\[ -h(x) (y(x)-f(x)) (y(x)-g(x)) \left (y(x)-\frac {a f(x)+b g(x)}{a+b}\right )-\frac {f'(x) (y(x)-g(x))}{f(x)-g(x)}-\frac {(y(x)-f(x)) g'(x)}{g(x)-f(x)}+y'(x)=0 \] ✓ Mathematica : cpu = 0.712988 (sec), leaf count = 354
\[\text {Solve}\left [-\frac {1}{3} (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3} \text {RootSum}\left [\text {$\#$1}^3 (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}-3 \text {$\#$1} a^2-3 \text {$\#$1} a b-3 \text {$\#$1} b^2+(a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}\& ,\frac {\log \left (\frac {\frac {-2 a f(x) h(x)-a g(x) h(x)-b f(x) h(x)-2 b g(x) h(x)}{a+b}+3 h(x) y(x)}{\sqrt [3]{\frac {(f(x)-g(x))^3 \left (2 a^3 h(x)^3+3 a^2 b h(x)^3-3 a b^2 h(x)^3-2 b^3 h(x)^3\right )}{(a+b)^3}}}-\text {$\#$1}\right )}{-\text {$\#$1}^2 (a-b)^{2/3} (2 a+b)^{2/3} (a+2 b)^{2/3}+a^2+a b+b^2}\& \right ]=\int _1^x \frac {\left (\frac {(f(K[1])-g(K[1]))^3 \left (2 a^3 h(K[1])^3+3 a^2 b h(K[1])^3-3 a b^2 h(K[1])^3-2 b^3 h(K[1])^3\right )}{(a+b)^3}\right )^{2/3}}{9 h(K[1])} \, dK[1]+c_1,y(x)\right ]\]
✓ Maple : cpu = 0.224 (sec), leaf count = 2348
\[ \left \{ y \left ( x \right ) =-{\frac {1}{9\,{a}^{3}+18\,{a}^{2}b+18\,a{b}^{2}+9\,{b}^{3}} \left ( 2\,g \left ( x \right ) {\it RootOf} \left ( -27\,\int ^{{\it \_Z}}\!{\frac { \left ( {a}^{2}+ab+{b}^{2} \right ) ^{3}}{4\,{{\it \_a}}^{3}{a}^{6}+12\,{{\it \_a}}^{3}{a}^{5}b-3\,{{\it \_a}}^{3}{a}^{4}{b}^{2}-26\,{{\it \_a}}^{3}{a}^{3}{b}^{3}-3\,{{\it \_a}}^{3}{a}^{2}{b}^{4}+12\,{{\it \_a}}^{3}a{b}^{5}+4\,{{\it \_a}}^{3}{b}^{6}-27\,{\it \_a}\,{a}^{6}-81\,{\it \_a}\,{a}^{5}b-162\,{\it \_a}\,{a}^{4}{b}^{2}-189\,{\it \_a}\,{a}^{3}{b}^{3}-162\,{\it \_a}\,{a}^{2}{b}^{4}-81\,{\it \_a}\,a{b}^{5}-27\,{\it \_a}\,{b}^{6}+27\,{a}^{6}+81\,{a}^{5}b+162\,{a}^{4}{b}^{2}+189\,{a}^{3}{b}^{3}+162\,{a}^{2}{b}^{4}+81\,a{b}^{5}+27\,{b}^{6}}}{d{\it \_a}}+\int \!1/3\,{\frac { \left ( \left ( g \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( g \left ( x \right ) \right ) ^{2}ab+ \left ( g \left ( x \right ) \right ) ^{2}{b}^{2}-2\,g \left ( x \right ) f \left ( x \right ) {a}^{2}-2\,g \left ( x \right ) f \left ( x \right ) ab-2\,g \left ( x \right ) f \left ( x \right ) {b}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}ab+{b}^{2} \left ( f \left ( x \right ) \right ) ^{2} \right ) h \left ( x \right ) }{ \left ( a+b \right ) ^{2}}}\,{\rm d}x+{\it \_C1} \right ) {a}^{3}+3\,g \left ( x \right ) {\it RootOf} \left ( -27\,\int ^{{\it \_Z}}\!{\frac { \left ( {a}^{2}+ab+{b}^{2} \right ) ^{3}}{4\,{{\it \_a}}^{3}{a}^{6}+12\,{{\it \_a}}^{3}{a}^{5}b-3\,{{\it \_a}}^{3}{a}^{4}{b}^{2}-26\,{{\it \_a}}^{3}{a}^{3}{b}^{3}-3\,{{\it \_a}}^{3}{a}^{2}{b}^{4}+12\,{{\it \_a}}^{3}a{b}^{5}+4\,{{\it \_a}}^{3}{b}^{6}-27\,{\it \_a}\,{a}^{6}-81\,{\it \_a}\,{a}^{5}b-162\,{\it \_a}\,{a}^{4}{b}^{2}-189\,{\it \_a}\,{a}^{3}{b}^{3}-162\,{\it \_a}\,{a}^{2}{b}^{4}-81\,{\it \_a}\,a{b}^{5}-27\,{\it \_a}\,{b}^{6}+27\,{a}^{6}+81\,{a}^{5}b+162\,{a}^{4}{b}^{2}+189\,{a}^{3}{b}^{3}+162\,{a}^{2}{b}^{4}+81\,a{b}^{5}+27\,{b}^{6}}}{d{\it \_a}}+\int \!1/3\,{\frac { \left ( \left ( g \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( g \left ( x \right ) \right ) ^{2}ab+ \left ( g \left ( x \right ) \right ) ^{2}{b}^{2}-2\,g \left ( x \right ) f \left ( x \right ) {a}^{2}-2\,g \left ( x \right ) f \left ( x \right ) ab-2\,g \left ( x \right ) f \left ( x \right ) {b}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}ab+{b}^{2} \left ( f \left ( x \right ) \right ) ^{2} \right ) h \left ( x \right ) }{ \left ( a+b \right ) ^{2}}}\,{\rm d}x+{\it \_C1} \right ) {a}^{2}b-3\,g \left ( x \right ) {\it RootOf} \left ( -27\,\int ^{{\it \_Z}}\!{\frac { \left ( {a}^{2}+ab+{b}^{2} \right ) ^{3}}{4\,{{\it \_a}}^{3}{a}^{6}+12\,{{\it \_a}}^{3}{a}^{5}b-3\,{{\it \_a}}^{3}{a}^{4}{b}^{2}-26\,{{\it \_a}}^{3}{a}^{3}{b}^{3}-3\,{{\it \_a}}^{3}{a}^{2}{b}^{4}+12\,{{\it \_a}}^{3}a{b}^{5}+4\,{{\it \_a}}^{3}{b}^{6}-27\,{\it \_a}\,{a}^{6}-81\,{\it \_a}\,{a}^{5}b-162\,{\it \_a}\,{a}^{4}{b}^{2}-189\,{\it \_a}\,{a}^{3}{b}^{3}-162\,{\it \_a}\,{a}^{2}{b}^{4}-81\,{\it \_a}\,a{b}^{5}-27\,{\it \_a}\,{b}^{6}+27\,{a}^{6}+81\,{a}^{5}b+162\,{a}^{4}{b}^{2}+189\,{a}^{3}{b}^{3}+162\,{a}^{2}{b}^{4}+81\,a{b}^{5}+27\,{b}^{6}}}{d{\it \_a}}+\int \!1/3\,{\frac { \left ( \left ( g \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( g \left ( x \right ) \right ) ^{2}ab+ \left ( g \left ( x \right ) \right ) ^{2}{b}^{2}-2\,g \left ( x \right ) f \left ( x \right ) {a}^{2}-2\,g \left ( x \right ) f \left ( x \right ) ab-2\,g \left ( x \right ) f \left ( x \right ) {b}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}ab+{b}^{2} \left ( f \left ( x \right ) \right ) ^{2} \right ) h \left ( x \right ) }{ \left ( a+b \right ) ^{2}}}\,{\rm d}x+{\it \_C1} \right ) a{b}^{2}-2\,g \left ( x \right ) {\it RootOf} \left ( -27\,\int ^{{\it \_Z}}\!{\frac { \left ( {a}^{2}+ab+{b}^{2} \right ) ^{3}}{4\,{{\it \_a}}^{3}{a}^{6}+12\,{{\it \_a}}^{3}{a}^{5}b-3\,{{\it \_a}}^{3}{a}^{4}{b}^{2}-26\,{{\it \_a}}^{3}{a}^{3}{b}^{3}-3\,{{\it \_a}}^{3}{a}^{2}{b}^{4}+12\,{{\it \_a}}^{3}a{b}^{5}+4\,{{\it \_a}}^{3}{b}^{6}-27\,{\it \_a}\,{a}^{6}-81\,{\it \_a}\,{a}^{5}b-162\,{\it \_a}\,{a}^{4}{b}^{2}-189\,{\it \_a}\,{a}^{3}{b}^{3}-162\,{\it \_a}\,{a}^{2}{b}^{4}-81\,{\it \_a}\,a{b}^{5}-27\,{\it \_a}\,{b}^{6}+27\,{a}^{6}+81\,{a}^{5}b+162\,{a}^{4}{b}^{2}+189\,{a}^{3}{b}^{3}+162\,{a}^{2}{b}^{4}+81\,a{b}^{5}+27\,{b}^{6}}}{d{\it \_a}}+\int \!1/3\,{\frac { \left ( \left ( g \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( g \left ( x \right ) \right ) ^{2}ab+ \left ( g \left ( x \right ) \right ) ^{2}{b}^{2}-2\,g \left ( x \right ) f \left ( x \right ) {a}^{2}-2\,g \left ( x \right ) f \left ( x \right ) ab-2\,g \left ( x \right ) f \left ( x \right ) {b}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}ab+{b}^{2} \left ( f \left ( x \right ) \right ) ^{2} \right ) h \left ( x \right ) }{ \left ( a+b \right ) ^{2}}}\,{\rm d}x+{\it \_C1} \right ) {b}^{3}-2\,f \left ( x \right ) {\it RootOf} \left ( -27\,\int ^{{\it \_Z}}\!{\frac { \left ( {a}^{2}+ab+{b}^{2} \right ) ^{3}}{4\,{{\it \_a}}^{3}{a}^{6}+12\,{{\it \_a}}^{3}{a}^{5}b-3\,{{\it \_a}}^{3}{a}^{4}{b}^{2}-26\,{{\it \_a}}^{3}{a}^{3}{b}^{3}-3\,{{\it \_a}}^{3}{a}^{2}{b}^{4}+12\,{{\it \_a}}^{3}a{b}^{5}+4\,{{\it \_a}}^{3}{b}^{6}-27\,{\it \_a}\,{a}^{6}-81\,{\it \_a}\,{a}^{5}b-162\,{\it \_a}\,{a}^{4}{b}^{2}-189\,{\it \_a}\,{a}^{3}{b}^{3}-162\,{\it \_a}\,{a}^{2}{b}^{4}-81\,{\it \_a}\,a{b}^{5}-27\,{\it \_a}\,{b}^{6}+27\,{a}^{6}+81\,{a}^{5}b+162\,{a}^{4}{b}^{2}+189\,{a}^{3}{b}^{3}+162\,{a}^{2}{b}^{4}+81\,a{b}^{5}+27\,{b}^{6}}}{d{\it \_a}}+\int \!1/3\,{\frac { \left ( \left ( g \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( g \left ( x \right ) \right ) ^{2}ab+ \left ( g \left ( x \right ) \right ) ^{2}{b}^{2}-2\,g \left ( x \right ) f \left ( x \right ) {a}^{2}-2\,g \left ( x \right ) f \left ( x \right ) ab-2\,g \left ( x \right ) f \left ( x \right ) {b}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}ab+{b}^{2} \left ( f \left ( x \right ) \right ) ^{2} \right ) h \left ( x \right ) }{ \left ( a+b \right ) ^{2}}}\,{\rm d}x+{\it \_C1} \right ) {a}^{3}-3\,f \left ( x \right ) {\it RootOf} \left ( -27\,\int ^{{\it \_Z}}\!{\frac { \left ( {a}^{2}+ab+{b}^{2} \right ) ^{3}}{4\,{{\it \_a}}^{3}{a}^{6}+12\,{{\it \_a}}^{3}{a}^{5}b-3\,{{\it \_a}}^{3}{a}^{4}{b}^{2}-26\,{{\it \_a}}^{3}{a}^{3}{b}^{3}-3\,{{\it \_a}}^{3}{a}^{2}{b}^{4}+12\,{{\it \_a}}^{3}a{b}^{5}+4\,{{\it \_a}}^{3}{b}^{6}-27\,{\it \_a}\,{a}^{6}-81\,{\it \_a}\,{a}^{5}b-162\,{\it \_a}\,{a}^{4}{b}^{2}-189\,{\it \_a}\,{a}^{3}{b}^{3}-162\,{\it \_a}\,{a}^{2}{b}^{4}-81\,{\it \_a}\,a{b}^{5}-27\,{\it \_a}\,{b}^{6}+27\,{a}^{6}+81\,{a}^{5}b+162\,{a}^{4}{b}^{2}+189\,{a}^{3}{b}^{3}+162\,{a}^{2}{b}^{4}+81\,a{b}^{5}+27\,{b}^{6}}}{d{\it \_a}}+\int \!1/3\,{\frac { \left ( \left ( g \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( g \left ( x \right ) \right ) ^{2}ab+ \left ( g \left ( x \right ) \right ) ^{2}{b}^{2}-2\,g \left ( x \right ) f \left ( x \right ) {a}^{2}-2\,g \left ( x \right ) f \left ( x \right ) ab-2\,g \left ( x \right ) f \left ( x \right ) {b}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}ab+{b}^{2} \left ( f \left ( x \right ) \right ) ^{2} \right ) h \left ( x \right ) }{ \left ( a+b \right ) ^{2}}}\,{\rm d}x+{\it \_C1} \right ) {a}^{2}b+3\,f \left ( x \right ) {\it RootOf} \left ( -27\,\int ^{{\it \_Z}}\!{\frac { \left ( {a}^{2}+ab+{b}^{2} \right ) ^{3}}{4\,{{\it \_a}}^{3}{a}^{6}+12\,{{\it \_a}}^{3}{a}^{5}b-3\,{{\it \_a}}^{3}{a}^{4}{b}^{2}-26\,{{\it \_a}}^{3}{a}^{3}{b}^{3}-3\,{{\it \_a}}^{3}{a}^{2}{b}^{4}+12\,{{\it \_a}}^{3}a{b}^{5}+4\,{{\it \_a}}^{3}{b}^{6}-27\,{\it \_a}\,{a}^{6}-81\,{\it \_a}\,{a}^{5}b-162\,{\it \_a}\,{a}^{4}{b}^{2}-189\,{\it \_a}\,{a}^{3}{b}^{3}-162\,{\it \_a}\,{a}^{2}{b}^{4}-81\,{\it \_a}\,a{b}^{5}-27\,{\it \_a}\,{b}^{6}+27\,{a}^{6}+81\,{a}^{5}b+162\,{a}^{4}{b}^{2}+189\,{a}^{3}{b}^{3}+162\,{a}^{2}{b}^{4}+81\,a{b}^{5}+27\,{b}^{6}}}{d{\it \_a}}+\int \!1/3\,{\frac { \left ( \left ( g \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( g \left ( x \right ) \right ) ^{2}ab+ \left ( g \left ( x \right ) \right ) ^{2}{b}^{2}-2\,g \left ( x \right ) f \left ( x \right ) {a}^{2}-2\,g \left ( x \right ) f \left ( x \right ) ab-2\,g \left ( x \right ) f \left ( x \right ) {b}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}ab+{b}^{2} \left ( f \left ( x \right ) \right ) ^{2} \right ) h \left ( x \right ) }{ \left ( a+b \right ) ^{2}}}\,{\rm d}x+{\it \_C1} \right ) a{b}^{2}+2\,f \left ( x \right ) {\it RootOf} \left ( -27\,\int ^{{\it \_Z}}\!{\frac { \left ( {a}^{2}+ab+{b}^{2} \right ) ^{3}}{4\,{{\it \_a}}^{3}{a}^{6}+12\,{{\it \_a}}^{3}{a}^{5}b-3\,{{\it \_a}}^{3}{a}^{4}{b}^{2}-26\,{{\it \_a}}^{3}{a}^{3}{b}^{3}-3\,{{\it \_a}}^{3}{a}^{2}{b}^{4}+12\,{{\it \_a}}^{3}a{b}^{5}+4\,{{\it \_a}}^{3}{b}^{6}-27\,{\it \_a}\,{a}^{6}-81\,{\it \_a}\,{a}^{5}b-162\,{\it \_a}\,{a}^{4}{b}^{2}-189\,{\it \_a}\,{a}^{3}{b}^{3}-162\,{\it \_a}\,{a}^{2}{b}^{4}-81\,{\it \_a}\,a{b}^{5}-27\,{\it \_a}\,{b}^{6}+27\,{a}^{6}+81\,{a}^{5}b+162\,{a}^{4}{b}^{2}+189\,{a}^{3}{b}^{3}+162\,{a}^{2}{b}^{4}+81\,a{b}^{5}+27\,{b}^{6}}}{d{\it \_a}}+\int \!1/3\,{\frac { \left ( \left ( g \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( g \left ( x \right ) \right ) ^{2}ab+ \left ( g \left ( x \right ) \right ) ^{2}{b}^{2}-2\,g \left ( x \right ) f \left ( x \right ) {a}^{2}-2\,g \left ( x \right ) f \left ( x \right ) ab-2\,g \left ( x \right ) f \left ( x \right ) {b}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}{a}^{2}+ \left ( f \left ( x \right ) \right ) ^{2}ab+{b}^{2} \left ( f \left ( x \right ) \right ) ^{2} \right ) h \left ( x \right ) }{ \left ( a+b \right ) ^{2}}}\,{\rm d}x+{\it \_C1} \right ) {b}^{3}-3\,{a}^{3}g \left ( x \right ) -9\,{a}^{2}g \left ( x \right ) b-9\,ag \left ( x \right ) {b}^{2}-6\,{b}^{3}g \left ( x \right ) -6\,{a}^{3}f \left ( x \right ) -9\,{a}^{2}f \left ( x \right ) b-9\,af \left ( x \right ) {b}^{2}-3\,{b}^{3}f \left ( x \right ) \right ) } \right \} \]