2.6.5.6 ✓ Maple. Time used: 0.004 (sec). Leaf size: 28

ode:=diff(y(x),x) = x*(y(x)^2*a+b); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y = \frac {\tan \left (\frac {\sqrt {b a}\, \left (x^{2}+2 c_1 \right )}{2}\right ) \sqrt {b a}}{a} \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
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=x \left (a y \left (x \right )^{2}+b \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=x \left (a y \left (x \right )^{2}+b \right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d x}y \left (x \right )}{a y \left (x \right )^{2}+b}=x \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {\frac {d}{d x}y \left (x \right )}{a y \left (x \right )^{2}+b}d x =\int x d x +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\arctan \left (\frac {a y \left (x \right )}{\sqrt {b a}}\right )}{\sqrt {b a}}=\frac {x^{2}}{2}+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \left (x \right ) \\ {} & {} & y \left (x \right )=\frac {\tan \left (\frac {x^{2} \sqrt {b a}}{2}+\mathit {C1} \sqrt {b a}\right ) \sqrt {b a}}{a} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y \left (x \right )=\frac {\tan \left (\sqrt {b a}\, \left (\frac {x^{2}}{2}+\mathit {C1} \right )\right ) \sqrt {b a}}{a} \\ \bullet & {} & \textrm {Redefine the integration constant(s)}\hspace {3pt} \\ {} & {} & y \left (x \right )=\frac {\tan \left (\mathit {C1} +\frac {x^{2} \sqrt {b a}}{2}\right ) \sqrt {b a}}{a} \end {array} \]