\[ a y(x) y''(x)+b y'(x)^2-\frac {y(x) y'(x)}{\sqrt {c^2+x^2}}=0 \] ✓ Mathematica : cpu = 0.637251 (sec), leaf count = 211
DSolve[-((y[x]*Derivative[1][y][x])/Sqrt[c^2 + x^2]) + b*Derivative[1][y][x]^2 + a*y[x]*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_2 \exp \left (\int _1^x-\frac {\left (\frac {K[2]}{\sqrt {c^2+K[2]^2}}+1\right )^{\left .\frac {1}{2}\right /a}}{\left (1-\frac {K[2]}{\sqrt {c^2+K[2]^2}}\right )^{\left .\frac {1}{2}\right /a} \int _1^{K[2]}\frac {\exp \left (\frac {\frac {1}{2} \log \left (\frac {K[1]}{\sqrt {c^2+K[1]^2}}+1\right )-\frac {1}{2} \log \left (1-\frac {K[1]}{\sqrt {c^2+K[1]^2}}\right )}{a}\right ) \left (-\sqrt {c^2+K[1]^2} a-b \sqrt {c^2+K[1]^2}\right )}{a \sqrt {c^2+K[1]^2}}dK[1]-c_1 \left (1-\frac {K[2]}{\sqrt {c^2+K[2]^2}}\right )^{\left .\frac {1}{2}\right /a}}dK[2]\right )\right \}\right \}\] ✓ Maple : cpu = 1.903 (sec), leaf count = 75
dsolve(a*y(x)*diff(diff(y(x),x),x)+b*diff(y(x),x)^2-y(x)*diff(y(x),x)/(c^2+x^2)^(1/2)=0,y(x))
\[y \left (x \right ) = \left (\frac {a}{\left (a +b \right ) \left (\frac {c_{1} 2^{\frac {1}{a}} a \,x^{\frac {1}{a}+1} \hypergeom \left (\left [-\frac {1}{2 a}, -\frac {1}{2 a}-\frac {1}{2}\right ], \left [1-\frac {1}{a}\right ], -\frac {c^{2}}{x^{2}}\right )}{a +1}+c_{2}\right )}\right )^{-\frac {a}{a +b}}\]