##### 4.2.35 $$y'(x)=y(x)^2 (a+b x y(x))$$

ODE
$y'(x)=y(x)^2 (a+b x y(x))$ ODE Classiﬁcation

[[_homogeneous, class G], _Abel]

Book solution method
Abel ODE, First kind

Mathematica
cpu = 0.417614 (sec), leaf count = 94

$\text {Solve}\left [\frac {a^2 \log \left (\frac {b x y(x) (a+b x y(x))+b}{b^2 x^2 y(x)^2}\right )+\frac {2 a^2 \tan ^{-1}\left (\frac {a+2 b x y(x)}{a \sqrt {\frac {4 b}{a^2}-1}}\right )}{\sqrt {\frac {4 b}{a^2}-1}}+2 a^2 \log (x)+2 b c_1}{b}=0,y(x)\right ]$

Maple
cpu = 0.344 (sec), leaf count = 103

$\left [y \left (x \right ) = \frac {{\mathrm e}^{\RootOf \left (2 \sqrt {a^{2}-4 b}\, a \arctanh \left (\frac {2 b \,{\mathrm e}^{\textit {\_Z}}+a}{\sqrt {a^{2}-4 b}}\right )-\ln \left (x^{2} \left (b \,{\mathrm e}^{2 \textit {\_Z}}+a \,{\mathrm e}^{\textit {\_Z}}+1\right )\right ) a^{2}+2 \textit {\_C1} \,a^{2}+2 \textit {\_Z} \,a^{2}+4 \ln \left (x^{2} \left (b \,{\mathrm e}^{2 \textit {\_Z}}+a \,{\mathrm e}^{\textit {\_Z}}+1\right )\right ) b -8 \textit {\_C1} b -8 \textit {\_Z} b \right )}}{x}\right ]$ Mathematica raw input

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

Mathematica raw output

Solve[((2*a^2*ArcTan[(a + 2*b*x*y[x])/(a*Sqrt[-1 + (4*b)/a^2])])/Sqrt[-1 + (4*b)
/a^2] + 2*b*C[1] + 2*a^2*Log[x] + a^2*Log[(b + b*x*y[x]*(a + b*x*y[x]))/(b^2*x^2
*y[x]^2)])/b == 0, y[x]]

Maple raw input

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

Maple raw output

[y(x) = exp(RootOf(2*(a^2-4*b)^(1/2)*a*arctanh((2*b*exp(_Z)+a)/(a^2-4*b)^(1/2))-
ln(x^2*(b*exp(2*_Z)+a*exp(_Z)+1))*a^2+2*_C1*a^2+2*_Z*a^2+4*ln(x^2*(b*exp(2*_Z)+a
*exp(_Z)+1))*b-8*_C1*b-8*_Z*b))/x]