ODE
\[ y''(x)=a \left (y'(x)^2+1\right )^{3/2} (b+c x+y(x)) \] ODE Classification
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
Book solution method
TO DO
Mathematica ✗
cpu = 100.3 (sec), leaf count = 0 , could not solve
DSolve[Derivative[2][y][x] == a*(b + c*x + y[x])*(1 + Derivative[1][y][x]^2)^(3/2), y[x], x]
Maple ✓
cpu = 0.695 (sec), leaf count = 271
\[ \left \{ x-\int ^{cx+y \left ( x \right ) }\!{\frac {1}{ \left ( 4\,{c}^{2}+4 \right ) c} \left ( -2\, \left ( {\it \_f}\,b+1/2\,{{\it \_f}}^{2}-{\it \_C1} \right ) a\sqrt {-4\,{c}^{2} \left ( \left ( {\it \_f}\,b+1/2\,{{\it \_f}}^{2}-{\it \_C1} \right ) ^{2}{a}^{2}-{c}^{2}-1 \right ) }+4\,{c}^{2} \left ( \left ( {\it \_f}\,b+1/2\,{{\it \_f}}^{2}-{\it \_C1} \right ) ^{2}{a}^{2}-{c}^{2}-1 \right ) \right ) \left ( \left ( {\it \_f}\,b+{\frac {{{\it \_f}}^{2}}{2}}-{\it \_C1} \right ) ^{2}{a}^{2}-{c}^{2}-1 \right ) ^{-1}}{d{\it \_f}}-{\it \_C2}=0,x-\int ^{cx+y \left ( x \right ) }\!{\frac {1}{ \left ( 4\,{c}^{2}+4 \right ) c} \left ( 2\, \left ( {\it \_f}\,b+1/2\,{{\it \_f}}^{2}-{\it \_C1} \right ) a\sqrt {-4\,{c}^{2} \left ( \left ( {\it \_f}\,b+1/2\,{{\it \_f}}^{2}-{\it \_C1} \right ) ^{2}{a}^{2}-{c}^{2}-1 \right ) }+4\,{c}^{2} \left ( \left ( {\it \_f}\,b+1/2\,{{\it \_f}}^{2}-{\it \_C1} \right ) ^{2}{a}^{2}-{c}^{2}-1 \right ) \right ) \left ( \left ( {\it \_f}\,b+{\frac {{{\it \_f}}^{2}}{2}}-{\it \_C1} \right ) ^{2}{a}^{2}-{c}^{2}-1 \right ) ^{-1}}{d{\it \_f}}-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[y''[x] == a*(b + c*x + y[x])*(1 + y'[x]^2)^(3/2),y[x],x]
Mathematica raw output
DSolve[Derivative[2][y][x] == a*(b + c*x + y[x])*(1 + Derivative[1][y][x]^2)^(3/2), y[x], x]
Maple raw input
dsolve(diff(diff(y(x),x),x) = a*(b+c*x+y(x))*(1+diff(y(x),x)^2)^(3/2), y(x),'implicit')
Maple raw output
x-Intat(1/4*(-2*(_f*b+1/2*_f^2-_C1)*a*(-4*c^2*((_f*b+1/2*_f^2-_C1)^2*a^2-c^2-1))^(1/2)+4*c^2*((_f*b+1/2*_f^2-_C1)^2*a^2-c^2-1))/(c^2+1)/c/((_f*b+1/2*_f^2-_C1)^2*a^2-c^2-1),_f = c*x+y(x))-_C2 = 0, x-Intat(1/4*(2*(_f*b+1/2*_f^2-_C1)*a*(-4*c^2*((_f*b+1/2*_f^2-_C1)^2*a^2-c^2-1))^(1/2)+4*c^2*((_f*b+1/2*_f^2-_C1)^2*a^2-c^2-1))/(c^2+1)/c/((_f*b+1/2*_f^2-_C1)^2*a^2-c^2-1),_f = c*x+y(x))-_C2 = 0