##### 4.13.8 $$\left (a^2+x^2+y(x)^2\right ) y'(x)+b^2+x^2+2 x y(x)=0$$

ODE
$\left (a^2+x^2+y(x)^2\right ) y'(x)+b^2+x^2+2 x y(x)=0$ ODE Classiﬁcation

[_exact, _rational]

Book solution method
Exact equation

Mathematica
cpu = 0.436356 (sec), leaf count = 438

$\left \{\left \{y(x)\to \frac {\sqrt [3]{2} \left (\sqrt {4 \left (a^2+x^2\right )^3+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+3 c_1\right ){}^{2/3}-2 a^2-2 x^2}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+3 c_1}}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \left (a^2+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+3 c_1}}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+3 c_1}}{2 \sqrt [3]{2}}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \left (a^2+x^2\right )}{2^{2/3} \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+3 c_1}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{\sqrt {4 \left (a^2+x^2\right )^3+\left (3 b^2 x+x^3-3 c_1\right ){}^2}-3 b^2 x-x^3+3 c_1}}{2 \sqrt [3]{2}}\right \}\right \}$

Maple
cpu = 0.044 (sec), leaf count = 900

$\left [y \left (x \right ) = \frac {\left (-12 b^{2} x -4 x^{3}-12 \textit {\_C1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} x \textit {\_C1} +6 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {2 \left (a^{2}+x^{2}\right )}{\left (-12 b^{2} x -4 x^{3}-12 \textit {\_C1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} x \textit {\_C1} +6 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}, y \left (x \right ) = -\frac {\left (-12 b^{2} x -4 x^{3}-12 \textit {\_C1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} x \textit {\_C1} +6 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {a^{2}+x^{2}}{\left (-12 b^{2} x -4 x^{3}-12 \textit {\_C1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} x \textit {\_C1} +6 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-12 b^{2} x -4 x^{3}-12 \textit {\_C1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} x \textit {\_C1} +6 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 a^{2}+2 x^{2}}{\left (-12 b^{2} x -4 x^{3}-12 \textit {\_C1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} x \textit {\_C1} +6 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (-12 b^{2} x -4 x^{3}-12 \textit {\_C1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} x \textit {\_C1} +6 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {a^{2}+x^{2}}{\left (-12 b^{2} x -4 x^{3}-12 \textit {\_C1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} x \textit {\_C1} +6 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-12 b^{2} x -4 x^{3}-12 \textit {\_C1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} x \textit {\_C1} +6 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {2 a^{2}+2 x^{2}}{\left (-12 b^{2} x -4 x^{3}-12 \textit {\_C1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} x \textit {\_C1} +6 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ]$ Mathematica raw input

DSolve[b^2 + x^2 + 2*x*y[x] + (a^2 + x^2 + y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (-2*a^2 - 2*x^2 + 2^(1/3)*(-3*b^2*x - x^3 + Sqrt[4*(a^2 + x^2)^3 + (3*
b^2*x + x^3 - 3*C[1])^2] + 3*C[1])^(2/3))/(2^(2/3)*(-3*b^2*x - x^3 + Sqrt[4*(a^2
 + x^2)^3 + (3*b^2*x + x^3 - 3*C[1])^2] + 3*C[1])^(1/3))}, {y[x] -> ((1 + I*Sqrt
[3])*(a^2 + x^2))/(2^(2/3)*(-3*b^2*x - x^3 + Sqrt[4*(a^2 + x^2)^3 + (3*b^2*x + x
^3 - 3*C[1])^2] + 3*C[1])^(1/3)) + ((I/2)*(I + Sqrt[3])*(-3*b^2*x - x^3 + Sqrt[4
*(a^2 + x^2)^3 + (3*b^2*x + x^3 - 3*C[1])^2] + 3*C[1])^(1/3))/2^(1/3)}, {y[x] ->
 ((1 - I*Sqrt[3])*(a^2 + x^2))/(2^(2/3)*(-3*b^2*x - x^3 + Sqrt[4*(a^2 + x^2)^3 +
 (3*b^2*x + x^3 - 3*C[1])^2] + 3*C[1])^(1/3)) - ((I/2)*(-I + Sqrt[3])*(-3*b^2*x
- x^3 + Sqrt[4*(a^2 + x^2)^3 + (3*b^2*x + x^3 - 3*C[1])^2] + 3*C[1])^(1/3))/2^(1
/3)}}

Maple raw input

dsolve((a^2+x^2+y(x)^2)*diff(y(x),x)+b^2+x^2+2*x*y(x) = 0, y(x))

Maple raw output

[y(x) = 1/2*(-12*b^2*x-4*x^3-12*_C1+4*(4*a^6+12*a^4*x^2+12*a^2*x^4+9*b^4*x^2+6*b
^2*x^4+5*x^6+18*_C1*b^2*x+6*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-2*(a^2+x^2)/(-12*b^2*x
-4*x^3-12*_C1+4*(4*a^6+12*a^4*x^2+12*a^2*x^4+9*b^4*x^2+6*b^2*x^4+5*x^6+18*_C1*b^
2*x+6*_C1*x^3+9*_C1^2)^(1/2))^(1/3), y(x) = -1/4*(-12*b^2*x-4*x^3-12*_C1+4*(4*a^
6+12*a^4*x^2+12*a^2*x^4+9*b^4*x^2+6*b^2*x^4+5*x^6+18*_C1*b^2*x+6*_C1*x^3+9*_C1^2
)^(1/2))^(1/3)+(a^2+x^2)/(-12*b^2*x-4*x^3-12*_C1+4*(4*a^6+12*a^4*x^2+12*a^2*x^4+
9*b^4*x^2+6*b^2*x^4+5*x^6+18*_C1*b^2*x+6*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-1/2*I*3^(
1/2)*(1/2*(-12*b^2*x-4*x^3-12*_C1+4*(4*a^6+12*a^4*x^2+12*a^2*x^4+9*b^4*x^2+6*b^2
*x^4+5*x^6+18*_C1*b^2*x+6*_C1*x^3+9*_C1^2)^(1/2))^(1/3)+2*(a^2+x^2)/(-12*b^2*x-4
*x^3-12*_C1+4*(4*a^6+12*a^4*x^2+12*a^2*x^4+9*b^4*x^2+6*b^2*x^4+5*x^6+18*_C1*b^2*
x+6*_C1*x^3+9*_C1^2)^(1/2))^(1/3)), y(x) = -1/4*(-12*b^2*x-4*x^3-12*_C1+4*(4*a^6
+12*a^4*x^2+12*a^2*x^4+9*b^4*x^2+6*b^2*x^4+5*x^6+18*_C1*b^2*x+6*_C1*x^3+9*_C1^2)
^(1/2))^(1/3)+(a^2+x^2)/(-12*b^2*x-4*x^3-12*_C1+4*(4*a^6+12*a^4*x^2+12*a^2*x^4+9
*b^4*x^2+6*b^2*x^4+5*x^6+18*_C1*b^2*x+6*_C1*x^3+9*_C1^2)^(1/2))^(1/3)+1/2*I*3^(1
/2)*(1/2*(-12*b^2*x-4*x^3-12*_C1+4*(4*a^6+12*a^4*x^2+12*a^2*x^4+9*b^4*x^2+6*b^2*
x^4+5*x^6+18*_C1*b^2*x+6*_C1*x^3+9*_C1^2)^(1/2))^(1/3)+2*(a^2+x^2)/(-12*b^2*x-4*
x^3-12*_C1+4*(4*a^6+12*a^4*x^2+12*a^2*x^4+9*b^4*x^2+6*b^2*x^4+5*x^6+18*_C1*b^2*x
+6*_C1*x^3+9*_C1^2)^(1/2))^(1/3))]