ODE
\[ x \left (x^2+1\right ) y'''(x)+3 \left (2 x^2+1\right ) y''(x)-12 y(x)=0 \] ODE Classification
[[_3rd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.432699 (sec), leaf count = 69
\[\left \{\left \{y(x)\to \frac {1}{6} \left (c_1 \left (4 x^2+2\right )+2 c_2 x \sqrt {x^2+1}-3 c_3 x \sqrt {x^2+1} \tanh ^{-1}\left (\sqrt {x^2+1}\right )+3 c_3 x+\frac {c_3}{x}\right )\right \}\right \}\]
Maple ✓
cpu = 0.317 (sec), leaf count = 56
\[\left [y \left (x \right ) = \textit {\_C1} x \sqrt {x^{2}+1}+\frac {\textit {\_C2} \left (3 x^{2} \sqrt {x^{2}+1}\, \arctanh \left (\frac {1}{\sqrt {x^{2}+1}}\right )-3 x^{2}-1\right )}{x}+\textit {\_C3} \left (2 x^{2}+1\right )\right ]\] Mathematica raw input
DSolve[-12*y[x] + 3*(1 + 2*x^2)*y''[x] + x*(1 + x^2)*y'''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> ((2 + 4*x^2)*C[1] + 2*x*Sqrt[1 + x^2]*C[2] + C[3]/x + 3*x*C[3] - 3*x*Sqrt[1 + x^2]*ArcTanh[Sqrt[1 + x^2]]*C[3])/6}}
Maple raw input
dsolve(x*(x^2+1)*diff(diff(diff(y(x),x),x),x)+3*(2*x^2+1)*diff(diff(y(x),x),x)-12*y(x) = 0, y(x))
Maple raw output
[y(x) = _C1*x*(x^2+1)^(1/2)+_C2*(3*x^2*(x^2+1)^(1/2)*arctanh(1/(x^2+1)^(1/2))-3*x^2-1)/x+_C3*(2*x^2+1)]