Chapter 7 - Techniques of Integration - 7.5 Strategy for Integration - 7.5 Exercises - Page 548: 57

$$ \frac{3}{7}(x+c)^{7/3}- \frac{3c}{4}(x+c)^{4/3}+c_1$$

Given \begin{aligned} \int x\sqrt[3]{x+c}dx \end{aligned} Let $$u^3=x+c\ \ \ \to \ \ 3u^2du=dx$$ then \begin{aligned} \int x\sqrt[3]{x+c}dx &= \int (u^3-c) (u)(3u^2)du\\ &=\int (3u^6- 3cu^3)du\\ &=\frac{3}{7}u^7- \frac{3c}{4}u^4+c_1\\ &= \frac{3}{7}(x+c)^{7/3}- \frac{3c}{4}(x+c)^{4/3}+c_1 \end{aligned}