2.51   ODE No. 51

1. Problem in Latex
2. Mathematica input
3. Maple input

$$-h(x)(y(x)-f(x))(y(x)-g(x))(y(x)-\frac{af(x)+bg(x)}{a+b})-\frac{f^{\prime }(x)(y(x)-g(x))}{f(x)-g(x)}-\frac{(y(x)-f(x))g^{\prime }(x)}{g(x)-f(x)}+y^{\prime }(x)=0$$ ✓ Mathematica : cpu = 0.742014 (sec), leaf count = 354

$$\text{Solve}[-\frac{1}{3}(a-b{)}^{2/3}(2a+b{)}^{2/3}(a+2b{)}^{2/3}\text{RootSum}[{\mathrm{\#}\text{1}}^3(a-b{)}^{2/3}(2a+b{)}^{2/3}(a+2b{)}^{2/3}-3\mathrm{\#}\text{1}{a}^2-3\mathrm{\#}\text{1}ab-3\mathrm{\#}\text{1}{b}^2+(a-b{)}^{2/3}(2a+b{)}^{2/3}(a+2b{)}^{2/3}\mathrm{\&},\frac{\log (\frac{\frac{-2af(x)h(x)-ag(x)h(x)-bf(x)h(x)-2bg(x)h(x)}{a+b}+3h(x)y(x)}{\sqrt[3]{\frac{(f(x)-g(x){)}^3(2a^{3}h(x{)}^3+3a^{2}bh(x{)}^3-3a{b}^{2}h(x{)}^3-2b^{3}h(x{)}^3)}{(a+b{)}^3}}}-\mathrm{\#}\text{1})}{-{\mathrm{\#}\text{1}}^2(a-b{)}^{2/3}(2a+b{)}^{2/3}(a+2b{)}^{2/3}+a^2+ab+b^2}\mathrm{\&}]={\int }_1^x\frac{{\left(\frac{(f(K[1])-g(K[1]){)}^3(2a^{3}h(K[1]{)}^3+3a^{2}bh(K[1]{)}^3-3a{b}^{2}h(K[1]{)}^3-2b^{3}h(K[1]{)}^3)}{(a+b{)}^3}\right)}^{2/3}}{9h(K[1])}dK[1]+c_1,y(x)]$$

✓ Maple : cpu = 0.221 (sec), leaf count = 2348

$$\{y\left(x\right)=-\frac{1}{9a^3+18a^{2}b+18a{b}^2+9b^3}(2g\left(x\right)\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-27{\int }^{\mathit{\_}\mathit{Z}}\frac{{(a^2+ab+b^2)}^3}{4{\mathit{\_}\mathit{a}}^3{a}^6+12{\mathit{\_}\mathit{a}}^3{a}^{5}b-3{\mathit{\_}\mathit{a}}^3{a}^4{b}^2-26{\mathit{\_}\mathit{a}}^3{a}^3{b}^3-3{\mathit{\_}\mathit{a}}^3{a}^2{b}^4+12{\mathit{\_}\mathit{a}}^{3}a{b}^5+4{\mathit{\_}\mathit{a}}^3{b}^6-27\mathit{\_}\mathit{a}{a}^6-81\mathit{\_}\mathit{a}{a}^{5}b-162\mathit{\_}\mathit{a}{a}^4{b}^2-189\mathit{\_}\mathit{a}{a}^3{b}^3-162\mathit{\_}\mathit{a}{a}^2{b}^4-81\mathit{\_}\mathit{a}a{b}^5-27\mathit{\_}\mathit{a}{b}^6+27a^6+81a^{5}b+162a^4{b}^2+189a^3{b}^3+162a^2{b}^4+81a{b}^5+27b^6}d\mathit{\_}\mathit{a}+\int 1/3\frac{({\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}^2-2g\left(x\right)f\left(x\right)a^2-2g\left(x\right)f\left(x\right)ab-2g\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)h\left(x\right)}{{(a+b)}^2}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1})a^3+3g\left(x\right)\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-27{\int }^{\mathit{\_}\mathit{Z}}\frac{{(a^2+ab+b^2)}^3}{4{\mathit{\_}\mathit{a}}^3{a}^6+12{\mathit{\_}\mathit{a}}^3{a}^{5}b-3{\mathit{\_}\mathit{a}}^3{a}^4{b}^2-26{\mathit{\_}\mathit{a}}^3{a}^3{b}^3-3{\mathit{\_}\mathit{a}}^3{a}^2{b}^4+12{\mathit{\_}\mathit{a}}^{3}a{b}^5+4{\mathit{\_}\mathit{a}}^3{b}^6-27\mathit{\_}\mathit{a}{a}^6-81\mathit{\_}\mathit{a}{a}^{5}b-162\mathit{\_}\mathit{a}{a}^4{b}^2-189\mathit{\_}\mathit{a}{a}^3{b}^3-162\mathit{\_}\mathit{a}{a}^2{b}^4-81\mathit{\_}\mathit{a}a{b}^5-27\mathit{\_}\mathit{a}{b}^6+27a^6+81a^{5}b+162a^4{b}^2+189a^3{b}^3+162a^2{b}^4+81a{b}^5+27b^6}d\mathit{\_}\mathit{a}+\int 1/3\frac{({\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}^2-2g\left(x\right)f\left(x\right)a^2-2g\left(x\right)f\left(x\right)ab-2g\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)h\left(x\right)}{{(a+b)}^2}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1})a^{2}b-3g\left(x\right)\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-27{\int }^{\mathit{\_}\mathit{Z}}\frac{{(a^2+ab+b^2)}^3}{4{\mathit{\_}\mathit{a}}^3{a}^6+12{\mathit{\_}\mathit{a}}^3{a}^{5}b-3{\mathit{\_}\mathit{a}}^3{a}^4{b}^2-26{\mathit{\_}\mathit{a}}^3{a}^3{b}^3-3{\mathit{\_}\mathit{a}}^3{a}^2{b}^4+12{\mathit{\_}\mathit{a}}^{3}a{b}^5+4{\mathit{\_}\mathit{a}}^3{b}^6-27\mathit{\_}\mathit{a}{a}^6-81\mathit{\_}\mathit{a}{a}^{5}b-162\mathit{\_}\mathit{a}{a}^4{b}^2-189\mathit{\_}\mathit{a}{a}^3{b}^3-162\mathit{\_}\mathit{a}{a}^2{b}^4-81\mathit{\_}\mathit{a}a{b}^5-27\mathit{\_}\mathit{a}{b}^6+27a^6+81a^{5}b+162a^4{b}^2+189a^3{b}^3+162a^2{b}^4+81a{b}^5+27b^6}d\mathit{\_}\mathit{a}+\int 1/3\frac{({\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}^2-2g\left(x\right)f\left(x\right)a^2-2g\left(x\right)f\left(x\right)ab-2g\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)h\left(x\right)}{{(a+b)}^2}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1})a{b}^2-2g\left(x\right)\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-27{\int }^{\mathit{\_}\mathit{Z}}\frac{{(a^2+ab+b^2)}^3}{4{\mathit{\_}\mathit{a}}^3{a}^6+12{\mathit{\_}\mathit{a}}^3{a}^{5}b-3{\mathit{\_}\mathit{a}}^3{a}^4{b}^2-26{\mathit{\_}\mathit{a}}^3{a}^3{b}^3-3{\mathit{\_}\mathit{a}}^3{a}^2{b}^4+12{\mathit{\_}\mathit{a}}^{3}a{b}^5+4{\mathit{\_}\mathit{a}}^3{b}^6-27\mathit{\_}\mathit{a}{a}^6-81\mathit{\_}\mathit{a}{a}^{5}b-162\mathit{\_}\mathit{a}{a}^4{b}^2-189\mathit{\_}\mathit{a}{a}^3{b}^3-162\mathit{\_}\mathit{a}{a}^2{b}^4-81\mathit{\_}\mathit{a}a{b}^5-27\mathit{\_}\mathit{a}{b}^6+27a^6+81a^{5}b+162a^4{b}^2+189a^3{b}^3+162a^2{b}^4+81a{b}^5+27b^6}d\mathit{\_}\mathit{a}+\int 1/3\frac{({\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}^2-2g\left(x\right)f\left(x\right)a^2-2g\left(x\right)f\left(x\right)ab-2g\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)h\left(x\right)}{{(a+b)}^2}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1})b^3-2f\left(x\right)\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-27{\int }^{\mathit{\_}\mathit{Z}}\frac{{(a^2+ab+b^2)}^3}{4{\mathit{\_}\mathit{a}}^3{a}^6+12{\mathit{\_}\mathit{a}}^3{a}^{5}b-3{\mathit{\_}\mathit{a}}^3{a}^4{b}^2-26{\mathit{\_}\mathit{a}}^3{a}^3{b}^3-3{\mathit{\_}\mathit{a}}^3{a}^2{b}^4+12{\mathit{\_}\mathit{a}}^{3}a{b}^5+4{\mathit{\_}\mathit{a}}^3{b}^6-27\mathit{\_}\mathit{a}{a}^6-81\mathit{\_}\mathit{a}{a}^{5}b-162\mathit{\_}\mathit{a}{a}^4{b}^2-189\mathit{\_}\mathit{a}{a}^3{b}^3-162\mathit{\_}\mathit{a}{a}^2{b}^4-81\mathit{\_}\mathit{a}a{b}^5-27\mathit{\_}\mathit{a}{b}^6+27a^6+81a^{5}b+162a^4{b}^2+189a^3{b}^3+162a^2{b}^4+81a{b}^5+27b^6}d\mathit{\_}\mathit{a}+\int 1/3\frac{({\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}^2-2g\left(x\right)f\left(x\right)a^2-2g\left(x\right)f\left(x\right)ab-2g\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)h\left(x\right)}{{(a+b)}^2}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1})a^3-3f\left(x\right)\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-27{\int }^{\mathit{\_}\mathit{Z}}\frac{{(a^2+ab+b^2)}^3}{4{\mathit{\_}\mathit{a}}^3{a}^6+12{\mathit{\_}\mathit{a}}^3{a}^{5}b-3{\mathit{\_}\mathit{a}}^3{a}^4{b}^2-26{\mathit{\_}\mathit{a}}^3{a}^3{b}^3-3{\mathit{\_}\mathit{a}}^3{a}^2{b}^4+12{\mathit{\_}\mathit{a}}^{3}a{b}^5+4{\mathit{\_}\mathit{a}}^3{b}^6-27\mathit{\_}\mathit{a}{a}^6-81\mathit{\_}\mathit{a}{a}^{5}b-162\mathit{\_}\mathit{a}{a}^4{b}^2-189\mathit{\_}\mathit{a}{a}^3{b}^3-162\mathit{\_}\mathit{a}{a}^2{b}^4-81\mathit{\_}\mathit{a}a{b}^5-27\mathit{\_}\mathit{a}{b}^6+27a^6+81a^{5}b+162a^4{b}^2+189a^3{b}^3+162a^2{b}^4+81a{b}^5+27b^6}d\mathit{\_}\mathit{a}+\int 1/3\frac{({\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}^2-2g\left(x\right)f\left(x\right)a^2-2g\left(x\right)f\left(x\right)ab-2g\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)h\left(x\right)}{{(a+b)}^2}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1})a^{2}b+3f\left(x\right)\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-27{\int }^{\mathit{\_}\mathit{Z}}\frac{{(a^2+ab+b^2)}^3}{4{\mathit{\_}\mathit{a}}^3{a}^6+12{\mathit{\_}\mathit{a}}^3{a}^{5}b-3{\mathit{\_}\mathit{a}}^3{a}^4{b}^2-26{\mathit{\_}\mathit{a}}^3{a}^3{b}^3-3{\mathit{\_}\mathit{a}}^3{a}^2{b}^4+12{\mathit{\_}\mathit{a}}^{3}a{b}^5+4{\mathit{\_}\mathit{a}}^3{b}^6-27\mathit{\_}\mathit{a}{a}^6-81\mathit{\_}\mathit{a}{a}^{5}b-162\mathit{\_}\mathit{a}{a}^4{b}^2-189\mathit{\_}\mathit{a}{a}^3{b}^3-162\mathit{\_}\mathit{a}{a}^2{b}^4-81\mathit{\_}\mathit{a}a{b}^5-27\mathit{\_}\mathit{a}{b}^6+27a^6+81a^{5}b+162a^4{b}^2+189a^3{b}^3+162a^2{b}^4+81a{b}^5+27b^6}d\mathit{\_}\mathit{a}+\int 1/3\frac{({\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}^2-2g\left(x\right)f\left(x\right)a^2-2g\left(x\right)f\left(x\right)ab-2g\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)h\left(x\right)}{{(a+b)}^2}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1})a{b}^2+2f\left(x\right)\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(-27{\int }^{\mathit{\_}\mathit{Z}}\frac{{(a^2+ab+b^2)}^3}{4{\mathit{\_}\mathit{a}}^3{a}^6+12{\mathit{\_}\mathit{a}}^3{a}^{5}b-3{\mathit{\_}\mathit{a}}^3{a}^4{b}^2-26{\mathit{\_}\mathit{a}}^3{a}^3{b}^3-3{\mathit{\_}\mathit{a}}^3{a}^2{b}^4+12{\mathit{\_}\mathit{a}}^{3}a{b}^5+4{\mathit{\_}\mathit{a}}^3{b}^6-27\mathit{\_}\mathit{a}{a}^6-81\mathit{\_}\mathit{a}{a}^{5}b-162\mathit{\_}\mathit{a}{a}^4{b}^2-189\mathit{\_}\mathit{a}{a}^3{b}^3-162\mathit{\_}\mathit{a}{a}^2{b}^4-81\mathit{\_}\mathit{a}a{b}^5-27\mathit{\_}\mathit{a}{b}^6+27a^6+81a^{5}b+162a^4{b}^2+189a^3{b}^3+162a^2{b}^4+81a{b}^5+27b^6}d\mathit{\_}\mathit{a}+\int 1/3\frac{({\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}^2-2g\left(x\right)f\left(x\right)a^2-2g\left(x\right)f\left(x\right)ab-2g\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)h\left(x\right)}{{(a+b)}^2}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1})b^3-3a^{3}g\left(x\right)-9a^{2}g\left(x\right)b-9ag\left(x\right)b^2-6b^{3}g\left(x\right)-6a^{3}f\left(x\right)-9a^{2}f\left(x\right)b-9af\left(x\right)b^2-3b^{3}f\left(x\right))\}$$