ODE
\[ \left (a^2+x^2\right ) y'(x)=\left (\sqrt {a^2+x^2}+x\right ) (b+y(x)) \] ODE Classification
[_separable]
Book solution method
Separable ODE, Neither variable missing
Mathematica ✓
cpu = 0.413936 (sec), leaf count = 60
\[\left \{\left \{y(x)\to \frac {\left (x \left (\sqrt {a^2+x^2}+x\right )+a^2\right ) \left (b x+a^2 c_1 \sqrt {a^2+x^2}\right )}{a^2 \sqrt {a^2+x^2}}\right \}\right \}\]
Maple ✓
cpu = 0.021 (sec), leaf count = 40
\[\left [y \left (x \right ) = \left (\frac {x b}{\sqrt {a^{2}+x^{2}}\, a^{2}}+\textit {\_C1} \right ) \left (x \sqrt {a^{2}+x^{2}}+a^{2}+x^{2}\right )\right ]\] Mathematica raw input
DSolve[(a^2 + x^2)*y'[x] == (x + Sqrt[a^2 + x^2])*(b + y[x]),y[x],x]
Mathematica raw output
{{y[x] -> ((a^2 + x*(x + Sqrt[a^2 + x^2]))*(b*x + a^2*Sqrt[a^2 + x^2]*C[1]))/(a^2*Sqrt[a^2 + x^2])}}
Maple raw input
dsolve((a^2+x^2)*diff(y(x),x) = (b+y(x))*(x+(a^2+x^2)^(1/2)), y(x))
Maple raw output
[y(x) = (1/(a^2+x^2)^(1/2)/a^2*x*b+_C1)*(x*(a^2+x^2)^(1/2)+a^2+x^2)]