2} \left( \frac{u^{5/2}}{5} - \frac{b u^{3/2}}{3} \right) + C $$ 7. **Replace $u$ with original expression:** $$ = \frac{2}{a^2} \left( \frac{(ax+b)^{5/2}}{5} - \frac{b (ax+b)^{3/2}}{3} \right) + C $$ **Final answer:** $$ \int x \sqrt{ax+b} \, dx = \frac{2}{a^2} \left( \frac{(ax+b)^{5/2}}{5} - \frac{b (ax+b)^{3/2}}{3} \right) + C $$