Solve ∫ 2at(a+x)=c

$$\int 2 a t ( a + x ) = c$$

Solve for a

$$\left\{\begin{matrix}a=\frac{\sqrt{4t^{2}x^{4}+Сtx+8ctx}-2tx^{2}}{4tx}\text{; }a=\frac{-\sqrt{4t^{2}x^{4}+Сtx+8ctx}-2tx^{2}}{4tx}\text{, }&\left(x\neq 0\text{ and }c\neq \frac{С}{2}\text{ and }t=-\frac{С_{1}}{4x^{3}}-\frac{2c}{x^{3}}\right)\text{ or }\left(x\neq 0\text{ and }t\geq -\frac{С_{2}}{4x^{3}}-\frac{2c}{x^{3}}\text{ and }t>0\right)\text{ or }\left(x\neq 0\text{ and }t\leq -\frac{С_{3}}{4x^{3}}-\frac{2c}{x^{3}}\text{ and }t<0\right)\\a\in \mathrm{R}\text{, }&c=С\text{ and }\left(x=0\text{ or }t=0\right)\end{matrix}\right.$$

Solve for c

$$c=atx\left(x+2a\right)+С$$