ODE
\[ x^2 (a+y(x))^2 y'(x)=\left (x^2+1\right ) \left (a^2+y(x)^2\right ) \] ODE Classification
[_separable]
Book solution method
Separable ODE, Neither variable missing
Mathematica ✓
cpu = 0.597967 (sec), leaf count = 30
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [a \log \left (\text {$\#$1}^2+a^2\right )+\text {$\#$1}\& \right ]\left [x-\frac {1}{x}+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.088 (sec), leaf count = 92
\[\left [y \left (x \right ) = \frac {-a x \RootOf \left (\textit {\_Z}^{2} a^{2} x^{2}-2 \textit {\_C1} \textit {\_Z} a \,x^{2}-2 \textit {\_Z} a \,x^{3}+\textit {\_C1}^{2} x^{2}+2 \textit {\_C1} \,x^{3}+a^{2} x^{2}+x^{4}-{\mathrm e}^{\textit {\_Z}} x^{2}+2 a x \textit {\_Z} -2 x \textit {\_C1} -2 x^{2}+1\right )+x \textit {\_C1} +x^{2}-1}{x}\right ]\] Mathematica raw input
DSolve[x^2*(a + y[x])^2*y'[x] == (1 + x^2)*(a^2 + y[x]^2),y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[a*Log[a^2 + #1^2] + #1 & ][-x^(-1) + x + C[1]]}}
Maple raw input
dsolve(x^2*(a+y(x))^2*diff(y(x),x) = (x^2+1)*(y(x)^2+a^2), y(x))
Maple raw output
[y(x) = (-a*x*RootOf(_Z^2*a^2*x^2-2*_C1*_Z*a*x^2-2*_Z*a*x^3+_C1^2*x^2+2*_C1*x^3+a^2*x^2+x^4-exp(_Z)*x^2+2*a*x*_Z-2*x*_C1-2*x^2+1)+x*_C1+x^2-1)/x]