##### 4.17.16 $$y'(x)^2+(2 y(x)+1) y'(x)+(y(x)-1) y(x)=0$$

ODE
$y'(x)^2+(2 y(x)+1) y'(x)+(y(x)-1) y(x)=0$ ODE Classiﬁcation

[_quadrature]

Book solution method
Missing Variables ODE, Independent variable missing, Use new variable

Mathematica
cpu = 0.304349 (sec), leaf count = 1373

$\left \{\left \{y(x)\to \frac {e^{-2 x} \left (128 e^x \left (12 e^x+e^{2 c_1}\right )+64 \sqrt [3]{24 \sqrt {3} \sqrt {-e^{7 x+4 c_1} \left (-27 e^x+e^{2 c_1}\right ){}^3}+540 e^{4 (x+c_1)}+5832 e^{5 x+2 c_1}-e^{3 x+6 c_1}}+\frac {64 e^{2 (x+c_1)} \left (216 e^x+e^{2 c_1}\right )}{\sqrt [3]{24 \sqrt {3} \sqrt {-e^{7 x+4 c_1} \left (-27 e^x+e^{2 c_1}\right ){}^3}+540 e^{4 (x+c_1)}+5832 e^{5 x+2 c_1}-e^{3 x+6 c_1}}}\right )}{1536}\right \},\left \{y(x)\to \frac {e^{-2 x} \left (256 e^x \left (12 e^x+e^{2 c_1}\right )+64 i \left (i+\sqrt {3}\right ) \sqrt [3]{24 \sqrt {3} \sqrt {-e^{7 x+4 c_1} \left (-27 e^x+e^{2 c_1}\right ){}^3}+540 e^{4 (x+c_1)}+5832 e^{5 x+2 c_1}-e^{3 x+6 c_1}}-\frac {64 i \left (-i+\sqrt {3}\right ) e^{2 (x+c_1)} \left (216 e^x+e^{2 c_1}\right )}{\sqrt [3]{24 \sqrt {3} \sqrt {-e^{7 x+4 c_1} \left (-27 e^x+e^{2 c_1}\right ){}^3}+540 e^{4 (x+c_1)}+5832 e^{5 x+2 c_1}-e^{3 x+6 c_1}}}\right )}{3072}\right \},\left \{y(x)\to \frac {e^{-2 x} \left (256 e^x \left (12 e^x+e^{2 c_1}\right )-64 \left (1+i \sqrt {3}\right ) \sqrt [3]{24 \sqrt {3} \sqrt {-e^{7 x+4 c_1} \left (-27 e^x+e^{2 c_1}\right ){}^3}+540 e^{4 (x+c_1)}+5832 e^{5 x+2 c_1}-e^{3 x+6 c_1}}+\frac {64 i \left (i+\sqrt {3}\right ) e^{2 (x+c_1)} \left (216 e^x+e^{2 c_1}\right )}{\sqrt [3]{24 \sqrt {3} \sqrt {-e^{7 x+4 c_1} \left (-27 e^x+e^{2 c_1}\right ){}^3}+540 e^{4 (x+c_1)}+5832 e^{5 x+2 c_1}-e^{3 x+6 c_1}}}\right )}{3072}\right \},\left \{y(x)\to \frac {e^{-2 (x+2 c_1)} \left (128 e^{x+2 c_1} \left (1+12 e^{x+2 c_1}\right )+64 \sqrt [3]{24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}+5832 e^{5 (x+2 c_1)}-e^{3 x+6 c_1}+540 e^{4 x+8 c_1}}+\frac {64 e^{2 x+4 c_1} \left (1+216 e^{x+2 c_1}\right )}{\sqrt [3]{24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}+5832 e^{5 (x+2 c_1)}-e^{3 x+6 c_1}+540 e^{4 x+8 c_1}}}\right )}{1536}\right \},\left \{y(x)\to \frac {e^{-2 (x+2 c_1)} \left (256 e^{x+2 c_1} \left (1+12 e^{x+2 c_1}\right )+64 i \left (i+\sqrt {3}\right ) \sqrt [3]{24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}+5832 e^{5 (x+2 c_1)}-e^{3 x+6 c_1}+540 e^{4 x+8 c_1}}-\frac {64 i \left (-i+\sqrt {3}\right ) e^{2 x+4 c_1} \left (1+216 e^{x+2 c_1}\right )}{\sqrt [3]{24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}+5832 e^{5 (x+2 c_1)}-e^{3 x+6 c_1}+540 e^{4 x+8 c_1}}}\right )}{3072}\right \},\left \{y(x)\to \frac {e^{-2 (x+2 c_1)} \left (256 e^{x+2 c_1} \left (1+12 e^{x+2 c_1}\right )-64 \left (1+i \sqrt {3}\right ) \sqrt [3]{24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}+5832 e^{5 (x+2 c_1)}-e^{3 x+6 c_1}+540 e^{4 x+8 c_1}}+\frac {64 i \left (i+\sqrt {3}\right ) e^{2 x+4 c_1} \left (1+216 e^{x+2 c_1}\right )}{\sqrt [3]{24 \sqrt {3} \sqrt {e^{7 (x+2 c_1)} \left (-1+27 e^{x+2 c_1}\right ){}^3}+5832 e^{5 (x+2 c_1)}-e^{3 x+6 c_1}+540 e^{4 x+8 c_1}}}\right )}{3072}\right \}\right \}$

Maple
cpu = 7.802 (sec), leaf count = 143

$\left [x +\frac {3 \ln \left (y \left (x \right )-1\right )}{2}-\frac {\ln \left (y \left (x \right )\right )}{2}+\frac {3 \ln \left (\sqrt {8 y \left (x \right )+1}+3\right )}{2}-\frac {\ln \left (\sqrt {8 y \left (x \right )+1}+1\right )}{2}+\frac {\ln \left (\sqrt {8 y \left (x \right )+1}-1\right )}{2}-\frac {3 \ln \left (\sqrt {8 y \left (x \right )+1}-3\right )}{2}-\textit {\_C1} = 0, x +\frac {3 \ln \left (y \left (x \right )-1\right )}{2}-\frac {\ln \left (y \left (x \right )\right )}{2}-\frac {3 \ln \left (\sqrt {8 y \left (x \right )+1}+3\right )}{2}+\frac {\ln \left (\sqrt {8 y \left (x \right )+1}+1\right )}{2}-\frac {\ln \left (\sqrt {8 y \left (x \right )+1}-1\right )}{2}+\frac {3 \ln \left (\sqrt {8 y \left (x \right )+1}-3\right )}{2}-\textit {\_C1} = 0\right ]$ Mathematica raw input

DSolve[(-1 + y[x])*y[x] + (1 + 2*y[x])*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> (128*E^x*(12*E^x + E^(2*C[1])) + (64*E^(2*(x + C[1]))*(216*E^x + E^(2*
C[1])))/(540*E^(4*(x + C[1])) + 5832*E^(5*x + 2*C[1]) - E^(3*x + 6*C[1]) + 24*Sq
rt[3]*Sqrt[-(E^(7*x + 4*C[1])*(-27*E^x + E^(2*C[1]))^3)])^(1/3) + 64*(540*E^(4*(
x + C[1])) + 5832*E^(5*x + 2*C[1]) - E^(3*x + 6*C[1]) + 24*Sqrt[3]*Sqrt[-(E^(7*x
 + 4*C[1])*(-27*E^x + E^(2*C[1]))^3)])^(1/3))/(1536*E^(2*x))}, {y[x] -> (256*E^x
*(12*E^x + E^(2*C[1])) - ((64*I)*(-I + Sqrt[3])*E^(2*(x + C[1]))*(216*E^x + E^(2
*C[1])))/(540*E^(4*(x + C[1])) + 5832*E^(5*x + 2*C[1]) - E^(3*x + 6*C[1]) + 24*S
qrt[3]*Sqrt[-(E^(7*x + 4*C[1])*(-27*E^x + E^(2*C[1]))^3)])^(1/3) + (64*I)*(I + S
qrt[3])*(540*E^(4*(x + C[1])) + 5832*E^(5*x + 2*C[1]) - E^(3*x + 6*C[1]) + 24*Sq
rt[3]*Sqrt[-(E^(7*x + 4*C[1])*(-27*E^x + E^(2*C[1]))^3)])^(1/3))/(3072*E^(2*x))}
, {y[x] -> (256*E^x*(12*E^x + E^(2*C[1])) + ((64*I)*(I + Sqrt[3])*E^(2*(x + C[1]
))*(216*E^x + E^(2*C[1])))/(540*E^(4*(x + C[1])) + 5832*E^(5*x + 2*C[1]) - E^(3*
x + 6*C[1]) + 24*Sqrt[3]*Sqrt[-(E^(7*x + 4*C[1])*(-27*E^x + E^(2*C[1]))^3)])^(1/
3) - 64*(1 + I*Sqrt[3])*(540*E^(4*(x + C[1])) + 5832*E^(5*x + 2*C[1]) - E^(3*x +
 6*C[1]) + 24*Sqrt[3]*Sqrt[-(E^(7*x + 4*C[1])*(-27*E^x + E^(2*C[1]))^3)])^(1/3))
/(3072*E^(2*x))}, {y[x] -> (128*E^(x + 2*C[1])*(1 + 12*E^(x + 2*C[1])) + (64*E^(
2*x + 4*C[1])*(1 + 216*E^(x + 2*C[1])))/(5832*E^(5*(x + 2*C[1])) - E^(3*x + 6*C[
1]) + 540*E^(4*x + 8*C[1]) + 24*Sqrt[3]*Sqrt[E^(7*(x + 2*C[1]))*(-1 + 27*E^(x +
2*C[1]))^3])^(1/3) + 64*(5832*E^(5*(x + 2*C[1])) - E^(3*x + 6*C[1]) + 540*E^(4*x
 + 8*C[1]) + 24*Sqrt[3]*Sqrt[E^(7*(x + 2*C[1]))*(-1 + 27*E^(x + 2*C[1]))^3])^(1/
3))/(1536*E^(2*(x + 2*C[1])))}, {y[x] -> (256*E^(x + 2*C[1])*(1 + 12*E^(x + 2*C[
1])) - ((64*I)*(-I + Sqrt[3])*E^(2*x + 4*C[1])*(1 + 216*E^(x + 2*C[1])))/(5832*E
^(5*(x + 2*C[1])) - E^(3*x + 6*C[1]) + 540*E^(4*x + 8*C[1]) + 24*Sqrt[3]*Sqrt[E^
(7*(x + 2*C[1]))*(-1 + 27*E^(x + 2*C[1]))^3])^(1/3) + (64*I)*(I + Sqrt[3])*(5832
*E^(5*(x + 2*C[1])) - E^(3*x + 6*C[1]) + 540*E^(4*x + 8*C[1]) + 24*Sqrt[3]*Sqrt[
E^(7*(x + 2*C[1]))*(-1 + 27*E^(x + 2*C[1]))^3])^(1/3))/(3072*E^(2*(x + 2*C[1])))
}, {y[x] -> (256*E^(x + 2*C[1])*(1 + 12*E^(x + 2*C[1])) + ((64*I)*(I + Sqrt[3])*
E^(2*x + 4*C[1])*(1 + 216*E^(x + 2*C[1])))/(5832*E^(5*(x + 2*C[1])) - E^(3*x + 6
*C[1]) + 540*E^(4*x + 8*C[1]) + 24*Sqrt[3]*Sqrt[E^(7*(x + 2*C[1]))*(-1 + 27*E^(x
 + 2*C[1]))^3])^(1/3) - 64*(1 + I*Sqrt[3])*(5832*E^(5*(x + 2*C[1])) - E^(3*x + 6
*C[1]) + 540*E^(4*x + 8*C[1]) + 24*Sqrt[3]*Sqrt[E^(7*(x + 2*C[1]))*(-1 + 27*E^(x
 + 2*C[1]))^3])^(1/3))/(3072*E^(2*(x + 2*C[1])))}}

Maple raw input

dsolve(diff(y(x),x)^2+(1+2*y(x))*diff(y(x),x)+y(x)*(y(x)-1) = 0, y(x))

Maple raw output

[x+3/2*ln(y(x)-1)-1/2*ln(y(x))+3/2*ln((8*y(x)+1)^(1/2)+3)-1/2*ln((8*y(x)+1)^(1/2
)+1)+1/2*ln((8*y(x)+1)^(1/2)-1)-3/2*ln((8*y(x)+1)^(1/2)-3)-_C1 = 0, x+3/2*ln(y(x
)-1)-1/2*ln(y(x))-3/2*ln((8*y(x)+1)^(1/2)+3)+1/2*ln((8*y(x)+1)^(1/2)+1)-1/2*ln((
8*y(x)+1)^(1/2)-1)+3/2*ln((8*y(x)+1)^(1/2)-3)-_C1 = 0]