3.51   ODE No. 51

$$\boxed{{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)-(y\left(x\right)-f\left(x\right))(y\left(x\right)-g\left(x\right))(y\left(x\right)-\frac{af\left(x\right)+bg\left(x\right)}{a+b})h\left(x\right)-\frac{\left(\frac{\mathrm{d}}{\mathrm{d}x}f\left(x\right)\right)(y\left(x\right)-g\left(x\right))}{f\left(x\right)-g\left(x\right)}-\frac{\left(\frac{\mathrm{d}}{\mathrm{d}x}g\left(x\right)\right)(y\left(x\right)-f\left(x\right))}{g\left(x\right)-f\left(x\right)}=0}}$$

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

Mathematica: cpu = 0.753096 (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 = 0.156 (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))\}$$

Sage: cpu = 2.588 (sec), leaf count = 0 $$[[\frac{(a^2+ab+b^2)\log (3a^2+3ab+3b^2-\frac{3(a^3+3a^{2}b+3a{b}^2+b^3)(a^2+ab-2b^2)(\frac{(2a+b)f\left(x\right)+(a+2b)g\left(x\right)}{a+b}-3y\left(x\right))(\frac{((a^2+ab+b^2)f{\left(x\right)}^3-3(a^2+ab+b^2)f{\left(x\right)}^{2}g\left(x\right)+3(a^2+ab+b^2)f\left(x\right)g{\left(x\right)}^2-(a^2+ab+b^2)g{\left(x\right)}^3)h{\left(x\right)}^2+6(a^2+2ab+b^2)h\left(x\right)D[0]\left(f\right)\left(x\right)-6(a^2+2ab+b^2)h\left(x\right)D[0]\left(g\right)\left(x\right)+3((a^2+2ab+b^2)f\left(x\right)-(a^2+2ab+b^2)g\left(x\right))D[0]\left(h\right)\left(x\right)}{((a^2+2ab+b^2)f\left(x\right)-(a^2+2ab+b^2)g\left(x\right))h\left(x\right)}-\frac{3(a^6+6a^{5}b+15a^4{b}^2+20a^3{b}^3+15a^2{b}^4+6a{b}^5+b^6)(\frac{2((4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{5}D[0]\left(f\right)\left(x\right)-5(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{4}g\left(x\right)D[0]\left(f\right)\left(x\right)+10(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{3}g{\left(x\right)}^{2}D[0]\left(f\right)\left(x\right)-10(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{2}g{\left(x\right)}^{3}D[0]\left(f\right)\left(x\right)+5(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f\left(x\right)g{\left(x\right)}^{4}D[0]\left(f\right)\left(x\right)-(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)g{\left(x\right)}^{5}D[0]\left(f\right)\left(x\right)-(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{5}D[0]\left(g\right)\left(x\right)+5(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{4}g\left(x\right)D[0]\left(g\right)\left(x\right)-10(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{3}g{\left(x\right)}^{2}D[0]\left(g\right)\left(x\right)+10(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{2}g{\left(x\right)}^{3}D[0]\left(g\right)\left(x\right)-5(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f\left(x\right)g{\left(x\right)}^{4}D[0]\left(g\right)\left(x\right)+(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)g{\left(x\right)}^{5}D[0]\left(g\right)\left(x\right))h{\left(x\right)}^3}{a^6+6a^{5}b+15a^4{b}^2+20a^3{b}^3+15a^2{b}^4+6a{b}^5+b^6}+\frac{((4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^6-6(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{5}g\left(x\right)+15(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{4}g{\left(x\right)}^2-20(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{3}g{\left(x\right)}^3+15(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{2}g{\left(x\right)}^4-6(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f\left(x\right)g{\left(x\right)}^5+(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)g{\left(x\right)}^6)h{\left(x\right)}^{2}D[0]\left(h\right)\left(x\right)}{a^6+6a^{5}b+15a^4{b}^2+20a^3{b}^3+15a^2{b}^4+6a{b}^5+b^6})}{((4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^6-6(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{5}g\left(x\right)+15(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{4}g{\left(x\right)}^2-20(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{3}g{\left(x\right)}^3+15(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{2}g{\left(x\right)}^4-6(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f\left(x\right)g{\left(x\right)}^5+(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)g{\left(x\right)}^6)h{\left(x\right)}^3})}{((2a^3+3a^{2}b-3a{b}^2-2b^3)f{\left(x\right)}^3-3(2a^3+3a^{2}b-3a{b}^2-2b^3)f{\left(x\right)}^{2}g\left(x\right)+3(2a^3+3a^{2}b-3a{b}^2-2b^3)f\left(x\right)g{\left(x\right)}^2-(2a^3+3a^{2}b-3a{b}^2-2b^3)g{\left(x\right)}^3)h\left(x\right)})}{3(a^2+ab)}+\frac{(a^2+ab+b^2)\log (-3a^2-3ab-3b^2-\frac{3(a^3+3a^{2}b+3a{b}^2+b^3)(2a^2-ab-b^2)(\frac{(2a+b)f\left(x\right)+(a+2b)g\left(x\right)}{a+b}-3y\left(x\right))(\frac{((a^2+ab+b^2)f{\left(x\right)}^3-3(a^2+ab+b^2)f{\left(x\right)}^{2}g\left(x\right)+3(a^2+ab+b^2)f\left(x\right)g{\left(x\right)}^2-(a^2+ab+b^2)g{\left(x\right)}^3)h{\left(x\right)}^2+6(a^2+2ab+b^2)h\left(x\right)D[0]\left(f\right)\left(x\right)-6(a^2+2ab+b^2)h\left(x\right)D[0]\left(g\right)\left(x\right)+3((a^2+2ab+b^2)f\left(x\right)-(a^2+2ab+b^2)g\left(x\right))D[0]\left(h\right)\left(x\right)}{((a^2+2ab+b^2)f\left(x\right)-(a^2+2ab+b^2)g\left(x\right))h\left(x\right)}-\frac{3(a^6+6a^{5}b+15a^4{b}^2+20a^3{b}^3+15a^2{b}^4+6a{b}^5+b^6)(\frac{2((4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{5}D[0]\left(f\right)\left(x\right)-5(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{4}g\left(x\right)D[0]\left(f\right)\left(x\right)+10(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{3}g{\left(x\right)}^{2}D[0]\left(f\right)\left(x\right)-10(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{2}g{\left(x\right)}^{3}D[0]\left(f\right)\left(x\right)+5(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f\left(x\right)g{\left(x\right)}^{4}D[0]\left(f\right)\left(x\right)-(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)g{\left(x\right)}^{5}D[0]\left(f\right)\left(x\right)-(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{5}D[0]\left(g\right)\left(x\right)+5(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{4}g\left(x\right)D[0]\left(g\right)\left(x\right)-10(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{3}g{\left(x\right)}^{2}D[0]\left(g\right)\left(x\right)+10(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{2}g{\left(x\right)}^{3}D[0]\left(g\right)\left(x\right)-5(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f\left(x\right)g{\left(x\right)}^{4}D[0]\left(g\right)\left(x\right)+(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)g{\left(x\right)}^{5}D[0]\left(g\right)\left(x\right))h{\left(x\right)}^3}{a^6+6a^{5}b+15a^4{b}^2+20a^3{b}^3+15a^2{b}^4+6a{b}^5+b^6}+\frac{((4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^6-6(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{5}g\left(x\right)+15(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{4}g{\left(x\right)}^2-20(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{3}g{\left(x\right)}^3+15(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{2}g{\left(x\right)}^4-6(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f\left(x\right)g{\left(x\right)}^5+(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)g{\left(x\right)}^6)h{\left(x\right)}^{2}D[0]\left(h\right)\left(x\right)}{a^6+6a^{5}b+15a^4{b}^2+20a^3{b}^3+15a^2{b}^4+6a{b}^5+b^6})}{((4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^6-6(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{5}g\left(x\right)+15(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{4}g{\left(x\right)}^2-20(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{3}g{\left(x\right)}^3+15(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{2}g{\left(x\right)}^4-6(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f\left(x\right)g{\left(x\right)}^5+(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)g{\left(x\right)}^6)h{\left(x\right)}^3})}{((2a^3+3a^{2}b-3a{b}^2-2b^3)f{\left(x\right)}^3-3(2a^3+3a^{2}b-3a{b}^2-2b^3)f{\left(x\right)}^{2}g\left(x\right)+3(2a^3+3a^{2}b-3a{b}^2-2b^3)f\left(x\right)g{\left(x\right)}^2-(2a^3+3a^{2}b-3a{b}^2-2b^3)g{\left(x\right)}^3)h\left(x\right)})}{3(ab+b^2)}-\frac{(a^2+ab+b^2)\log (3a^2+3ab+3b^2+\frac{3(a^3+3a^{2}b+3a{b}^2+b^3)(2a^2+5ab+2b^2)(\frac{(2a+b)f\left(x\right)+(a+2b)g\left(x\right)}{a+b}-3y\left(x\right))(\frac{((a^2+ab+b^2)f{\left(x\right)}^3-3(a^2+ab+b^2)f{\left(x\right)}^{2}g\left(x\right)+3(a^2+ab+b^2)f\left(x\right)g{\left(x\right)}^2-(a^2+ab+b^2)g{\left(x\right)}^3)h{\left(x\right)}^2+6(a^2+2ab+b^2)h\left(x\right)D[0]\left(f\right)\left(x\right)-6(a^2+2ab+b^2)h\left(x\right)D[0]\left(g\right)\left(x\right)+3((a^2+2ab+b^2)f\left(x\right)-(a^2+2ab+b^2)g\left(x\right))D[0]\left(h\right)\left(x\right)}{((a^2+2ab+b^2)f\left(x\right)-(a^2+2ab+b^2)g\left(x\right))h\left(x\right)}-\frac{3(a^6+6a^{5}b+15a^4{b}^2+20a^3{b}^3+15a^2{b}^4+6a{b}^5+b^6)(\frac{2((4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{5}D[0]\left(f\right)\left(x\right)-5(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{4}g\left(x\right)D[0]\left(f\right)\left(x\right)+10(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{3}g{\left(x\right)}^{2}D[0]\left(f\right)\left(x\right)-10(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{2}g{\left(x\right)}^{3}D[0]\left(f\right)\left(x\right)+5(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f\left(x\right)g{\left(x\right)}^{4}D[0]\left(f\right)\left(x\right)-(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)g{\left(x\right)}^{5}D[0]\left(f\right)\left(x\right)-(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{5}D[0]\left(g\right)\left(x\right)+5(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{4}g\left(x\right)D[0]\left(g\right)\left(x\right)-10(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{3}g{\left(x\right)}^{2}D[0]\left(g\right)\left(x\right)+10(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{2}g{\left(x\right)}^{3}D[0]\left(g\right)\left(x\right)-5(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f\left(x\right)g{\left(x\right)}^{4}D[0]\left(g\right)\left(x\right)+(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)g{\left(x\right)}^{5}D[0]\left(g\right)\left(x\right))h{\left(x\right)}^3}{a^6+6a^{5}b+15a^4{b}^2+20a^3{b}^3+15a^2{b}^4+6a{b}^5+b^6}+\frac{((4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^6-6(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{5}g\left(x\right)+15(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{4}g{\left(x\right)}^2-20(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{3}g{\left(x\right)}^3+15(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{2}g{\left(x\right)}^4-6(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f\left(x\right)g{\left(x\right)}^5+(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)g{\left(x\right)}^6)h{\left(x\right)}^{2}D[0]\left(h\right)\left(x\right)}{a^6+6a^{5}b+15a^4{b}^2+20a^3{b}^3+15a^2{b}^4+6a{b}^5+b^6})}{((4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^6-6(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{5}g\left(x\right)+15(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{4}g{\left(x\right)}^2-20(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{3}g{\left(x\right)}^3+15(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{2}g{\left(x\right)}^4-6(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f\left(x\right)g{\left(x\right)}^5+(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)g{\left(x\right)}^6)h{\left(x\right)}^3})}{((2a^3+3a^{2}b-3a{b}^2-2b^3)f{\left(x\right)}^3-3(2a^3+3a^{2}b-3a{b}^2-2b^3)f{\left(x\right)}^{2}g\left(x\right)+3(2a^3+3a^{2}b-3a{b}^2-2b^3)f\left(x\right)g{\left(x\right)}^2-(2a^3+3a^{2}b-3a{b}^2-2b^3)g{\left(x\right)}^3)h\left(x\right)})}{3ab}-\int \frac{((a^2+ab+b^2)f{\left(x\right)}^3-3(a^2+ab+b^2)f{\left(x\right)}^{2}g\left(x\right)+3(a^2+ab+b^2)f\left(x\right)g{\left(x\right)}^2-(a^2+ab+b^2)g{\left(x\right)}^3)h{\left(x\right)}^2+6(a^2+2ab+b^2)h\left(x\right)D[0]\left(f\right)\left(x\right)-6(a^2+2ab+b^2)h\left(x\right)D[0]\left(g\right)\left(x\right)+3((a^2+2ab+b^2)f\left(x\right)-(a^2+2ab+b^2)g\left(x\right))D[0]\left(h\right)\left(x\right)}{3((a^2+2ab+b^2)f\left(x\right)-(a^2+2ab+b^2)g\left(x\right))h\left(x\right)}dx+\frac{1}{3}\log (-\frac{((4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^6-6(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{5}g\left(x\right)+15(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{4}g{\left(x\right)}^2-20(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{3}g{\left(x\right)}^3+15(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f{\left(x\right)}^{2}g{\left(x\right)}^4-6(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)f\left(x\right)g{\left(x\right)}^5+(4a^6+12a^{5}b-3a^4{b}^2-26a^3{b}^3-3a^2{b}^4+12a{b}^5+4b^6)g{\left(x\right)}^6)h{\left(x\right)}^3}{729(a^6+6a^{5}b+15a^4{b}^2+20a^3{b}^3+15a^2{b}^4+6a{b}^5+b^6)})=c],\mathtt{\text{abel}}]$$