#### 2.252   ODE No. 252

$\left (x^2 y(x)-1\right ) y'(x)-x y(x)^2+1=0$ Mathematica : cpu = 11.0152 (sec), leaf count = 819

$\left \{\left \{y(x)\to \frac {6 x c_1-x}{6 c_1-1}+\frac {\sqrt [3]{-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+\sqrt {4 \left (54 x^2 c_1-9 x^2\right ){}^3+\left (-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+54\right ){}^2}+54}}{3 \sqrt [3]{2} (6 c_1-1)}-\frac {\sqrt [3]{2} \left (54 x^2 c_1-9 x^2\right )}{3 (6 c_1-1) \sqrt [3]{-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+\sqrt {4 \left (54 x^2 c_1-9 x^2\right ){}^3+\left (-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+54\right ){}^2}+54}}\right \},\left \{y(x)\to \frac {6 x c_1-x}{6 c_1-1}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+\sqrt {4 \left (54 x^2 c_1-9 x^2\right ){}^3+\left (-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+54\right ){}^2}+54}}{6 \sqrt [3]{2} (6 c_1-1)}+\frac {\left (1+i \sqrt {3}\right ) \left (54 x^2 c_1-9 x^2\right )}{3\ 2^{2/3} (6 c_1-1) \sqrt [3]{-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+\sqrt {4 \left (54 x^2 c_1-9 x^2\right ){}^3+\left (-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+54\right ){}^2}+54}}\right \},\left \{y(x)\to \frac {6 x c_1-x}{6 c_1-1}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+\sqrt {4 \left (54 x^2 c_1-9 x^2\right ){}^3+\left (-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+54\right ){}^2}+54}}{6 \sqrt [3]{2} (6 c_1-1)}+\frac {\left (1-i \sqrt {3}\right ) \left (54 x^2 c_1-9 x^2\right )}{3\ 2^{2/3} (6 c_1-1) \sqrt [3]{-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+\sqrt {4 \left (54 x^2 c_1-9 x^2\right ){}^3+\left (-1944 c_1{}^2 x^3+648 c_1 x^3-54 x^3+1944 c_1{}^2-648 c_1+54\right ){}^2}+54}}\right \}\right \}$ Maple : cpu = 0.597 (sec), leaf count = 1338

$\left \{ y \left ( x \right ) ={ \left ( \left ( \left ( -{\it \_C1}+80 \right ) {x}^{7}-160\,{x}^{4}+80\,x \right ) \sqrt [3]{4}\sqrt [3]{ \left ( -80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3} \right ) ^{2} \left ( -{\frac {1}{4}}+\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) {\it \_C1}}+ \left ( {{\it \_C1}}^{2}-80\,{\it \_C1} \right ) {x}^{8}+160\,{\it \_C1}\,{x}^{5}-80\,{\it \_C1}\,{x}^{2}+ \left ( {\it \_C1}\, \left ( -1+4\,\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) \left ( {\it \_C1}\,{x}^{6}-80\,{x}^{6}+160\,{x}^{3}-80 \right ) ^{2} \right ) ^{{\frac {2}{3}}} \right ) \left ( \left ( 80+ \left ( -{\it \_C1}+80 \right ) {x}^{6}-160\,{x}^{3} \right ) \sqrt [3]{4}\sqrt [3]{ \left ( -80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3} \right ) ^{2} \left ( -{\frac {1}{4}}+\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) {\it \_C1}}+{x}^{2} \left ( \left ( {{\it \_C1}}^{2}-80\,{\it \_C1} \right ) {x}^{8}+160\,{\it \_C1}\,{x}^{5}-80\,{\it \_C1}\,{x}^{2}+ \left ( {\it \_C1}\, \left ( -1+4\,\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) \left ( {\it \_C1}\,{x}^{6}-80\,{x}^{6}+160\,{x}^{3}-80 \right ) ^{2} \right ) ^{{\frac {2}{3}}} \right ) \right ) ^{-1}},y \left ( x \right ) ={ \left ( \left ( \left ( -2\,{\it \_C1}+160 \right ) {x}^{7}-320\,{x}^{4}+160\,x \right ) \sqrt [3]{4}\sqrt [3]{ \left ( -80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3} \right ) ^{2} \left ( -{\frac {1}{4}}+\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) {\it \_C1}}+ \left ( -i{\it \_C1}\, \left ( {\it \_C1}-80 \right ) {x}^{8}-160\,i{\it \_C1}\,{x}^{5}+80\,i{\it \_C1}\,{x}^{2}+i \left ( {\it \_C1}\, \left ( -1+4\,\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) \left ( {\it \_C1}\,{x}^{6}-80\,{x}^{6}+160\,{x}^{3}-80 \right ) ^{2} \right ) ^{{\frac {2}{3}}} \right ) \sqrt {3}+ \left ( -{{\it \_C1}}^{2}+80\,{\it \_C1} \right ) {x}^{8}-160\,{\it \_C1}\,{x}^{5}+80\,{\it \_C1}\,{x}^{2}- \left ( {\it \_C1}\, \left ( -1+4\,\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) \left ( {\it \_C1}\,{x}^{6}-80\,{x}^{6}+160\,{x}^{3}-80 \right ) ^{2} \right ) ^{{\frac {2}{3}}} \right ) \left ( \left ( 160+ \left ( -2\,{\it \_C1}+160 \right ) {x}^{6}-320\,{x}^{3} \right ) \sqrt [3]{4}\sqrt [3]{ \left ( -80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3} \right ) ^{2} \left ( -{\frac {1}{4}}+\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) {\it \_C1}}+{x}^{2} \left ( \left ( -i{\it \_C1}\, \left ( {\it \_C1}-80 \right ) {x}^{8}-160\,i{\it \_C1}\,{x}^{5}+80\,i{\it \_C1}\,{x}^{2}+i \left ( {\it \_C1}\, \left ( -1+4\,\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) \left ( {\it \_C1}\,{x}^{6}-80\,{x}^{6}+160\,{x}^{3}-80 \right ) ^{2} \right ) ^{{\frac {2}{3}}} \right ) \sqrt {3}+ \left ( -{{\it \_C1}}^{2}+80\,{\it \_C1} \right ) {x}^{8}-160\,{\it \_C1}\,{x}^{5}+80\,{\it \_C1}\,{x}^{2}- \left ( {\it \_C1}\, \left ( -1+4\,\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) \left ( {\it \_C1}\,{x}^{6}-80\,{x}^{6}+160\,{x}^{3}-80 \right ) ^{2} \right ) ^{{\frac {2}{3}}} \right ) \right ) ^{-1}},y \left ( x \right ) ={ \left ( \left ( \left ( 2\,{\it \_C1}-160 \right ) {x}^{7}+320\,{x}^{4}-160\,x \right ) \sqrt [3]{4}\sqrt [3]{ \left ( -80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3} \right ) ^{2} \left ( -{\frac {1}{4}}+\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) {\it \_C1}}+ \left ( -i{\it \_C1}\, \left ( {\it \_C1}-80 \right ) {x}^{8}-160\,i{\it \_C1}\,{x}^{5}+80\,i{\it \_C1}\,{x}^{2}+i \left ( {\it \_C1}\, \left ( -1+4\,\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) \left ( {\it \_C1}\,{x}^{6}-80\,{x}^{6}+160\,{x}^{3}-80 \right ) ^{2} \right ) ^{{\frac {2}{3}}} \right ) \sqrt {3}+ \left ( {{\it \_C1}}^{2}-80\,{\it \_C1} \right ) {x}^{8}+160\,{\it \_C1}\,{x}^{5}-80\,{\it \_C1}\,{x}^{2}+ \left ( {\it \_C1}\, \left ( -1+4\,\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) \left ( {\it \_C1}\,{x}^{6}-80\,{x}^{6}+160\,{x}^{3}-80 \right ) ^{2} \right ) ^{{\frac {2}{3}}} \right ) \left ( \left ( -160+ \left ( 2\,{\it \_C1}-160 \right ) {x}^{6}+320\,{x}^{3} \right ) \sqrt [3]{4}\sqrt [3]{ \left ( -80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3} \right ) ^{2} \left ( -{\frac {1}{4}}+\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) {\it \_C1}}+{x}^{2} \left ( \left ( -i{\it \_C1}\, \left ( {\it \_C1}-80 \right ) {x}^{8}-160\,i{\it \_C1}\,{x}^{5}+80\,i{\it \_C1}\,{x}^{2}+i \left ( {\it \_C1}\, \left ( -1+4\,\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) \left ( {\it \_C1}\,{x}^{6}-80\,{x}^{6}+160\,{x}^{3}-80 \right ) ^{2} \right ) ^{{\frac {2}{3}}} \right ) \sqrt {3}+ \left ( {{\it \_C1}}^{2}-80\,{\it \_C1} \right ) {x}^{8}+160\,{\it \_C1}\,{x}^{5}-80\,{\it \_C1}\,{x}^{2}+ \left ( {\it \_C1}\, \left ( -1+4\,\sqrt {{\frac {-5\,{x}^{6}+10\,{x}^{3}-5}{-80+ \left ( {\it \_C1}-80 \right ) {x}^{6}+160\,{x}^{3}}}} \right ) \left ( {\it \_C1}\,{x}^{6}-80\,{x}^{6}+160\,{x}^{3}-80 \right ) ^{2} \right ) ^{{\frac {2}{3}}} \right ) \right ) ^{-1}} \right \}$