indefinite integral

$${f}{\left({\mathtt{x}}\right)} = \left(\left({\mathtt{4}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{x}}}}{\mathtt{\,-\,}}{\frac{{\mathtt{3}}}{{\mathtt{2}}}}\right){\mathtt{\,\times\,}}\left({\frac{{\mathtt{2}}}{{\mathtt{x}}}}{\mathtt{\,\small\textbf+\,}}{{\mathtt{x}}}^{{\mathtt{2}}}\right)\right)$$

$$\\\int((4x^{0.5}-1.5)(2x^{-1}+x^2))dx\\\\
=\int((4x^{0.5})(2x^{-1}+x^2)-1.5(2x^{-1}+x^2))dx\\\\
=\int(8x^{-0.5}+4x^{2.5}-3x^{-1}-1.5x^2)dx\\\\
etc$$

Find the indefinite integral of the function f(x)=(4sqrt(x)-3/2)(2/x+x^2)

$$\\ \small{\text{
$f(x)=
\left(4\sqrt{x}-\frac{3}{2}\right)\cdot
\left(\frac{2}{x}+x^2\right)
$}}\\\\
\small{\text{
$f(x)=4\sqrt{x} \cdot \frac{2}{x}
+ 4\sqrt{x} \cdot x^2
- \frac{3}{2} \cdot \frac{2}{x}
- \frac{3}{2} \cdot x^2
$}}\\\\
\small{\text{
$f(x)=8 x^{\frac{1}{2}} \cdot x^{-1}
+ 4 x^{\frac{1}{2}} \cdot x^2
- 3 \cdot x^{-1}
- \frac{3}{2} \cdot x^2
$}}\\\\
\boxed{
\small{\text{
$f(x)=8 x^{-\frac{1}{2}}
+ 4 x^{\frac{5}{2}}
- 3 x^{-1}
- \frac{3}{2} \cdot x^2
$}}}\\\\
\int{(8 x^{-\frac{1}{2}}
+ 4 x^{\frac{5}{2}}
- 3 x^{-1}
- \frac{3}{2} \cdot x^2
)}\ dx \\\\
\small{\text{
$
=
8\int{(x^{-\frac{1}{2}} ) }\ dx
+
4\int{( x^{\frac{5}{2}} ) }\ dx
-
3\int{( x^{-1} ) }\ dx
-
\frac{3}{2} \int{(x^2 ) }\ dx
$
}}\\
\boxed{\int{(x^{-\frac{1}{2}} ) }\ dx = 2\sqrt{x} +c \qquad
\int{(x^{ \frac{5}{2}} ) }\ dx = \frac{2}{7}x^3\sqrt{x} +c } \\
\boxed{\int{(x^{-1}) }\ dx = \ln{(x)} +c \qquad
\int{(x^{2}) }\ dx = \frac{x^3}{3} +c } \\\\\\
\small{\text{
$
\int{(8 x^{-\frac{1}{2}}
+ 4 x^{\frac{5}{2}}
- 3 x^{-1}
- \frac{3}{2} \cdot x^2
)}\ dx
=16\sqrt{x}
+ \frac{8}{7}x^3\sqrt{x}
-3\ln{(x)}
-\frac{x^3}{2}
+c
$
}}$$