#### 2.875   ODE No. 875

$y'(x)=\frac {x^5 \left (-\sqrt {x^2+y(x)^2}\right )+x^4 y(x) \sqrt {x^2+y(x)^2}+x y(x)+y(x)}{x (x+1)}$ Mathematica : cpu = 0.239628 (sec), leaf count = 497

$\left \{\left \{y(x)\to \frac {x-2 \sqrt {x^2 \tanh ^2\left (\frac {1}{12} \left (-12 \sqrt {2} c_1-3 \sqrt {2} x^4+4 \sqrt {2} x^3-6 \sqrt {2} x^2+12 \sqrt {2} x-12 \sqrt {2} \log (x+1)+25 \sqrt {2}\right )\right )-x^2 \tanh ^4\left (\frac {1}{12} \left (-12 \sqrt {2} c_1-3 \sqrt {2} x^4+4 \sqrt {2} x^3-6 \sqrt {2} x^2+12 \sqrt {2} x-12 \sqrt {2} \log (x+1)+25 \sqrt {2}\right )\right )}}{2 \tanh ^2\left (\frac {1}{12} \left (-12 \sqrt {2} c_1-3 \sqrt {2} x^4+4 \sqrt {2} x^3-6 \sqrt {2} x^2+12 \sqrt {2} x-12 \sqrt {2} \log (x+1)+25 \sqrt {2}\right )\right )-1}\right \},\left \{y(x)\to \frac {2 \sqrt {x^2 \tanh ^2\left (\frac {1}{12} \left (-12 \sqrt {2} c_1-3 \sqrt {2} x^4+4 \sqrt {2} x^3-6 \sqrt {2} x^2+12 \sqrt {2} x-12 \sqrt {2} \log (x+1)+25 \sqrt {2}\right )\right )-x^2 \tanh ^4\left (\frac {1}{12} \left (-12 \sqrt {2} c_1-3 \sqrt {2} x^4+4 \sqrt {2} x^3-6 \sqrt {2} x^2+12 \sqrt {2} x-12 \sqrt {2} \log (x+1)+25 \sqrt {2}\right )\right )}+x}{2 \tanh ^2\left (\frac {1}{12} \left (-12 \sqrt {2} c_1-3 \sqrt {2} x^4+4 \sqrt {2} x^3-6 \sqrt {2} x^2+12 \sqrt {2} x-12 \sqrt {2} \log (x+1)+25 \sqrt {2}\right )\right )-1}\right \}\right \}$ Maple : cpu = 0.174 (sec), leaf count = 73

$\left \{ \ln \left ( 2\,{\frac {x \left ( \sqrt {2\, \left ( y \left ( x \right ) \right ) ^{2}+2\,{x}^{2}}+y \left ( x \right ) +x \right ) }{y \left ( x \right ) -x}} \right ) +\sqrt {2}\ln \left ( 1+x \right ) +{\frac { \left ( 3\,{x}^{4}-4\,{x}^{3}+6\,{x}^{2}-12\,x \right ) \sqrt {2}}{12}}-{\it \_C1}-\ln \left ( x \right ) =0 \right \}$