ODE
\[ \left (a+b x^2\right ) y'(x)=A+B y(x)^2 \] ODE Classification
[_separable]
Book solution method
Separable ODE, Neither variable missing
Mathematica ✓
cpu = 0.0384604 (sec), leaf count = 55
\[\left \{\left \{y(x)\to \frac {\sqrt {A} \tan \left (\sqrt {A} \sqrt {B} \left (\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}+c_1\right )\right )}{\sqrt {B}}\right \}\right \}\]
Maple ✓
cpu = 0.012 (sec), leaf count = 37
\[ \left \{ {1\arctan \left ( {bx{\frac {1}{\sqrt {ab}}}} \right ) {\frac {1}{\sqrt {ab}}}}-{1\arctan \left ( {By \left ( x \right ) {\frac {1}{\sqrt {AB}}}} \right ) {\frac {1}{\sqrt {AB}}}}+{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[(a + b*x^2)*y'[x] == A + B*y[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> (Sqrt[A]*Tan[Sqrt[A]*Sqrt[B]*(ArcTan[(Sqrt[b]*x)/Sqrt[a]]/(Sqrt[a]*Sqrt[b]) + C[1])])/Sqrt[B]}}
Maple raw input
dsolve((b*x^2+a)*diff(y(x),x) = A+B*y(x)^2, y(x),'implicit')
Maple raw output
1/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))-1/(A*B)^(1/2)*arctan(y(x)*B/(A*B)^(1/2))+_C1 = 0