##### 4.15.6 $$x y'(x) \left (a+b x y(x)^3\right )+y(x) \left (a+c x^3 y(x)\right )=0$$

ODE
$x y'(x) \left (a+b x y(x)^3\right )+y(x) \left (a+c x^3 y(x)\right )=0$ ODE Classiﬁcation

[_rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.556459 (sec), leaf count = 484

$\left \{\left \{y(x)\to \frac {x \left (-c x^2+2 c_1\right )}{\sqrt [3]{3} \sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}}+\frac {\sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}}{3^{2/3} b x}\right \},\left \{y(x)\to \frac {i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}\right ){}^{2/3}+\sqrt [6]{3} \left (\sqrt {3}+3 i\right ) b x^2 \left (c x^2-2 c_1\right )}{6 b x \sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}}\right \},\left \{y(x)\to \frac {\sqrt [6]{3} \left (\sqrt {3}-3 i\right ) b x^2 \left (c x^2-2 c_1\right )-i \sqrt [3]{3} \left (\sqrt {3}-i\right ) \left (9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}\right ){}^{2/3}}{6 b x \sqrt [3]{9 a b^2 x^2+\sqrt {3} \sqrt {b^3 x^4 \left (27 a^2 b+x^2 \left (c x^2-2 c_1\right ){}^3\right )}}}\right \}\right \}$

Maple
cpu = 0.123 (sec), leaf count = 761

$\left [y \left (x \right ) = \frac {\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 \textit {\_C1} \,c^{2} x^{6}-12 \textit {\_C1}^{2} c \,x^{4}+8 \textit {\_C1}^{3} x^{2}-27 a^{2} b \right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}{3 b x}+\frac {\left (-c \,x^{2}+2 \textit {\_C1} \right ) x}{\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 \textit {\_C1} \,c^{2} x^{6}-12 \textit {\_C1}^{2} c \,x^{4}+8 \textit {\_C1}^{3} x^{2}-27 a^{2} b \right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}, y \left (x \right ) = -\frac {\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 \textit {\_C1} \,c^{2} x^{6}-12 \textit {\_C1}^{2} c \,x^{4}+8 \textit {\_C1}^{3} x^{2}-27 a^{2} b \right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}{6 b x}-\frac {\left (-c \,x^{2}+2 \textit {\_C1} \right ) x}{2 \left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 \textit {\_C1} \,c^{2} x^{6}-12 \textit {\_C1}^{2} c \,x^{4}+8 \textit {\_C1}^{3} x^{2}-27 a^{2} b \right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 \textit {\_C1} \,c^{2} x^{6}-12 \textit {\_C1}^{2} c \,x^{4}+8 \textit {\_C1}^{3} x^{2}-27 a^{2} b \right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}{3 b x}-\frac {\left (-c \,x^{2}+2 \textit {\_C1} \right ) x}{\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 \textit {\_C1} \,c^{2} x^{6}-12 \textit {\_C1}^{2} c \,x^{4}+8 \textit {\_C1}^{3} x^{2}-27 a^{2} b \right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 \textit {\_C1} \,c^{2} x^{6}-12 \textit {\_C1}^{2} c \,x^{4}+8 \textit {\_C1}^{3} x^{2}-27 a^{2} b \right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}{6 b x}-\frac {\left (-c \,x^{2}+2 \textit {\_C1} \right ) x}{2 \left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 \textit {\_C1} \,c^{2} x^{6}-12 \textit {\_C1}^{2} c \,x^{4}+8 \textit {\_C1}^{3} x^{2}-27 a^{2} b \right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 \textit {\_C1} \,c^{2} x^{6}-12 \textit {\_C1}^{2} c \,x^{4}+8 \textit {\_C1}^{3} x^{2}-27 a^{2} b \right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}{3 b x}-\frac {\left (-c \,x^{2}+2 \textit {\_C1} \right ) x}{\left (\left (27 a +3 \sqrt {-\frac {3 \left (-c^{3} x^{8}+6 \textit {\_C1} \,c^{2} x^{6}-12 \textit {\_C1}^{2} c \,x^{4}+8 \textit {\_C1}^{3} x^{2}-27 a^{2} b \right )}{b}}\right ) b^{2} x^{2}\right )^{\frac {1}{3}}}\right )}{2}\right ]$ Mathematica raw input

DSolve[y[x]*(a + c*x^3*y[x]) + x*(a + b*x*y[x]^3)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> (9*a*b^2*x^2 + Sqrt[3]*Sqrt[b^3*x^4*(27*a^2*b + x^2*(c*x^2 - 2*C[1])^3
)])^(1/3)/(3^(2/3)*b*x) + (x*(-(c*x^2) + 2*C[1]))/(3^(1/3)*(9*a*b^2*x^2 + Sqrt[3
]*Sqrt[b^3*x^4*(27*a^2*b + x^2*(c*x^2 - 2*C[1])^3)])^(1/3))}, {y[x] -> (I*3^(1/3
)*(I + Sqrt[3])*(9*a*b^2*x^2 + Sqrt[3]*Sqrt[b^3*x^4*(27*a^2*b + x^2*(c*x^2 - 2*C
[1])^3)])^(2/3) + 3^(1/6)*(3*I + Sqrt[3])*b*x^2*(c*x^2 - 2*C[1]))/(6*b*x*(9*a*b^
2*x^2 + Sqrt[3]*Sqrt[b^3*x^4*(27*a^2*b + x^2*(c*x^2 - 2*C[1])^3)])^(1/3))}, {y[x
] -> ((-I)*3^(1/3)*(-I + Sqrt[3])*(9*a*b^2*x^2 + Sqrt[3]*Sqrt[b^3*x^4*(27*a^2*b
+ x^2*(c*x^2 - 2*C[1])^3)])^(2/3) + 3^(1/6)*(-3*I + Sqrt[3])*b*x^2*(c*x^2 - 2*C[
1]))/(6*b*x*(9*a*b^2*x^2 + Sqrt[3]*Sqrt[b^3*x^4*(27*a^2*b + x^2*(c*x^2 - 2*C[1])
^3)])^(1/3))}}

Maple raw input

dsolve(x*(a+b*x*y(x)^3)*diff(y(x),x)+(a+c*x^3*y(x))*y(x) = 0, y(x))

Maple raw output

[y(x) = 1/3/b/x*((27*a+3*(-3*(-c^3*x^8+6*_C1*c^2*x^6-12*_C1^2*c*x^4+8*_C1^3*x^2-
27*a^2*b)/b)^(1/2))*b^2*x^2)^(1/3)+(-c*x^2+2*_C1)*x/((27*a+3*(-3*(-c^3*x^8+6*_C1
*c^2*x^6-12*_C1^2*c*x^4+8*_C1^3*x^2-27*a^2*b)/b)^(1/2))*b^2*x^2)^(1/3), y(x) = -
1/6/b/x*((27*a+3*(-3*(-c^3*x^8+6*_C1*c^2*x^6-12*_C1^2*c*x^4+8*_C1^3*x^2-27*a^2*b
)/b)^(1/2))*b^2*x^2)^(1/3)-1/2*(-c*x^2+2*_C1)*x/((27*a+3*(-3*(-c^3*x^8+6*_C1*c^2
*x^6-12*_C1^2*c*x^4+8*_C1^3*x^2-27*a^2*b)/b)^(1/2))*b^2*x^2)^(1/3)-1/2*I*3^(1/2)
*(1/3/b/x*((27*a+3*(-3*(-c^3*x^8+6*_C1*c^2*x^6-12*_C1^2*c*x^4+8*_C1^3*x^2-27*a^2
*b)/b)^(1/2))*b^2*x^2)^(1/3)-(-c*x^2+2*_C1)*x/((27*a+3*(-3*(-c^3*x^8+6*_C1*c^2*x
^6-12*_C1^2*c*x^4+8*_C1^3*x^2-27*a^2*b)/b)^(1/2))*b^2*x^2)^(1/3)), y(x) = -1/6/b
/x*((27*a+3*(-3*(-c^3*x^8+6*_C1*c^2*x^6-12*_C1^2*c*x^4+8*_C1^3*x^2-27*a^2*b)/b)^
(1/2))*b^2*x^2)^(1/3)-1/2*(-c*x^2+2*_C1)*x/((27*a+3*(-3*(-c^3*x^8+6*_C1*c^2*x^6-
12*_C1^2*c*x^4+8*_C1^3*x^2-27*a^2*b)/b)^(1/2))*b^2*x^2)^(1/3)+1/2*I*3^(1/2)*(1/3
/b/x*((27*a+3*(-3*(-c^3*x^8+6*_C1*c^2*x^6-12*_C1^2*c*x^4+8*_C1^3*x^2-27*a^2*b)/b
)^(1/2))*b^2*x^2)^(1/3)-(-c*x^2+2*_C1)*x/((27*a+3*(-3*(-c^3*x^8+6*_C1*c^2*x^6-12
*_C1^2*c*x^4+8*_C1^3*x^2-27*a^2*b)/b)^(1/2))*b^2*x^2)^(1/3))]