3.323   ODE No. 323

$$\boxed{{\displaystyle (ax{\left(y\left(x\right)\right)}^3+c)x\frac{\mathrm{d}}{\mathrm{d}x}y\left(x\right)+(b{x}^{3}y\left(x\right)+c)y\left(x\right)=0}}$$

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

Mathematica: cpu = 0.032004 (sec), leaf count = 463 $$\{\{y(x)\to \frac{\sqrt[3]{54a^{2}c{x}^2+\sqrt{2916a^4{c}^2{x}^4+108a^3{x}^3(b{x}^3-2c_{1}x){}^3}}}{3\sqrt[3]{2}ax}-\frac{\sqrt[3]{2}(b{x}^3-2c_{1}x)}{\sqrt[3]{54a^{2}c{x}^2+\sqrt{2916a^4{c}^2{x}^4+108a^3{x}^3(b{x}^3-2c_{1}x){}^3}}}\},\{y(x)\to \frac{(1+i\sqrt{3})(b{x}^3-2c_{1}x)}{2^{2/3}\sqrt[3]{54a^{2}c{x}^2+\sqrt{2916a^4{c}^2{x}^4+108a^3{x}^3(b{x}^3-2c_{1}x){}^3}}}-\frac{(1-i\sqrt{3})\sqrt[3]{54a^{2}c{x}^2+\sqrt{2916a^4{c}^2{x}^4+108a^3{x}^3(b{x}^3-2c_{1}x){}^3}}}{6\sqrt[3]{2}ax}\},\{y(x)\to \frac{(1-i\sqrt{3})(b{x}^3-2c_{1}x)}{2^{2/3}\sqrt[3]{54a^{2}c{x}^2+\sqrt{2916a^4{c}^2{x}^4+108a^3{x}^3(b{x}^3-2c_{1}x){}^3}}}-\frac{(1+i\sqrt{3})\sqrt[3]{54a^{2}c{x}^2+\sqrt{2916a^4{c}^2{x}^4+108a^3{x}^3(b{x}^3-2c_{1}x){}^3}}}{6\sqrt[3]{2}ax}\}\}$$

Maple: cpu = 0.093 (sec), leaf count = 761 $$\{y\left(x\right)=\frac{1}{3ax}\sqrt[3]{(27c+3\sqrt{-\frac{-3b^3{x}^8+18\mathit{\_}\mathit{C}\mathit{1}{b}^2{x}^6-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}b{x}^4+24{\mathit{\_}\mathit{C}\mathit{1}}^3{x}^2-81c^{2}a}{a}})a^2{x}^2}+x(-b{x}^2+2\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt[3]{(27c+3\sqrt{-\frac{-3b^3{x}^8+18\mathit{\_}\mathit{C}\mathit{1}{b}^2{x}^6-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}b{x}^4+24{\mathit{\_}\mathit{C}\mathit{1}}^3{x}^2-81c^{2}a}{a}})a^2{x}^2}},y\left(x\right)=-\frac{1}{6ax}\sqrt[3]{(27c+3\sqrt{-\frac{-3b^3{x}^8+18\mathit{\_}\mathit{C}\mathit{1}{b}^2{x}^6-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}b{x}^4+24{\mathit{\_}\mathit{C}\mathit{1}}^3{x}^2-81c^{2}a}{a}})a^2{x}^2}-\frac{x(-b{x}^2+2\mathit{\_}\mathit{C}\mathit{1})}{2}\frac{1}{\sqrt[3]{(27c+3\sqrt{-\frac{-3b^3{x}^8+18\mathit{\_}\mathit{C}\mathit{1}{b}^2{x}^6-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}b{x}^4+24{\mathit{\_}\mathit{C}\mathit{1}}^3{x}^2-81c^{2}a}{a}})a^2{x}^2}}-\frac{i}{2}\sqrt{3}(\frac{1}{3ax}\sqrt[3]{(27c+3\sqrt{-\frac{-3b^3{x}^8+18\mathit{\_}\mathit{C}\mathit{1}{b}^2{x}^6-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}b{x}^4+24{\mathit{\_}\mathit{C}\mathit{1}}^3{x}^2-81c^{2}a}{a}})a^2{x}^2}-x(-b{x}^2+2\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt[3]{(27c+3\sqrt{-\frac{-3b^3{x}^8+18\mathit{\_}\mathit{C}\mathit{1}{b}^2{x}^6-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}b{x}^4+24{\mathit{\_}\mathit{C}\mathit{1}}^3{x}^2-81c^{2}a}{a}})a^2{x}^2}}),y\left(x\right)=-\frac{1}{6ax}\sqrt[3]{(27c+3\sqrt{-\frac{-3b^3{x}^8+18\mathit{\_}\mathit{C}\mathit{1}{b}^2{x}^6-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}b{x}^4+24{\mathit{\_}\mathit{C}\mathit{1}}^3{x}^2-81c^{2}a}{a}})a^2{x}^2}-\frac{x(-b{x}^2+2\mathit{\_}\mathit{C}\mathit{1})}{2}\frac{1}{\sqrt[3]{(27c+3\sqrt{-\frac{-3b^3{x}^8+18\mathit{\_}\mathit{C}\mathit{1}{b}^2{x}^6-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}b{x}^4+24{\mathit{\_}\mathit{C}\mathit{1}}^3{x}^2-81c^{2}a}{a}})a^2{x}^2}}+\frac{i}{2}\sqrt{3}(\frac{1}{3ax}\sqrt[3]{(27c+3\sqrt{-\frac{-3b^3{x}^8+18\mathit{\_}\mathit{C}\mathit{1}{b}^2{x}^6-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}b{x}^4+24{\mathit{\_}\mathit{C}\mathit{1}}^3{x}^2-81c^{2}a}{a}})a^2{x}^2}-x(-b{x}^2+2\mathit{\_}\mathit{C}\mathit{1})\frac{1}{\sqrt[3]{(27c+3\sqrt{-\frac{-3b^3{x}^8+18\mathit{\_}\mathit{C}\mathit{1}{b}^2{x}^6-36{\mathit{\_}\mathit{C}\mathit{1}}^{2}b{x}^4+24{\mathit{\_}\mathit{C}\mathit{1}}^3{x}^2-81c^{2}a}{a}})a^2{x}^2}})\}$$