ODE
\[ \left (x^2+1\right ) x^3 y'''(x)-2 \left (2 x^2+1\right ) x^2 y''(x)+2 \left (5 x^2+2\right ) x y'(x)-4 \left (3 x^2+1\right ) y(x)=0 \] ODE Classification
[[_3rd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.121642 (sec), leaf count = 46
\[\left \{\left \{y(x)\to \frac {1}{2} x \left (c_2 x^2-2 c_1 \left (x^2-3 x+1\right )-2 c_2 x+c_3 x+c_3 x \log (x)+c_2\right )\right \}\right \}\]
Maple ✓
cpu = 0.246 (sec), leaf count = 23
\[ \left \{ y \left ( x \right ) =x \left ( \ln \left ( x \right ) {\it \_C2}\,x+{\it \_C3}\,{x}^{2}+ \left ( {\it \_C1}+{\it \_C2} \right ) x+{\it \_C3} \right ) \right \} \] Mathematica raw input
DSolve[-4*(1 + 3*x^2)*y[x] + 2*x*(2 + 5*x^2)*y'[x] - 2*x^2*(1 + 2*x^2)*y''[x] + x^3*(1 + x^2)*y'''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (x*(-2*(1 - 3*x + x^2)*C[1] + C[2] - 2*x*C[2] + x^2*C[2] + x*C[3] + x*C[3]*Log[x]))/2}}
Maple raw input
dsolve(x^3*(x^2+1)*diff(diff(diff(y(x),x),x),x)-2*x^2*(2*x^2+1)*diff(diff(y(x),x),x)+2*x*(5*x^2+2)*diff(y(x),x)-4*(3*x^2+1)*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = x*(ln(x)*_C2*x+_C3*x^2+(_C1+_C2)*x+_C3)