##### 4.45.5 $$\left (x^2+1\right ) x^2 y'''(x)+8 x^3 y''(x)+10 x^2 y'(x)=3 x^2+2 x^2 \log (x)-1$$

ODE
$\left (x^2+1\right ) x^2 y'''(x)+8 x^3 y''(x)+10 x^2 y'(x)=3 x^2+2 x^2 \log (x)-1$ ODE Classiﬁcation

[[_3rd_order, _missing_y]]

Book solution method
TO DO

Mathematica
cpu = 0.623653 (sec), leaf count = 258

$\left \{\left \{y(x)\to \frac {1}{225} \left (-\frac {51 x}{x^2+1}-\frac {34 x}{\left (x^2+1\right )^2}-\frac {225 c_2 x}{x^2+1}-\frac {150 c_2 x}{\left (x^2+1\right )^2}-\frac {225 c_1}{4 \left (x^2+1\right )^2}-9 x+\frac {47}{x-i}+\frac {47}{x+i}+45 x \log (x)+60 i \log (-x+i)+\frac {171}{2} i \log (1-i x)-\frac {171}{2} i \log (1+i x)+\frac {30 \log (x)}{x-i}+\frac {30 \log (x)}{x+i}-\frac {30 i \log (x)}{(x-i)^2}+\frac {30 i \log (x)}{(x+i)^2}-60 i \log (x+i)+\frac {75 c_2}{x-i}+\frac {75 c_2}{x+i}+\frac {225}{2} i c_2 \log (1-i x)-\frac {225}{2} i c_2 \log (1+i x)-3 (17+75 c_2) \tan ^{-1}(x)\right )+c_3\right \}\right \}$

Maple
cpu = 0.334 (sec), leaf count = 230

$\left [y \left (x \right ) = -\frac {x}{25}+\frac {\left (-225 \textit {\_C1} +103\right ) x^{3}+\left (-675 \textit {\_C1} +69\right ) x -\frac {225 \textit {\_C2}}{4}}{225 \left (x^{2}+1\right )^{2}}+\frac {x \ln \left (x \right )}{5}+\frac {4 \ln \left (x \right ) \left (i \ln \left (i x +1\right ) x^{2}-i \ln \left (-i x +1\right ) x^{2}+i \ln \left (i x +1\right )-i \ln \left (-i x +1\right )-2 x \right )}{15 \left (x^{2}+1\right )}+\frac {2 i \ln \left (x \right ) \ln \left (i x +1\right )}{15}-\frac {2 i \ln \left (x \right ) \ln \left (-i x +1\right )}{15}-\frac {4 x}{15 \left (x^{2}+1\right )}-\frac {2 \ln \left (x \right ) \left (3 i \ln \left (i x +1\right ) x^{4}-3 i \ln \left (-i x +1\right ) x^{4}+6 i \ln \left (i x +1\right ) x^{2}-6 i \ln \left (-i x +1\right ) x^{2}-6 x^{3}+3 i \ln \left (i x +1\right )-3 i \ln \left (-i x +1\right )-10 x \right )}{15 \left (x^{2}+1\right )^{2}}+\textit {\_C3}\right ]$ Mathematica raw input

DSolve[10*x^2*y'[x] + 8*x^3*y''[x] + x^2*(1 + x^2)*y'''[x] == -1 + 3*x^2 + 2*x^2*Log[x],y[x],x]

Mathematica raw output

{{y[x] -> C[3] + (-9*x + 47/(-I + x) + 47/(I + x) - (34*x)/(1 + x^2)^2 - (51*x)/
(1 + x^2) - (225*C[1])/(4*(1 + x^2)^2) + (75*C[2])/(-I + x) + (75*C[2])/(I + x)
- (150*x*C[2])/(1 + x^2)^2 - (225*x*C[2])/(1 + x^2) - 3*ArcTan[x]*(17 + 75*C[2])
 + (60*I)*Log[I - x] + ((171*I)/2)*Log[1 - I*x] + ((225*I)/2)*C[2]*Log[1 - I*x]
- ((171*I)/2)*Log[1 + I*x] - ((225*I)/2)*C[2]*Log[1 + I*x] + 45*x*Log[x] - ((30*
I)*Log[x])/(-I + x)^2 + (30*Log[x])/(-I + x) + ((30*I)*Log[x])/(I + x)^2 + (30*L
og[x])/(I + x) - (60*I)*Log[I + x])/225}}

Maple raw input

dsolve(x^2*(x^2+1)*diff(diff(diff(y(x),x),x),x)+8*x^3*diff(diff(y(x),x),x)+10*x^2*diff(y(x),x) = 2*x^2*ln(x)-1+3*x^2, y(x))

Maple raw output

[y(x) = -1/25*x+1/225*((-225*_C1+103)*x^3+(-675*_C1+69)*x-225/4*_C2)/(x^2+1)^2+1
/5*x*ln(x)+4/15*ln(x)*(I*ln(1+I*x)*x^2-I*ln(1-I*x)*x^2+I*ln(1+I*x)-I*ln(1-I*x)-2
*x)/(x^2+1)+2/15*I*ln(x)*ln(1+I*x)-2/15*I*ln(x)*ln(1-I*x)-4/15*x/(x^2+1)-2/15*ln
(x)*(3*I*ln(1+I*x)*x^4-3*I*ln(1-I*x)*x^4+6*I*ln(1+I*x)*x^2-6*I*ln(1-I*x)*x^2-6*x
^3+3*I*ln(1+I*x)-3*I*ln(1-I*x)-10*x)/(x^2+1)^2+_C3]