\[ \left (a^2-2 a x+y(x)^2\right ) y'(x)^2+2 a y(x) y'(x)+y(x)^2=0 \] ✓ Mathematica : cpu = 23.0669 (sec), leaf count = 403
DSolve[y[x]^2 + 2*a*y[x]*Derivative[1][y][x] + (a^2 - 2*a*x + y[x]^2)*Derivative[1][y][x]^2 == 0,y[x],x]
\[\left \{\text {Solve}\left [\left \{y(x)=\frac {-\sqrt {-a K[1]^2 \left (a K[1]^2-2 x K[1]^2-2 x\right )}-a K[1]}{K[1]^2+1},x=\frac {a K[1]^2 \text {arctanh}\left (\sqrt {K[1]^2+1}\right )^2+a \text {arctanh}\left (\sqrt {K[1]^2+1}\right )^2-2 a \sqrt {K[1]^2+1} \text {arctanh}\left (\sqrt {K[1]^2+1}\right )+2 a c_1 K[1]^2 \text {arctanh}\left (\sqrt {K[1]^2+1}\right )+2 a c_1 \text {arctanh}\left (\sqrt {K[1]^2+1}\right )+a K[1]^2+a c_1{}^2 K[1]^2-2 a c_1 \sqrt {K[1]^2+1}+a+a c_1{}^2}{2 \left (K[1]^2+1\right )}\right \},\{y(x),K[1]\}\right ],\text {Solve}\left [\left \{y(x)=\frac {\sqrt {-a K[2]^2 \left (a K[2]^2-2 x K[2]^2-2 x\right )}-a K[2]}{K[2]^2+1},x=\frac {a K[2]^2 \text {arctanh}\left (\sqrt {K[2]^2+1}\right )^2+a \text {arctanh}\left (\sqrt {K[2]^2+1}\right )^2-2 a \sqrt {K[2]^2+1} \text {arctanh}\left (\sqrt {K[2]^2+1}\right )+2 a c_1 K[2]^2 \text {arctanh}\left (\sqrt {K[2]^2+1}\right )+2 a c_1 \text {arctanh}\left (\sqrt {K[2]^2+1}\right )+a K[2]^2+a c_1{}^2 K[2]^2-2 a c_1 \sqrt {K[2]^2+1}+a+a c_1{}^2}{2 \left (K[2]^2+1\right )}\right \},\{y(x),K[2]\}\right ]\right \}\] ✓ Maple : cpu = 0.811 (sec), leaf count = 109
dsolve((y(x)^2-2*a*x+a^2)*diff(y(x),x)^2+2*a*y(x)*diff(y(x),x)+y(x)^2 = 0,y(x))
\[\left [x \left (\textit {\_T} \right ) = \frac {{\operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )}^{2} \sqrt {\textit {\_T}^{2}+1}\, a^{2}+\left (-2 a c_{1} \sqrt {\textit {\_T}^{2}+1}-2 a^{2}\right ) \operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )+\left (a^{2}+c_{1}^{2}\right ) \sqrt {\textit {\_T}^{2}+1}+2 c_{1} a}{2 \sqrt {\textit {\_T}^{2}+1}\, a}, y \left (\textit {\_T} \right ) = \frac {\left (-a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )+c_{1}\right ) \textit {\_T}}{\sqrt {\textit {\_T}^{2}+1}}\right ]\]