\[ a y(x) y'(x)+f(x)+x y(x) y''(x)+x y'(x)^2=0 \] ✓ Mathematica : cpu = 321.185 (sec), leaf count = 118
\[\left \{\left \{y(x)\to -\sqrt {2} \sqrt {\int _1^x \left (c_1 K[3]^{-a}+K[3]^{-a} \left (\int _1^{K[3]} K[2]^{a-1} (-f(K[2])) \, dK[2]\right )\right ) \, dK[3]+c_2}\right \},\left \{y(x)\to \sqrt {2} \sqrt {\int _1^x \left (c_1 K[3]^{-a}+K[3]^{-a} \left (\int _1^{K[3]} K[2]^{a-1} (-f(K[2])) \, dK[2]\right )\right ) \, dK[3]+c_2}\right \}\right \}\] ✓ Maple : cpu = 0.096 (sec), leaf count = 114
\[ \left \{ y \left ( x \right ) ={\frac {\sqrt {2}}{a-1}\sqrt { \left ( a-1 \right ) \left ( {x}^{1-a}\int \!{\frac {{x}^{a}f \left ( x \right ) }{x}}\,{\rm d}x+{x}^{1-a}{\it \_C1}-\int \!f \left ( x \right ) \,{\rm d}x-{\it \_C2} \right ) }},y \left ( x \right ) =-{\frac {\sqrt {2}}{a-1}\sqrt { \left ( a-1 \right ) \left ( {x}^{1-a}\int \!{\frac {{x}^{a}f \left ( x \right ) }{x}}\,{\rm d}x+{x}^{1-a}{\it \_C1}-\int \!f \left ( x \right ) \,{\rm d}x-{\it \_C2} \right ) }} \right \} \]