Chapter 7 - Section 7.6 - Integration Using Tables and Computer Algebra Systems - 7.6 Exercises - Page 513: 12

$$\int x\sqrt{x+x^{4}}dx=\frac{x^{2}}{2}\sqrt{2+x^{4}}+ln(x^{2}+\sqrt{2+x^{4}})+C$$

Using entry #21: $\int \sqrt{a^{2}+u^{2}}du=\frac{u}{2}\sqrt{a^{2}+u^{2}}+\frac{a^{2}}{2}ln(u+\sqrt{a^{2}+u^{2}})+C$ $$\int x\sqrt{x+x^{4}}dx=\frac{1}{2}\int \sqrt{2+(x^{2})^{2}}d(x^{2})$$ $$=\frac{x^{2}}{2}\sqrt{2+x^{4}}+ln(x^{2}+\sqrt{2+x^{4}})+C$$