\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) -a \left ( c+bx+y \left ( x \right ) \right ) \left ( \left ( {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) \right ) ^{2}+1 \right ) ^{3/2}=0} \]
Mathematica: cpu = 100.341242 (sec), leaf count = 33 \[ \text {DSolve}\left [y''(x)-a \left (y'(x)^2+1\right )^{3/2} (b x+c+y(x))=0,y(x),x\right ] \]
Maple: cpu = 0.499 (sec), leaf count = 771 \[ \left \{ y \left ( x \right ) =-bx+{\it RootOf} \left ( -x+\int ^{{\it \_Z}}\!{\frac {1}{ \left ( {{\it \_f}}^{4}{a}^{2}+4\,{{\it \_f}}^{3}{a} ^{2}c+4\,{{\it \_f}}^{2}{a}^{2}{c}^{2}-4\,{\it \_C1}\,{{\it \_f}}^{2}{ a}^{2}-8\,{\it \_C1}\,{\it \_f}\,{a}^{2}c+4\,{{\it \_C1}}^{2}{a}^{2}-4 \,{b}^{2}-4 \right ) \left ( {b}^{2}+1 \right ) b} \left ( 4\,{a}^{2}{b}^ {2}{c}^{2}{{\it \_f}}^{2}+4\,{a}^{2}{b}^{2}c{{\it \_f}}^{3}+{a}^{2}{b} ^{2}{{\it \_f}}^{4}-8\,{\it \_C1}\,{a}^{2}{b}^{2}c{\it \_f}-4\,{\it \_C1}\,{a}^{2}{b}^{2}{{\it \_f}}^{2}+4\,{{\it \_C1}}^{2}{a}^{2}{b}^{2} -2\,\sqrt {-{b}^{2} \left ( {{\it \_f}}^{4}{a}^{2}+4\,{{\it \_f}}^{3}{a }^{2}c+4\,{{\it \_f}}^{2}{a}^{2}{c}^{2}-4\,{\it \_C1}\,{{\it \_f}}^{2} {a}^{2}-8\,{\it \_C1}\,{\it \_f}\,{a}^{2}c+4\,{{\it \_C1}}^{2}{a}^{2}- 4\,{b}^{2}-4 \right ) }ac{\it \_f}-\sqrt {-{b}^{2} \left ( {{\it \_f}}^{ 4}{a}^{2}+4\,{{\it \_f}}^{3}{a}^{2}c+4\,{{\it \_f}}^{2}{a}^{2}{c}^{2}- 4\,{\it \_C1}\,{{\it \_f}}^{2}{a}^{2}-8\,{\it \_C1}\,{\it \_f}\,{a}^{2 }c+4\,{{\it \_C1}}^{2}{a}^{2}-4\,{b}^{2}-4 \right ) }a{{\it \_f}}^{2}-4 \,{b}^{4}+2\,\sqrt {-{b}^{2} \left ( {{\it \_f}}^{4}{a}^{2}+4\,{{\it \_f}}^{3}{a}^{2}c+4\,{{\it \_f}}^{2}{a}^{2}{c}^{2}-4\,{\it \_C1}\,{{ \it \_f}}^{2}{a}^{2}-8\,{\it \_C1}\,{\it \_f}\,{a}^{2}c+4\,{{\it \_C1} }^{2}{a}^{2}-4\,{b}^{2}-4 \right ) }{\it \_C1}\,a-4\,{b}^{2} \right ) }{ d{\it \_f}}+{\it \_C2} \right ) ,y \left ( x \right ) =-bx+{\it RootOf} \left ( -x+\int ^{{\it \_Z}}\!-{\frac {1}{ \left ( {{\it \_f}}^{4}{a}^{ 2}+4\,{{\it \_f}}^{3}{a}^{2}c+4\,{{\it \_f}}^{2}{a}^{2}{c}^{2}-4\,{ \it \_C1}\,{{\it \_f}}^{2}{a}^{2}-8\,{\it \_C1}\,{\it \_f}\,{a}^{2}c+4 \,{{\it \_C1}}^{2}{a}^{2}-4\,{b}^{2}-4 \right ) \left ( {b}^{2}+1 \right ) b} \left ( -{a}^{2}{b}^{2}{{\it \_f}}^{4}-4\,{a}^{2}{b}^{2}c{{ \it \_f}}^{3}-4\,{a}^{2}{b}^{2}{c}^{2}{{\it \_f}}^{2}+4\,{\it \_C1}\,{ a}^{2}{b}^{2}{{\it \_f}}^{2}+8\,{\it \_C1}\,{a}^{2}{b}^{2}c{\it \_f}-4 \,{{\it \_C1}}^{2}{a}^{2}{b}^{2}-\sqrt {-{b}^{2} \left ( {{\it \_f}}^{4 }{a}^{2}+4\,{{\it \_f}}^{3}{a}^{2}c+4\,{{\it \_f}}^{2}{a}^{2}{c}^{2}-4 \,{\it \_C1}\,{{\it \_f}}^{2}{a}^{2}-8\,{\it \_C1}\,{\it \_f}\,{a}^{2} c+4\,{{\it \_C1}}^{2}{a}^{2}-4\,{b}^{2}-4 \right ) }a{{\it \_f}}^{2}-2 \,\sqrt {-{b}^{2} \left ( {{\it \_f}}^{4}{a}^{2}+4\,{{\it \_f}}^{3}{a}^ {2}c+4\,{{\it \_f}}^{2}{a}^{2}{c}^{2}-4\,{\it \_C1}\,{{\it \_f}}^{2}{a }^{2}-8\,{\it \_C1}\,{\it \_f}\,{a}^{2}c+4\,{{\it \_C1}}^{2}{a}^{2}-4 \,{b}^{2}-4 \right ) }ac{\it \_f}+4\,{b}^{4}+2\,\sqrt {-{b}^{2} \left ( {{\it \_f}}^{4}{a}^{2}+4\,{{\it \_f}}^{3}{a}^{2}c+4\,{{\it \_f}}^{2}{a }^{2}{c}^{2}-4\,{\it \_C1}\,{{\it \_f}}^{2}{a}^{2}-8\,{\it \_C1}\,{ \it \_f}\,{a}^{2}c+4\,{{\it \_C1}}^{2}{a}^{2}-4\,{b}^{2}-4 \right ) }{ \it \_C1}\,a+4\,{b}^{2} \right ) }{d{\it \_f}}+{\it \_C2} \right ) \right \} \]