\[ \boxed { {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) ={\frac { \left ( \left ( {x}^{2}+1 \right ) ^{3/2}{x}^{2}+ \left ( {x}^{2}+1 \right ) ^{3/2}+ \left ( y \left ( x \right ) \right ) ^{2} \left ( {x}^{2}+1 \right ) ^{3/2}+{x}^{2} \left ( y \left ( x \right ) \right ) ^{3}+ \left ( y \left ( x \right ) \right ) ^{3} \right ) x}{ \left ( {x}^{2}+1 \right ) ^{3}}}=0} \]
Mathematica: cpu = 0.183023 (sec), leaf count = 148 \[ \text {Solve}\left [-\frac {19}{3} \text {RootSum}\left [-19 \text {$\#$1}^3+6 \sqrt [3]{38} \text {$\#$1}-19\& ,\frac {\log \left (\frac {\frac {3 x y(x)}{\left (x^2+1\right )^2}+\frac {x}{\left (x^2+1\right )^{3/2}}}{\sqrt [3]{38} \sqrt [3]{\frac {x^3}{\left (x^2+1\right )^{9/2}}}}-\text {$\#$1}\right )}{2 \sqrt [3]{38}-19 \text {$\#$1}^2}\& \right ]=c_1+\frac {19^{2/3} \left (\frac {x^3}{\left (x^2+1\right )^{9/2}}\right )^{2/3} \left (x^2+1\right )^3 \log \left (x^2+1\right )}{9 \sqrt [3]{2} x^2},y(x)\right ] \]
Maple: cpu = 0.062 (sec), leaf count = 89 \[ \left \{ y \left ( x \right ) ={\frac {19\,{\it RootOf} \left ( -1296\, \int ^{{\it \_Z}}\! \left ( 361\,{{\it \_a}}^{3}-432\,{\it \_a}+432 \right ) ^{-1}{d{\it \_a}}+2\,\ln \left ( {x}^{2}+1 \right ) +3\,{\it \_C1} \right ) {x}^{2}-6\,{x}^{2}+19\,{\it RootOf} \left ( -1296\,\int ^ {{\it \_Z}}\! \left ( 361\,{{\it \_a}}^{3}-432\,{\it \_a}+432 \right ) ^ {-1}{d{\it \_a}}+2\,\ln \left ( {x}^{2}+1 \right ) +3\,{\it \_C1} \right ) -6}{18}{\frac {1}{\sqrt {{x}^{2}+1}}}} \right \} \]