ODE
\[ y''(x)=a \sqrt {b y(x)^2+y'(x)^2} \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.705606 (sec), leaf count = 74
\[\text {Solve}\left [x=\int _1^{y(x)}\frac {1}{\text {InverseFunction}\left [\int \frac {\text {$\#$1}}{K[1] \left (\frac {\text {$\#$1}^2}{K[1]^2}-a \sqrt {\frac {\text {$\#$1}^2}{K[1]^2}+b}\right )}d\frac {\text {$\#$1}}{K[1]}\& \right ][c_1-\log (K[1])]}dK[1]+c_2,y(x)\right ]\]
Maple ✓
cpu = 1.396 (sec), leaf count = 36
\[\left [y \left (x \right ) = {\mathrm e}^{\int \RootOf \left (x -\left (\int _{}^{\textit {\_Z}}\frac {1}{a \sqrt {\textit {\_f}^{2}+b}-\textit {\_f}^{2}}d \textit {\_f} \right )+\textit {\_C1} \right )d x +\textit {\_C2}}\right ]\] Mathematica raw input
DSolve[y''[x] == a*Sqrt[b*y[x]^2 + y'[x]^2],y[x],x]
Mathematica raw output
Solve[x == C[2] + Inactive[Integrate][InverseFunction[Inactive[Integrate][#1/(K[1]*(#1^2/K[1]^2 - a*Sqrt[b + #1^2/K[1]^2])), #1/K[1]] & ][C[1] - Log[K[1]]]^(-1), {K[1], 1, y[x]}], y[x]]
Maple raw input
dsolve(diff(diff(y(x),x),x) = a*(b*y(x)^2+diff(y(x),x)^2)^(1/2), y(x))
Maple raw output
[y(x) = exp(Int(RootOf(x-Intat(1/(a*(_f^2+b)^(1/2)-_f^2),_f = _Z)+_C1),x)+_C2)]