Задача №1247

\[ \int\frac{\left(1+x^\frac{1}{2}\right)^3}{x^\frac{1}{3}}dx =\int\frac{1+3x^\frac{1}{2}+3x+x^\frac{3}{2}}{x^\frac{1}{3}}dx =\int\left(\frac{1}{x^\frac{1}{3}}+\frac{3x^\frac{1}{2}}{x^\frac{1}{3}}+\frac{3x}{x^\frac{1}{3}}+\frac{x^\frac{3}{2}}{x^\frac{1}{3}}\right)dx=\\ =\int\left(x^{-\frac{1}{3}}+3x^\frac{1}{6}+3x^\frac{2}{3}+x^\frac{7}{6}\right)dx =\frac{x^\frac{2}{3}}{\frac{2}{3}}+3\cdot\frac{x^\frac{7}{6}}{\frac{7}{6}}+3\cdot\frac{x^\frac{5}{3}}{\frac{5}{3}}+\frac{x^\frac{13}{6}}{\frac{13}{6}}+C=\\ =\frac{3\sqrt[3]{x^2}}{2}+\frac{18x\sqrt[6]{x}}{7}+\frac{9x\sqrt[3]{x^2}}{5}+\frac{6x^2\sqrt[6]{x}}{13}+C \]