##### 4.4.43 $$x y'(x)=a y(x)+b \left (x^2+1\right ) y(x)^3$$

ODE
$x y'(x)=a y(x)+b \left (x^2+1\right ) y(x)^3$ ODE Classiﬁcation

[_rational, _Bernoulli]

Book solution method
The Bernoulli ODE

Mathematica
cpu = 0.303954 (sec), leaf count = 103

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

Maple
cpu = 0.049 (sec), leaf count = 184

$\left [y \left (x \right ) = \frac {\sqrt {-\left (a b \,x^{2+2 a}+a b \,x^{2 a}-\textit {\_C1} \,a^{2}+b \,x^{2 a}-\textit {\_C1} a \right ) a \,x^{2 a} \left (1+a \right )}}{a b \,x^{2+2 a}+a b \,x^{2 a}-\textit {\_C1} \,a^{2}+b \,x^{2 a}-\textit {\_C1} a}, y \left (x \right ) = -\frac {\sqrt {-\left (a b \,x^{2+2 a}+a b \,x^{2 a}-\textit {\_C1} \,a^{2}+b \,x^{2 a}-\textit {\_C1} a \right ) a \,x^{2 a} \left (1+a \right )}}{a b \,x^{2+2 a}+a b \,x^{2 a}-\textit {\_C1} \,a^{2}+b \,x^{2 a}-\textit {\_C1} a}\right ]$ Mathematica raw input

DSolve[x*y'[x] == a*y[x] + b*(1 + x^2)*y[x]^3,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(x*diff(y(x),x) = a*y(x)+b*(x^2+1)*y(x)^3, y(x))

Maple raw output

[y(x) = (-(a*b*x^(2+2*a)+a*b*x^(2*a)-_C1*a^2+b*x^(2*a)-_C1*a)*a*x^(2*a)*(1+a))^(
1/2)/(a*b*x^(2+2*a)+a*b*x^(2*a)-_C1*a^2+b*x^(2*a)-_C1*a), y(x) = -(-(a*b*x^(2+2*
a)+a*b*x^(2*a)-_C1*a^2+b*x^(2*a)-_C1*a)*a*x^(2*a)*(1+a))^(1/2)/(a*b*x^(2+2*a)+a*
b*x^(2*a)-_C1*a^2+b*x^(2*a)-_C1*a)]