Find all the solutions of $x_t=x(t), t\in [0,T]$.
$$\frac{d^2 x_t}{dt^2}-A\cdot x_t\cdot \big(\frac{dx_t}{dt} \big)^2 + B\cdot \frac{dx_t}{dt} +A\cdot x_t^3+ (B-\frac{3}{2})\cdot x_t=0,$$ with $x_0=C$, $x_T=D$ and $\int_0^T x_tdt=0.$ Here $A, B, C, D$ are constants and $A>0$. $A$ might have to take discrete values.
$$\frac{d^2 x_t}{dt^2}-A\cdot x_t\cdot \big(\frac{dx_t}{dt} \big)^2 + B\cdot \frac{dx_t}{dt} +A\cdot x_t^3+ (B-\frac{3}{2})\cdot x_t=0,$$ This an ODE of autonomous kind. Usually one can reduce the order with change of variables : $\quad\begin{cases} x_t=x\\ \frac{dx}{dt}=u(x) \end{cases}$ $\quad\implies\quad \frac{d^2 x}{dt^2}=\frac{du}{dx}\frac{dx}{dt}=u\frac{du}{dx}$ $$u\frac{du}{dx}-A x u ^2 + Bu +A x^3+ (B-\frac{3}{2})x=0,$$ Let $\quad u(x)=\frac{1}{y(x)}$ $$-\frac{1}{y^3}\frac{dy}{dx}-A x \frac{1}{y^2} + B\frac{1}{y} +A x^3+ (B-\frac{3}{2})x=0,$$ $$\frac{dy}{dx}=-Axy+ B y^2+ \left(A x^3+(B-\frac{3}{2})x\right)y^3$$ This is an Abel's differential equation : $$y'=f_0(x)+f_1(x)y+f_2(x)y^2+f_3(x)y^3\qquad \begin{cases} f_0(x)=0\\ f_1(x)=-Ax\\ f_2(x)=B\\ f_3(x)=A x^3+(B-\frac{3}{2})x \end{cases}$$ The Abel's ODEs are generally not solvable on closed form in terms of finite number of standard functions.
I am afraid that you have to solve it approximately with series or more likely with numerical calculus. Nevertheless, have a look at this paper : https://www.hindawi.com/journals/ijmms/2011/387429/#sec2
This doesn't means that there is never a closed form solution. The equation can be solvable for particular values of the parameters $A,B$. For example, it is solvable in in term of elementary function in case of $B=\frac14$.