Evaluate the indefinite integral ${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(2x^4+\sqrt[\phantom{2}]{?}3x+1)dx.$

Select one:

$a.\frac{2}{5}{x}^5+\frac{2\sqrt[\phantom{2}]{?}3}{3}{x}^{\frac{3}{2}}+x$

$b.\frac{2}{5}{x}^5+\frac{1}{2}(\sqrt[\phantom{2}]{?}3x{)}^{\frac{3}{2}}+x+C$

$c.\frac{2}{5}{x}^5+\frac{2\sqrt[\phantom{2}]{?}3}{3}{x}^{\frac{3}{2}}+x+C$

$d.\frac{1}{2}{x}^4+\frac{2\sqrt[\phantom{2}]{3}}{3}{x}^{\frac{3}{2}}+x+C$