ODE
\[ x (x+1)^2 y(x) y''(x)=a (x+2) y(x)^2-2 \left (x^2+1\right ) y(x) y'(x)+x (x+1)^2 y'(x)^2 \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✗
cpu = 599.998 (sec), leaf count = 0 , timed out
$Aborted
Maple ✓
cpu = 1.267 (sec), leaf count = 168
\[ \left \{ -16\,{{\rm e}^{4\, \left ( 1+x \right ) ^{-1}}}\int \!{\frac { \left ( 2+x \right ) x}{ \left ( 1+x \right ) ^{2}}{\it Ei} \left ( 1,-4\,{\frac {x}{1+x}} \right ) {{\rm e}^{-4\,{\frac {x}{1+x}}}}}\,{\rm d}xax+16\,{\it Ei} \left ( 1,-4\,{\frac {x}{1+x}} \right ) a \left ( x+5/4 \right ) x{{\rm e}^{{\frac {4-4\,x}{1+x}}}}+64\,{{\rm e}^{-4\,{\frac {x}{1+x}}}}x \left ( a{\it Ei} \left ( 1,-4\, \left ( 1+x \right ) ^{-1} \right ) -{\it \_C2}/4 \right ) {\it Ei} \left ( 1,-4\,{\frac {x}{1+x}} \right ) +16\,a \left ( 1+x \right ) {\it Ei} \left ( 1,-4\, \left ( 1+x \right ) ^{-1} \right ) + \left ( -4\,a\ln \left ( 1+x \right ) x+4\,x\ln \left ( y \left ( x \right ) \right ) + \left ( 9\,a+{\it \_C1} \right ) x+5\,a \right ) {{\rm e}^{4\, \left ( 1+x \right ) ^{-1}}}-4\,{\it \_C2}\, \left ( 1+x \right ) =0 \right \} \] Mathematica raw input
DSolve[x*(1 + x)^2*y[x]*y''[x] == a*(2 + x)*y[x]^2 - 2*(1 + x^2)*y[x]*y'[x] + x*(1 + x)^2*y'[x]^2,y[x],x]
Mathematica raw output
$Aborted
Maple raw input
dsolve(x*(1+x)^2*y(x)*diff(diff(y(x),x),x) = x*(1+x)^2*diff(y(x),x)^2-2*(x^2+1)*y(x)*diff(y(x),x)+a*(2+x)*y(x)^2, y(x),'implicit')
Maple raw output
-16*exp(4/(1+x))*Int(Ei(1,-4*x/(1+x))*exp(-4*x/(1+x))*x*(2+x)/(1+x)^2,x)*a*x+16*Ei(1,-4*x/(1+x))*a*(x+5/4)*x*exp((4-4*x)/(1+x))+64*exp(-4*x/(1+x))*x*(a*Ei(1,-4/(1+x))-1/4*_C2)*Ei(1,-4*x/(1+x))+16*a*(1+x)*Ei(1,-4/(1+x))+(-4*a*ln(1+x)*x+4*x*ln(y(x))+(9*a+_C1)*x+5*a)*exp(4/(1+x))-4*_C2*(1+x) = 0