\[ -h(x) (y(x)-f(x)) (y(x)-g(x)) \left (y(x)-\frac {a f(x)+b g(x)}{a+b}\right )-\frac {f'(x) (y(x)-g(x))}{f(x)-g(x)}-\frac {(y(x)-f(x)) g'(x)}{g(x)-f(x)}+y'(x)=0 \] ✓ Mathematica : cpu = 0.864297 (sec), leaf count = 322
\[\text {Solve}\left [-\frac {1}{3} (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3} \text {RootSum}\left [\text {$\#$1}^3 (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}-3 \text {$\#$1} a^2-3 \text {$\#$1} a b-3 \text {$\#$1} b^2+(a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}\& ,\frac {\log \left (-\text {$\#$1}-\frac {h(x) (2 a f(x)+a g(x)-3 a y(x)+b f(x)+2 b g(x)-3 b y(x))}{(a+b) \sqrt [3]{\frac {\left (2 a^3+3 a^2 b-3 a b^2-2 b^3\right ) h(x)^3 (f(x)-g(x))^3}{(a+b)^3}}}\right )}{-\text {$\#$1}^2 (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}+a^2+a b+b^2}\& \right ]=\int _1^x \frac {\left (\frac {\left (2 a^3+3 a^2 b-3 a b^2-2 b^3\right ) h(K[1])^3 (f(K[1])-g(K[1]))^3}{(a+b)^3}\right )^{2/3}}{9 h(K[1])} \, dK[1]+c_1,y(x)\right ]\]
✓ Maple : cpu = 0.235 (sec), leaf count = 237
\[ \left \{ y \left ( x \right ) ={\frac {1}{9\,{a}^{3}+18\,{a}^{2}b+18\,a{b}^{2}+9\,{b}^{3}} \left ( 2\, \left ( a+2\,b \right ) \left ( a-b \right ) \left ( f \left ( x \right ) -g \left ( x \right ) \right ) \left ( b/2+a \right ) {\it RootOf} \left ( -27\,\int ^{{\it \_Z}}\!{\frac { \left ( {a}^{2}+ab+{b}^{2} \right ) ^{3}}{ \left ( 2\,{\it \_a}\,{a}^{2}-{\it \_a}\,ab-{\it \_a}\,{b}^{2}-3\,{a}^{2}-3\,ab-3\,{b}^{2} \right ) \left ( 2\,{\it \_a}\,{a}^{2}+5\,{\it \_a}\,ab+2\,{\it \_a}\,{b}^{2}-3\,{a}^{2}-3\,ab-3\,{b}^{2} \right ) \left ( {\it \_a}\,{a}^{2}+{\it \_a}\,ab-2\,{\it \_a}\,{b}^{2}+3\,{a}^{2}+3\,ab+3\,{b}^{2} \right ) }}{d{\it \_a}}+\int \!1/3\,{\frac { \left ( f \left ( x \right ) -g \left ( x \right ) \right ) ^{2} \left ( {a}^{2}+ab+{b}^{2} \right ) h \left ( x \right ) }{ \left ( a+b \right ) ^{2}}}\,{\rm d}x+{\it \_C1} \right ) +6\, \left ( {a}^{2}+ab+{b}^{2} \right ) \left ( \left ( b/2+a \right ) f \left ( x \right ) +1/2\,g \left ( x \right ) \left ( a+2\,b \right ) \right ) \right ) } \right \} \]