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 = 0.851392 (sec), leaf count = 70
\[\left \{\left \{y(x)\to c_2 \exp \left (\int _1^x\frac {e^{-\frac {4}{K[1]+1}} \left (4 c_1-16 a \text {Ei}\left (\frac {4}{K[1]+1}\right )+a e^{\frac {4}{K[1]+1}} (4 K[1]+5)\right )}{4 K[1]^2}dK[1]\right )\right \}\right \}\]
Maple ✓
cpu = 4.709 (sec), leaf count = 324
\[\left [y \left (x \right ) = \frac {{\mathrm e}^{-16 \expIntegral \left (1, -\frac {4 x}{x +1}\right ) \expIntegral \left (1, -\frac {4}{x +1}\right ) a \,{\mathrm e}^{-\frac {4}{x +1}-\frac {4 x}{x +1}}} {\mathrm e}^{-4 x \expIntegral \left (1, -\frac {4 x}{x +1}\right ) a \,{\mathrm e}^{-\frac {4}{x +1}-\frac {4 \left (x -1\right )}{x +1}}} {\mathrm e}^{4 \expIntegral \left (1, -\frac {4 x}{x +1}\right ) \textit {\_C2} \,{\mathrm e}^{-\frac {4}{x +1}-\frac {4 x}{x +1}}} {\mathrm e}^{-5 \expIntegral \left (1, -\frac {4 x}{x +1}\right ) a \,{\mathrm e}^{-\frac {4}{x +1}-\frac {4 \left (x -1\right )}{x +1}}} \left (x +1\right )^{a} {\mathrm e}^{\int \frac {8 \expIntegral \left (1, -\frac {4 x}{x +1}\right ) {\mathrm e}^{-\frac {4 x}{x +1}} a x}{\left (x +1\right )^{2}}d x} {\mathrm e}^{\int \frac {4 \expIntegral \left (1, -\frac {4 x}{x +1}\right ) {\mathrm e}^{-\frac {4 x}{x +1}} a \,x^{2}}{\left (x +1\right )^{2}}d x} {\mathrm e}^{-\frac {9 a}{4}} {\mathrm e}^{-4 \,{\mathrm e}^{-\frac {4}{x +1}} \expIntegral \left (1, -\frac {4}{x +1}\right ) a} {\mathrm e}^{-\frac {5 a}{4 x}} {\mathrm e}^{-\frac {4 \,{\mathrm e}^{-\frac {4}{x +1}} a \expIntegral \left (1, -\frac {4}{x +1}\right )}{x}} {\mathrm e}^{{\mathrm e}^{-\frac {4}{x +1}} \textit {\_C2}} {\mathrm e}^{\frac {{\mathrm e}^{-\frac {4}{x +1}} \textit {\_C2}}{x}}}{\textit {\_C1}}\right ]\] 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
{{y[x] -> E^Inactive[Integrate][(4*C[1] - 16*a*ExpIntegralEi[4/(1 + K[1])] + a*E^(4/(1 + K[1]))*(5 + 4*K[1]))/(4*E^(4/(1 + K[1]))*K[1]^2), {K[1], 1, x}]*C[2]}}
Maple raw input
dsolve(x*(x+1)^2*y(x)*diff(diff(y(x),x),x) = x*(x+1)^2*diff(y(x),x)^2-2*(x^2+1)*y(x)*diff(y(x),x)+a*(2+x)*y(x)^2, y(x))
Maple raw output
[y(x) = 1/exp(1/exp(4/(x+1))*Ei(1,-4*x/(x+1))*exp(-4*x/(x+1))*Ei(1,-4/(x+1))*a)^16/exp(x/exp(4/(x+1))*Ei(1,-4*x/(x+1))*exp(-4*(x-1)/(x+1))*a)^4*exp(1/exp(4/(x+1))*Ei(1,-4*x/(x+1))*exp(-4*x/(x+1))*_C2)^4/exp(1/exp(4/(x+1))*Ei(1,-4*x/(x+1))*exp(-4*(x-1)/(x+1))*a)^5*(x+1)^a*exp(a*Int(Ei(1,-4*x/(x+1))/exp(x/(x+1))^4*x/(x+1)^2,x))^8*exp(a*Int(Ei(1,-4*x/(x+1))/exp(x/(x+1))^4*x^2/(x+1)^2,x))^4/_C1*exp(-9/4*a)/exp(1/exp(4/(x+1))*Ei(1,-4/(x+1))*a)^4*exp(-5/4*a/x)/exp(1/x/exp(4/(x+1))*a*Ei(1,-4/(x+1)))^4*exp(1/exp(4/(x+1))*_C2)*exp(1/x/exp(4/(x+1))*_C2)]