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 = 0.283479 (sec), leaf count = 66
\[\left \{\left \{y(x)\to \int _1^x\int _1^{K[2]}\frac {f\left (\frac {K[1]}{\sqrt {c K[1]^2+2 b K[1]+a}}\right )}{\left (c K[1]^2+2 b K[1]+a\right )^{3/2}}dK[1]dK[2]+c_2 x+c_1\right \}\right \}\]
Maple ✓
cpu = 0.737 (sec), leaf count = 43
\[\left [y \left (x \right ) = \int \int \frac {f \left (\frac {x}{\sqrt {c \,x^{2}+2 b x +a}}\right )}{\left (c \,x^{2}+2 b x +a \right )^{\frac {3}{2}}}d x d x +\textit {\_C1} x +\textit {\_C2}\right ]\] 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] + Inactive[Integrate][Inactive[Integrate][f[K[1]/Sqrt[a + 2*b*K[1] + c*K[1]^2]]/(a + 2*b*K[1] + c*K[1]^2)^(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))
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)+_C1*x+_C2]