2.6.4.3 ✓ Maple. Time used: 0.069 (sec). Leaf size: 204

ode:=diff(diff(phi(x),x),x) = 4*Pi*n*c/(v__0^2+2*e/m*(phi(x)-V__0))^(1/2); 
dsolve(ode,phi(x), singsol=all);
 

\begin{gather*} \begin {aligned} e \int _{}^{\phi }\frac {\sqrt {\frac {2 \left (\textit {\_a} -V_{0} \right ) e +v_{0}^{2} m}{m}}}{4 \sqrt {\sqrt {\frac {2 \left (\textit {\_a} -V_{0} \right ) e +v_{0}^{2} m}{m}}\, \left (\frac {c_1 \sqrt {2 e \left (-\textit {\_a} +V_{0} \right )-v_{0}^{2} m}}{16}+\pi c \left (\left (\textit {\_a} -V_{0} \right ) e +\frac {v_{0}^{2} m}{2}\right ) n \right ) e}}d \textit {\_a} -x -c_2 &= 0\\ -e \int _{}^{\phi }\frac {\sqrt {\frac {2 \left (\textit {\_a} -V_{0} \right ) e +v_{0}^{2} m}{m}}}{4 \sqrt {\sqrt {\frac {2 \left (\textit {\_a} -V_{0} \right ) e +v_{0}^{2} m}{m}}\, \left (\frac {c_1 \sqrt {2 e \left (-\textit {\_a} +V_{0} \right )-v_{0}^{2} m}}{16}+\pi c \left (\left (\textit {\_a} -V_{0} \right ) e +\frac {v_{0}^{2} m}{2}\right ) n \right ) e}}d \textit {\_a} -x -c_2 &= 0\\ \end {aligned} \end{gather*}

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying 2nd order Liouville 
trying 2nd order WeierstrassP 
trying 2nd order JacobiSN 
differential order: 2; trying a linearization to 3rd order 
trying 2nd order ODE linearizable_by_differentiation 
trying 2nd order, 2 integrating factors of the form mu(x,y) 
trying differential order: 2; missing variables 
   -> Computing symmetries using: way = 3 
-> Calling odsolve with the ODE, diff(_b(_a),_a)*_b(_a)-4*Pi*n*c/(-(-m*v__0^2+2*V__0*e-2*_a*e)/m)^(1/2) = 0, _b(_a), 
HINT = [[-2/3*(-m*v__0^2+2*V__0*e-2*_a*e)/e, 1/3*_b]] 
   *** Sublevel 2 *** 
   symmetry methods on request 
   1st order, trying reduction of order with given symmetries: 
[-2/3*(-m*v__0^2+2*V__0*e-2*_a*e)/e, 1/3*_b] 
   1st order, trying the canonical coordinates of the invariance group 
      -> Calling odsolve with the ODE, diff(y(x),x) = 1/2*y(x)/(2*(x-V__0)*e+v__0^2*m)*e, y(x) 
         *** Sublevel 3 *** 
         Methods for first order ODEs: 
         --- Trying classification methods --- 
         trying a quadrature 
         trying 1st order linear 
         <- 1st order linear successful 
   <- 1st order, canonical coordinates successful 
<- differential order: 2; canonical coordinates successful 
<- differential order 2; missing variables successful