\[ y'(x)=\frac {y(x)^{3/2}}{x^2-2 x y(x)+y(x)^2+y(x)^{3/2}} \] ✓ Mathematica : cpu = 0.291109 (sec), leaf count = 2633
DSolve[Derivative[1][y][x] == y[x]^(3/2)/(x^2 - 2*x*y[x] + y[x]^(3/2) + y[x]^2),y[x],x]
\[\left \{\left \{y(x)\to \frac {2}{3} \left (x+e^{c_1}+2 e^{2 c_1}\right )-\frac {1}{3} \sqrt [3]{x^3+3 e^{c_1} x^2-12 e^{2 c_1} x^2+3 e^{2 c_1} x+12 e^{3 c_1} x+48 e^{4 c_1} x+e^{3 c_1}-30 e^{4 c_1}-96 e^{5 c_1}-64 e^{6 c_1}+6 \sqrt {3} \sqrt {-e^{4 c_1} x^3-3 e^{5 c_1} x^2+8 e^{6 c_1} x^2-3 e^{6 c_1} x-20 e^{7 c_1} x-16 e^{8 c_1} x-e^{7 c_1}-e^{8 c_1}}}+\frac {-x^2-2 e^{c_1} x+8 e^{2 c_1} x-e^{2 c_1}-16 e^{3 c_1}-16 e^{4 c_1}}{3 \sqrt [3]{x^3+3 e^{c_1} x^2-12 e^{2 c_1} x^2+3 e^{2 c_1} x+12 e^{3 c_1} x+48 e^{4 c_1} x+e^{3 c_1}-30 e^{4 c_1}-96 e^{5 c_1}-64 e^{6 c_1}+6 \sqrt {3} \sqrt {-e^{4 c_1} x^3-3 e^{5 c_1} x^2+8 e^{6 c_1} x^2-3 e^{6 c_1} x-20 e^{7 c_1} x-16 e^{8 c_1} x-e^{7 c_1}-e^{8 c_1}}}}\right \},\left \{y(x)\to \frac {2}{3} \left (x+e^{c_1}+2 e^{2 c_1}\right )+\frac {1}{6} \left (1-i \sqrt {3}\right ) \sqrt [3]{x^3+3 e^{c_1} x^2-12 e^{2 c_1} x^2+3 e^{2 c_1} x+12 e^{3 c_1} x+48 e^{4 c_1} x+e^{3 c_1}-30 e^{4 c_1}-96 e^{5 c_1}-64 e^{6 c_1}+6 \sqrt {3} \sqrt {-e^{4 c_1} x^3-3 e^{5 c_1} x^2+8 e^{6 c_1} x^2-3 e^{6 c_1} x-20 e^{7 c_1} x-16 e^{8 c_1} x-e^{7 c_1}-e^{8 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \left (-x^2-2 e^{c_1} x+8 e^{2 c_1} x-e^{2 c_1}-16 e^{3 c_1}-16 e^{4 c_1}\right )}{6 \sqrt [3]{x^3+3 e^{c_1} x^2-12 e^{2 c_1} x^2+3 e^{2 c_1} x+12 e^{3 c_1} x+48 e^{4 c_1} x+e^{3 c_1}-30 e^{4 c_1}-96 e^{5 c_1}-64 e^{6 c_1}+6 \sqrt {3} \sqrt {-e^{4 c_1} x^3-3 e^{5 c_1} x^2+8 e^{6 c_1} x^2-3 e^{6 c_1} x-20 e^{7 c_1} x-16 e^{8 c_1} x-e^{7 c_1}-e^{8 c_1}}}}\right \},\left \{y(x)\to \frac {2}{3} \left (x+e^{c_1}+2 e^{2 c_1}\right )+\frac {1}{6} \left (1+i \sqrt {3}\right ) \sqrt [3]{x^3+3 e^{c_1} x^2-12 e^{2 c_1} x^2+3 e^{2 c_1} x+12 e^{3 c_1} x+48 e^{4 c_1} x+e^{3 c_1}-30 e^{4 c_1}-96 e^{5 c_1}-64 e^{6 c_1}+6 \sqrt {3} \sqrt {-e^{4 c_1} x^3-3 e^{5 c_1} x^2+8 e^{6 c_1} x^2-3 e^{6 c_1} x-20 e^{7 c_1} x-16 e^{8 c_1} x-e^{7 c_1}-e^{8 c_1}}}-\frac {\left (1-i \sqrt {3}\right ) \left (-x^2-2 e^{c_1} x+8 e^{2 c_1} x-e^{2 c_1}-16 e^{3 c_1}-16 e^{4 c_1}\right )}{6 \sqrt [3]{x^3+3 e^{c_1} x^2-12 e^{2 c_1} x^2+3 e^{2 c_1} x+12 e^{3 c_1} x+48 e^{4 c_1} x+e^{3 c_1}-30 e^{4 c_1}-96 e^{5 c_1}-64 e^{6 c_1}+6 \sqrt {3} \sqrt {-e^{4 c_1} x^3-3 e^{5 c_1} x^2+8 e^{6 c_1} x^2-3 e^{6 c_1} x-20 e^{7 c_1} x-16 e^{8 c_1} x-e^{7 c_1}-e^{8 c_1}}}}\right \},\left \{y(x)\to \frac {2}{3} \left (x-e^{c_1}+2 e^{2 c_1}\right )-\frac {1}{3} \sqrt [3]{x^3-3 e^{c_1} x^2-12 e^{2 c_1} x^2+3 e^{2 c_1} x-12 e^{3 c_1} x+48 e^{4 c_1} x-e^{3 c_1}-30 e^{4 c_1}+96 e^{5 c_1}-64 e^{6 c_1}+6 \sqrt {3} \sqrt {-e^{4 c_1} x^3+3 e^{5 c_1} x^2+8 e^{6 c_1} x^2-3 e^{6 c_1} x+20 e^{7 c_1} x-16 e^{8 c_1} x+e^{7 c_1}-e^{8 c_1}}}+\frac {-x^2+2 e^{c_1} x+8 e^{2 c_1} x-e^{2 c_1}+16 e^{3 c_1}-16 e^{4 c_1}}{3 \sqrt [3]{x^3-3 e^{c_1} x^2-12 e^{2 c_1} x^2+3 e^{2 c_1} x-12 e^{3 c_1} x+48 e^{4 c_1} x-e^{3 c_1}-30 e^{4 c_1}+96 e^{5 c_1}-64 e^{6 c_1}+6 \sqrt {3} \sqrt {-e^{4 c_1} x^3+3 e^{5 c_1} x^2+8 e^{6 c_1} x^2-3 e^{6 c_1} x+20 e^{7 c_1} x-16 e^{8 c_1} x+e^{7 c_1}-e^{8 c_1}}}}\right \},\left \{y(x)\to \frac {2}{3} \left (x-e^{c_1}+2 e^{2 c_1}\right )+\frac {1}{6} \left (1-i \sqrt {3}\right ) \sqrt [3]{x^3-3 e^{c_1} x^2-12 e^{2 c_1} x^2+3 e^{2 c_1} x-12 e^{3 c_1} x+48 e^{4 c_1} x-e^{3 c_1}-30 e^{4 c_1}+96 e^{5 c_1}-64 e^{6 c_1}+6 \sqrt {3} \sqrt {-e^{4 c_1} x^3+3 e^{5 c_1} x^2+8 e^{6 c_1} x^2-3 e^{6 c_1} x+20 e^{7 c_1} x-16 e^{8 c_1} x+e^{7 c_1}-e^{8 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \left (-x^2+2 e^{c_1} x+8 e^{2 c_1} x-e^{2 c_1}+16 e^{3 c_1}-16 e^{4 c_1}\right )}{6 \sqrt [3]{x^3-3 e^{c_1} x^2-12 e^{2 c_1} x^2+3 e^{2 c_1} x-12 e^{3 c_1} x+48 e^{4 c_1} x-e^{3 c_1}-30 e^{4 c_1}+96 e^{5 c_1}-64 e^{6 c_1}+6 \sqrt {3} \sqrt {-e^{4 c_1} x^3+3 e^{5 c_1} x^2+8 e^{6 c_1} x^2-3 e^{6 c_1} x+20 e^{7 c_1} x-16 e^{8 c_1} x+e^{7 c_1}-e^{8 c_1}}}}\right \},\left \{y(x)\to \frac {2}{3} \left (x-e^{c_1}+2 e^{2 c_1}\right )+\frac {1}{6} \left (1+i \sqrt {3}\right ) \sqrt [3]{x^3-3 e^{c_1} x^2-12 e^{2 c_1} x^2+3 e^{2 c_1} x-12 e^{3 c_1} x+48 e^{4 c_1} x-e^{3 c_1}-30 e^{4 c_1}+96 e^{5 c_1}-64 e^{6 c_1}+6 \sqrt {3} \sqrt {-e^{4 c_1} x^3+3 e^{5 c_1} x^2+8 e^{6 c_1} x^2-3 e^{6 c_1} x+20 e^{7 c_1} x-16 e^{8 c_1} x+e^{7 c_1}-e^{8 c_1}}}-\frac {\left (1-i \sqrt {3}\right ) \left (-x^2+2 e^{c_1} x+8 e^{2 c_1} x-e^{2 c_1}+16 e^{3 c_1}-16 e^{4 c_1}\right )}{6 \sqrt [3]{x^3-3 e^{c_1} x^2-12 e^{2 c_1} x^2+3 e^{2 c_1} x-12 e^{3 c_1} x+48 e^{4 c_1} x-e^{3 c_1}-30 e^{4 c_1}+96 e^{5 c_1}-64 e^{6 c_1}+6 \sqrt {3} \sqrt {-e^{4 c_1} x^3+3 e^{5 c_1} x^2+8 e^{6 c_1} x^2-3 e^{6 c_1} x+20 e^{7 c_1} x-16 e^{8 c_1} x+e^{7 c_1}-e^{8 c_1}}}}\right \}\right \}\] ✓ Maple : cpu = 0.116 (sec), leaf count = 44
dsolve(diff(y(x),x) = y(x)^(3/2)/(y(x)^(3/2)+x^2-2*x*y(x)+y(x)^2),y(x))
\[\frac {2 \sqrt {y \left (x \right )}}{y \left (x \right )-x}+\frac {1}{y \left (x \right )-x}-\frac {2 x}{\left (y \left (x \right )-x \right ) \sqrt {y \left (x \right )}}-c_{1} = 0\]