##### 4.45.8 $$\left (x^2+1\right ) x^3 y'''(x)-2 \left (2 x^2+1\right ) x^2 y''(x)+2 \left (5 x^2+2\right ) x y'(x)-4 \left (3 x^2+1\right ) y(x)=0$$

ODE
$\left (x^2+1\right ) x^3 y'''(x)-2 \left (2 x^2+1\right ) x^2 y''(x)+2 \left (5 x^2+2\right ) x y'(x)-4 \left (3 x^2+1\right ) y(x)=0$ ODE Classiﬁcation

[[_3rd_order, _with_linear_symmetries]]

Book solution method
TO DO

Mathematica
cpu = 0.488672 (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 = 0.412 (sec), leaf count = 26

$[y \left (x \right ) = x^{2} \textit {\_C1} +\textit {\_C2} \,x^{2} \left (1+\ln \left (x \right )\right )+\textit {\_C3} \left (x^{3}+x \right )]$ 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))

Maple raw output

[y(x) = x^2*_C1+_C2*x^2*(1+ln(x))+_C3*(x^3+x)]