Evaluate the following integrals:

Given:

Applying: $(a+b)^{3}=a^{3}+b^{3}+3 a b^{2}+3 a^{2} b$

$\Rightarrow \int \frac{1+x^{2}+3 x^{2} \times 1+3 \times 1^{2} \times x}{\sqrt{x}} d x$

$\Rightarrow \int \frac{1+x^{2}+3 x^{2}+3 x}{\sqrt{x}} d x$

By Splitting, we get,

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

$\Rightarrow \frac{x^{\frac{1}{2}+1}}{-\frac{1}{2}+1}+\frac{x^{\frac{5}{2}+1}}{\frac{5}{2}+1}+3 \frac{x^{\frac{3}{2}+1}}{\frac{3}{2}+1}+\frac{3 x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+c$

$\Rightarrow \frac{x^{\frac{1}{2}}}{\frac{1}{2}}+\frac{x^{\frac{7}{2}}}{\frac{7}{2}}+\frac{3 x^{\frac{5}{2}}}{\frac{5}{2}}+\frac{3 x^{\frac{3}{2}}}{\frac{3}{2}}+c$