Using substitution, integrate x(2 + x))^1/2 where u^2 = 2 + x

u² = 2 + xx = u² - 2dx/du = 2u
ʃ x(2 + x))^1/2 dx= ʃ xu dx= ʃ u(u² - 2) dx= ʃ 2u² (u² - 2) du= ʃ 2u⁴ - 4u² du= [2/5u⁵ - 4/3u³ + C]= 2/5 (2 + x)^5/2 - 4/3 (2 + x)^3/2 + C