#### 2.179   ODE No. 179

$3 x \left (x^2-1\right ) y'(x)-\left (x^2+1\right ) y(x)+x y(x)^2-3 x=0$ Mathematica : cpu = 0.591139 (sec), leaf count = 2833

$\left \{\left \{y(x)\to \frac {3 \left (x^2-1\right ) \left (c_1 \left (\frac {e^{\int _1^x\text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {\#1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {\#1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {\#1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {\#1}+5\& ,1\right ]dK[1]} \text {Root}\left [125 x^8-164 x^6+70 x^4-20 x^2+\left (1296 x^{12}-5184 x^{10}+7776 x^8-5184 x^6+1296 x^4\right ) \text {\#1}^4+\left (-3456 x^{11}+12096 x^9-15552 x^7+8640 x^5-1728 x^3\right ) \text {\#1}^3+\left (3240 x^{10}-9504 x^8+9936 x^6-4320 x^4+648 x^2\right ) \text {\#1}^2+\left (-1200 x^9+2736 x^7-2160 x^5+720 x^3-96 x\right ) \text {\#1}+5\& ,1\right ]}{\sqrt [6]{x} \sqrt [3]{1-x^2}}-\frac {e^{\int _1^x\text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {\#1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {\#1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {\#1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {\#1}+5\& ,1\right ]dK[1]}}{6 x^{7/6} \sqrt [3]{1-x^2}}+\frac {2 e^{\int _1^x\text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {\#1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {\#1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {\#1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {\#1}+5\& ,1\right ]dK[1]} x^{5/6}}{3 \left (1-x^2\right )^{4/3}}\right )+\frac {e^{\int _1^x\text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {\#1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {\#1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {\#1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {\#1}+5\& ,1\right ]dK[1]} \text {Root}\left [125 x^8-164 x^6+70 x^4-20 x^2+\left (1296 x^{12}-5184 x^{10}+7776 x^8-5184 x^6+1296 x^4\right ) \text {\#1}^4+\left (-3456 x^{11}+12096 x^9-15552 x^7+8640 x^5-1728 x^3\right ) \text {\#1}^3+\left (3240 x^{10}-9504 x^8+9936 x^6-4320 x^4+648 x^2\right ) \text {\#1}^2+\left (-1200 x^9+2736 x^7-2160 x^5+720 x^3-96 x\right ) \text {\#1}+5\& ,1\right ] \int _1^xe^{-2 \int _1^{K[2]}\text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {\#1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {\#1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {\#1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {\#1}+5\& ,1\right ]dK[1]}dK[2]}{\sqrt [6]{x} \sqrt [3]{1-x^2}}-\frac {e^{\int _1^x\text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {\#1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {\#1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {\#1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {\#1}+5\& ,1\right ]dK[1]} \int _1^xe^{-2 \int _1^{K[2]}\text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {\#1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {\#1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {\#1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {\#1}+5\& ,1\right ]dK[1]}dK[2]}{6 x^{7/6} \sqrt [3]{1-x^2}}+\frac {2 e^{\int _1^x\text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {\#1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {\#1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {\#1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {\#1}+5\& ,1\right ]dK[1]} x^{5/6} \int _1^xe^{-2 \int _1^{K[2]}\text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {\#1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {\#1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {\#1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {\#1}+5\& ,1\right ]dK[1]}dK[2]}{3 \left (1-x^2\right )^{4/3}}+\frac {e^{-\int _1^x\text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {\#1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {\#1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {\#1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {\#1}+5\& ,1\right ]dK[1]}}{\sqrt [6]{x} \sqrt [3]{1-x^2}}\right )}{\frac {e^{\int _1^x\text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {\#1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {\#1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {\#1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {\#1}+5\& ,1\right ]dK[1]} c_1}{\sqrt [6]{x} \sqrt [3]{1-x^2}}+\frac {e^{\int _1^x\text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {\#1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {\#1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {\#1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {\#1}+5\& ,1\right ]dK[1]} \int _1^xe^{-2 \int _1^{K[2]}\text {Root}\left [125 K[1]^8-164 K[1]^6+70 K[1]^4-20 K[1]^2+\left (1296 K[1]^{12}-5184 K[1]^{10}+7776 K[1]^8-5184 K[1]^6+1296 K[1]^4\right ) \text {\#1}^4+\left (-3456 K[1]^{11}+12096 K[1]^9-15552 K[1]^7+8640 K[1]^5-1728 K[1]^3\right ) \text {\#1}^3+\left (3240 K[1]^{10}-9504 K[1]^8+9936 K[1]^6-4320 K[1]^4+648 K[1]^2\right ) \text {\#1}^2+\left (-1200 K[1]^9+2736 K[1]^7-2160 K[1]^5+720 K[1]^3-96 K[1]\right ) \text {\#1}+5\& ,1\right ]dK[1]}dK[2]}{\sqrt [6]{x} \sqrt [3]{1-x^2}}}\right \}\right \}$ Maple : cpu = 0.11 (sec), leaf count = 112

$\left \{ y \left ( x \right ) =35\,{\frac {1}{\sqrt [3]{x} \left ( 8\,{x}^{2/3}{\mbox {_2F_1}(5/6,7/6;\,4/3;\,{x}^{2})}{\it \_C1}+8\,{\mbox {_2F_1}(1/2,5/6;\,2/3;\,{x}^{2})} \right ) } \left ( {\it \_C1}\, \left ( {\frac {8\,{x}^{2}}{7}}-{\frac {16}{35}} \right ) {\mbox {_2F_1}(5/6,7/6;\,4/3;\,{x}^{2})}+ \left ( {x}^{4}-{x}^{2} \right ) {\it \_C1}\,{\mbox {_2F_1}({\frac {11}{6}},{\frac {13}{6}};\,7/3;\,{x}^{2})}+ \left ( -6/7\,{x}^{4/3}+6/7\,{x}^{10/3} \right ) {\mbox {_2F_1}(3/2,{\frac {11}{6}};\,5/3;\,{x}^{2})}+{\frac {24\,{\mbox {_2F_1}(1/2,5/6;\,2/3;\,{x}^{2})}{x}^{4/3}}{35}} \right ) } \right \}$