\frac{d}{dx}\left(\frac{dy}{dx}\right)=6x+3

\begin{align*}
\frac{d}{dx}\left(\frac{dy}{dx}\right) & =6x+3  \\\  \\
let\:u & =\frac{dy}{dx} \\\ \\
\frac{du}{dx} & =6x+3 \\\ \\
Integrate,\\
\int \frac{du}{dx} & =\int (6x+3)dx \\\ \\
\int \frac{du}{dx} & =6\int xdx+3\int dx \\\ \\
u & =\frac{6x^2}{2}+3x+C_1 \\\ \\
u & =3x^2+3x+C_1 \\\ \\
Substitute, \\
\frac{dy}{dx} & =3x^2+3x+C_1 \\\ \\
dy & =\left(3x^2+3x+C_1\right)dx \\\ \\
Integrate,\\
\int dy & =\int (3x^2+3x+C_1)dx \\\ \\
\int dy & =3\int x^2dx+3\int xdx+C_1\int dx \\\ \\
y & =\frac{3x^3}{3}+\frac{3x^2}{2}+C_1x+C_2 \\\ \\
Simplify, \\
y & =x^3+\frac{3x^2}{2}+C_1x+C_2 \\
\end{align*}