2.6.6.6 ✓ Maple. Time used: 0.003 (sec). Leaf size: 12

ode:=diff(n(x),x) = (n(x)^2+1)*x; 
dsolve(ode,n(x), singsol=all);
 

\begin{align*} n = \tan \left (\frac {x^{2}}{2}+c_1 \right ) \end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
<- separable successful