ODE
\[ y''(x) \left (a+2 b x+c x^2\right )^{3/2}=f\left (\frac {x}{\sqrt {a+2 b x+c x^2}}\right ) \] ODE Classification
[[_2nd_order, _quadrature]]
Book solution method
TO DO
Mathematica ✓
cpu = 2.38103 (sec), leaf count = 64
\[\left \{\left \{y(x)\to \int _1^x \left (\int _1^{K[2]} \frac {f\left (\frac {K[1]}{\sqrt {2 b K[1]+c K[1]^2+a}}\right )}{(K[1] (c K[1]+2 b)+a)^{3/2}} \, dK[1]\right ) \, dK[2]+c_2 x+c_1\right \}\right \}\]
Maple ✓
cpu = 0.115 (sec), leaf count = 43
\[ \left \{ y \left ( x \right ) =\int \!\!\!\int \!{1f \left ( {x{\frac {1}{\sqrt {c{x}^{2}+2\,bx+a}}}} \right ) \left ( c{x}^{2}+2\,bx+a \right ) ^{-{\frac {3}{2}}}}\,{\rm d}x\,{\rm d}x+x{\it \_C1}+{\it \_C2} \right \} \] Mathematica raw input
DSolve[(a + 2*b*x + c*x^2)^(3/2)*y''[x] == f[x/Sqrt[a + 2*b*x + c*x^2]],y[x],x]
Mathematica raw output
{{y[x] -> C[1] + x*C[2] + Integrate[Integrate[f[K[1]/Sqrt[a + 2*b*K[1] + c*K[1]^2]]/(a + K[1]*(2*b + c*K[1]))^(3/2), {K[1], 1, K[2]}], {K[2], 1, x}]}}
Maple raw input
dsolve((c*x^2+2*b*x+a)^(3/2)*diff(diff(y(x),x),x) = f(x/(c*x^2+2*b*x+a)^(1/2)), y(x),'implicit')
Maple raw output
y(x) = Int(Int(f(x/(c*x^2+2*b*x+a)^(1/2))/(c*x^2+2*b*x+a)^(3/2),x),x)+x*_C1+_C2