4.19.50 $ax{y}^{\prime }(x)+by(x)+y(x)y^{\prime }(x{)}^2=0$

ODE
$$ax{y}^{\prime }(x)+by(x)+y(x)y^{\prime }(x{)}^2=0$$ ODE Classification

[[_homogeneous, `class A`], _rational, _dAlembert]

Book solution method
No Missing Variables ODE, Solve for $x$

Mathematica ✓
cpu = 0.467269 (sec), leaf count = 245

$$\{\text{Solve}[\frac{1}{8}(\frac{2a{\tanh }^{-1}\left(\frac{\sqrt{a^2-\frac{4by(x{)}^2}{x^2}}}{a}\right)-2(a+2b){\tanh }^{-1}\left(\frac{\sqrt{a^2-\frac{4by(x{)}^2}{x^2}}}{a+2b}\right)+a\log (a+b+\frac{y(x{)}^2}{x^2})+2b\log (a+b+\frac{y(x{)}^2}{x^2})+a\log \left(\frac{y(x{)}^2}{x^2}\right)}{a+b}+4\log (x))=c_1,y(x)],\text{Solve}[\frac{1}{8}(\frac{-2a{\tanh }^{-1}\left(\frac{\sqrt{a^2-\frac{4by(x{)}^2}{x^2}}}{a}\right)+2(a+2b){\tanh }^{-1}\left(\frac{\sqrt{a^2-\frac{4by(x{)}^2}{x^2}}}{a+2b}\right)+a\log (a+b+\frac{y(x{)}^2}{x^2})+2b\log (a+b+\frac{y(x{)}^2}{x^2})+a\log \left(\frac{y(x{)}^2}{x^2}\right)}{a+b}+4\log (x))=c_1,y(x)]\}$$

Maple ✓
cpu = 0.057 (sec), leaf count = 112

$$\{[x\left(\mathit{\_}\mathit{T}\right)=({\mathit{\_}\mathit{T}}^2+b)\mathit{\_}\mathit{C}\mathit{1}{({\mathit{\_}\mathit{T}}^2+a+b)}^{-\frac{a}{2a+2b}}{\left({\mathit{\_}\mathit{T}}^{\frac{a}{a+b}}\right)}^{-1}{\left({({\mathit{\_}\mathit{T}}^2+a+b)}^{\frac{b}{a+b}}\right)}^{-1},y\left(\mathit{\_}\mathit{T}\right)=-\mathit{\_}\mathit{C}\mathit{1}\mathit{\_}\mathit{T}a{({\mathit{\_}\mathit{T}}^2+a+b)}^{-\frac{a}{2a+2b}}{\left({\mathit{\_}\mathit{T}}^{\frac{a}{a+b}}\right)}^{-1}{\left({({\mathit{\_}\mathit{T}}^2+a+b)}^{\frac{b}{a+b}}\right)}^{-1}]\}$$ Mathematica raw input

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

Mathematica raw output

{Solve[(4*Log[x] + (2*a*ArcTanh[Sqrt[a^2 - (4*b*y[x]^2)/x^2]/a] - 2*(a + 2*b)*Ar
cTanh[Sqrt[a^2 - (4*b*y[x]^2)/x^2]/(a + 2*b)] + a*Log[y[x]^2/x^2] + a*Log[a + b 
+ y[x]^2/x^2] + 2*b*Log[a + b + y[x]^2/x^2])/(a + b))/8 == C[1], y[x]], Solve[(4
*Log[x] + (-2*a*ArcTanh[Sqrt[a^2 - (4*b*y[x]^2)/x^2]/a] + 2*(a + 2*b)*ArcTanh[Sq
rt[a^2 - (4*b*y[x]^2)/x^2]/(a + 2*b)] + a*Log[y[x]^2/x^2] + a*Log[a + b + y[x]^2
/x^2] + 2*b*Log[a + b + y[x]^2/x^2])/(a + b))/8 == C[1], y[x]]}

Maple raw input

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

Maple raw output

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