1. **State the problem:** We need to evaluate the integral $$\int x^3 \sqrt{1+x^2} \, dx$$. 2. **Identify a substitution:** Let $$u = 1 + x^2$$. Then, $$du = 2x \, dx$$ or $$x \, dx = \frac{du}{2}$$. 3. **Rewrite the integral:** Note that $$x^3 = x^2 \cdot x = (u - 1) \cdot x$$. So, $$\int x^3 \sqrt{1+x^2} \, dx = \int (u - 1) \sqrt{u} \cdot x \, dx = \int (u - 1) \sqrt{u} \cdot \frac{du}{2} = \frac{1}{2} \int (u - 1) u^{1/2} \, du$$. 4. **Simplify the integrand:** $$(u - 1) u^{1/2} = u^{3/2} - u^{1/2}$$. 5. **Integrate term-by-term:** $$\frac{1}{2} \int (u^{3/2} - u^{1/2}) \, du = \frac{1}{2} \left( \int u^{3/2} \, du - \int u^{1/2} \, du \right)$$. 6. **Compute each integral:** $$\int u^{3/2} \, du = \frac{u^{5/2}}{\frac{5}{2}} = \frac{2}{5} u^{5/2}$$ $$\int u^{1/2} \, du = \frac{u^{3/2}}{\frac{3}{2}} = \frac{2}{3} u^{3/2}$$. 7. **Substitute back:** $$\frac{1}{2} \left( \frac{2}{5} u^{5/2} - \frac{2}{3} u^{3/2} \right) + C = \frac{1}{5} u^{5/2} - \frac{1}{3} u^{3/2} + C$$. 8. **Replace $$u$$ with $$1 + x^2$$:** $$\boxed{\frac{1}{5} (1+x^2)^{5/2} - \frac{1}{3} (1+x^2)^{3/2} + C}$$. This is the evaluated integral.