Evaluate the following integrals:

$\int\left\{\sqrt{x}\left(a x^{2}+b x+c\right)\right\} d x$

$\Rightarrow \int\left(\sqrt{x a x}^{2}+\sqrt{x b x}+\sqrt{x c}\right) d x$

By Splitting, we get,

$\Rightarrow \mathrm{a} \int \mathrm{x}^{2} \times \mathrm{x}^{\frac{1}{2}} \mathrm{dx}+\mathrm{b} \int \mathrm{x}^{1} \times \mathrm{x}^{\frac{1}{2}} \mathrm{dx}+\mathrm{c} \int \mathrm{x}^{1 / 2} \mathrm{dx}$

$\Rightarrow \mathrm{a} \int \mathrm{x}^{\frac{5}{2}} \mathrm{dx}+\mathrm{b} \int \mathrm{x}^{\frac{2}{2}} \mathrm{dx}+\mathrm{c} \int \mathrm{x}^{\frac{1}{2}} \mathrm{dx}$

By using the formula

$\int x^{n} d x=\frac{x^{n+1}}{n+1}$

$\Rightarrow \frac{\mathrm{ax}^{\frac{5}{2}+1}}{\frac{5}{2}+1}+\frac{\mathrm{bx}^{\frac{3}{2}+1}}{\frac{3}{2}+1}+\frac{\mathrm{cx}^{\frac{1}{2}+1}}{\frac{1}{2}+1}+\mathrm{c}$

$\Rightarrow \frac{\mathrm{ax}^{\frac{7}{2}}}{7 / 2}+\frac{\mathrm{bx}^{\frac{5}{2}}}{5 / 2}+\frac{\mathrm{cx}^{\frac{2}{2}}}{3 / 2}+\mathrm{c}$