ODE No. 1656

\[ y''(x)-a \left (y'(x)^2+1\right )^{3/2} (b x+c+y(x))=0 \] ✓ Mathematica : cpu = 37.0914 (sec), leaf count = 9706

DSolve[-(a*(c + b*x + y[x])*(1 + Derivative[1][y][x]^2)^(3/2)) + Derivative[2][y][x] == 0,y[x],x]
 

\[\left \{\text {Solve}\left [\int _1^x\left (\frac {1}{b^2+1}-\frac {b \sqrt {-a^2 \left (c^2+2 b K[1] c+2 y(x) c+b^2 K[1]^2+y(x)^2-2 c_1+2 b K[1] y(x)\right ){}^2 \left (a^2 c^4+4 a^2 b K[1] c^3+4 a^2 y(x) c^3+6 a^2 b^2 K[1]^2 c^2+6 a^2 y(x)^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[1] y(x) c^2+4 a^2 b^3 K[1]^3 c+4 a^2 y(x)^3 c+12 a^2 b K[1] y(x)^2 c-8 a^2 b c_1 K[1] c+12 a^2 b^2 K[1]^2 y(x) c-8 a^2 c_1 y(x) c+a^2 b^4 K[1]^4+a^2 y(x)^4+4 a^2 b K[1] y(x)^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[1]^2+6 a^2 b^2 K[1]^2 y(x)^2-4 a^2 c_1 y(x)^2+4 a^2 b^3 K[1]^3 y(x)-8 a^2 b c_1 K[1] y(x)-4\right )}}{\left (b^2+1\right ) \left (a^2 c^4+4 a^2 b K[1] c^3+4 a^2 y(x) c^3+6 a^2 b^2 K[1]^2 c^2+6 a^2 y(x)^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[1] y(x) c^2+4 a^2 b^3 K[1]^3 c+4 a^2 y(x)^3 c+12 a^2 b K[1] y(x)^2 c-8 a^2 b c_1 K[1] c+12 a^2 b^2 K[1]^2 y(x) c-8 a^2 c_1 y(x) c+a^2 b^4 K[1]^4+a^2 y(x)^4+4 a^2 b K[1] y(x)^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[1]^2+6 a^2 b^2 K[1]^2 y(x)^2-4 a^2 c_1 y(x)^2+4 a^2 b^3 K[1]^3 y(x)-8 a^2 b c_1 K[1] y(x)-4\right )}\right )dK[1]+\int _1^{y(x)}\left (\frac {\sqrt {-a^2 \left (c^2+2 b x c+2 K[2] c+b^2 x^2+K[2]^2-2 c_1+2 b x K[2]\right ){}^2 \left (a^2 c^4+4 a^2 b x c^3+4 a^2 K[2] c^3+6 a^2 b^2 x^2 c^2+6 a^2 K[2]^2 c^2-4 a^2 c_1 c^2+12 a^2 b x K[2] c^2+4 a^2 b^3 x^3 c+4 a^2 K[2]^3 c+12 a^2 b x K[2]^2 c-8 a^2 b x c_1 c+12 a^2 b^2 x^2 K[2] c-8 a^2 c_1 K[2] c+a^2 b^4 x^4+a^2 K[2]^4+4 a^2 b x K[2]^3-4 b^2+4 a^2 c_1{}^2+6 a^2 b^2 x^2 K[2]^2-4 a^2 c_1 K[2]^2-4 a^2 b^2 x^2 c_1+4 a^2 b^3 x^3 K[2]-8 a^2 b x c_1 K[2]-4\right )}}{\left (-b^2-1\right ) \left (a^2 c^4+4 a^2 b x c^3+4 a^2 K[2] c^3+6 a^2 b^2 x^2 c^2+6 a^2 K[2]^2 c^2-4 a^2 c_1 c^2+12 a^2 b x K[2] c^2+4 a^2 b^3 x^3 c+4 a^2 K[2]^3 c+12 a^2 b x K[2]^2 c-8 a^2 b x c_1 c+12 a^2 b^2 x^2 K[2] c-8 a^2 c_1 K[2] c+a^2 b^4 x^4+a^2 K[2]^4+4 a^2 b x K[2]^3-4 b^2+4 a^2 c_1{}^2+6 a^2 b^2 x^2 K[2]^2-4 a^2 c_1 K[2]^2-4 a^2 b^2 x^2 c_1+4 a^2 b^3 x^3 K[2]-8 a^2 b x c_1 K[2]-4\right )}-\frac {\int _1^x\left (\frac {b \left (4 a^2 c^3+12 a^2 b K[1] c^2+12 a^2 K[2] c^2+12 a^2 b^2 K[1]^2 c+12 a^2 K[2]^2 c-8 a^2 c_1 c+24 a^2 b K[1] K[2] c+4 a^2 b^3 K[1]^3+4 a^2 K[2]^3+12 a^2 b K[1] K[2]^2-8 a^2 b c_1 K[1]+12 a^2 b^2 K[1]^2 K[2]-8 a^2 c_1 K[2]\right ) \sqrt {-a^2 \left (c^2+2 b K[1] c+2 K[2] c+b^2 K[1]^2+K[2]^2-2 c_1+2 b K[1] K[2]\right ){}^2 \left (a^2 c^4+4 a^2 b K[1] c^3+4 a^2 K[2] c^3+6 a^2 b^2 K[1]^2 c^2+6 a^2 K[2]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[1] K[2] c^2+4 a^2 b^3 K[1]^3 c+4 a^2 K[2]^3 c+12 a^2 b K[1] K[2]^2 c-8 a^2 b c_1 K[1] c+12 a^2 b^2 K[1]^2 K[2] c-8 a^2 c_1 K[2] c+a^2 b^4 K[1]^4+a^2 K[2]^4+4 a^2 b K[1] K[2]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[1]^2+6 a^2 b^2 K[1]^2 K[2]^2-4 a^2 c_1 K[2]^2+4 a^2 b^3 K[1]^3 K[2]-8 a^2 b c_1 K[1] K[2]-4\right )}}{\left (b^2+1\right ) \left (a^2 c^4+4 a^2 b K[1] c^3+4 a^2 K[2] c^3+6 a^2 b^2 K[1]^2 c^2+6 a^2 K[2]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[1] K[2] c^2+4 a^2 b^3 K[1]^3 c+4 a^2 K[2]^3 c+12 a^2 b K[1] K[2]^2 c-8 a^2 b c_1 K[1] c+12 a^2 b^2 K[1]^2 K[2] c-8 a^2 c_1 K[2] c+a^2 b^4 K[1]^4+a^2 K[2]^4+4 a^2 b K[1] K[2]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[1]^2+6 a^2 b^2 K[1]^2 K[2]^2-4 a^2 c_1 K[2]^2+4 a^2 b^3 K[1]^3 K[2]-8 a^2 b c_1 K[1] K[2]-4\right ){}^2}-\frac {b \left (-\left (c^2+2 b K[1] c+2 K[2] c+b^2 K[1]^2+K[2]^2-2 c_1+2 b K[1] K[2]\right ){}^2 \left (4 a^2 c^3+12 a^2 b K[1] c^2+12 a^2 K[2] c^2+12 a^2 b^2 K[1]^2 c+12 a^2 K[2]^2 c-8 a^2 c_1 c+24 a^2 b K[1] K[2] c+4 a^2 b^3 K[1]^3+4 a^2 K[2]^3+12 a^2 b K[1] K[2]^2-8 a^2 b c_1 K[1]+12 a^2 b^2 K[1]^2 K[2]-8 a^2 c_1 K[2]\right ) a^2-2 (2 c+2 b K[1]+2 K[2]) \left (c^2+2 b K[1] c+2 K[2] c+b^2 K[1]^2+K[2]^2-2 c_1+2 b K[1] K[2]\right ) \left (a^2 c^4+4 a^2 b K[1] c^3+4 a^2 K[2] c^3+6 a^2 b^2 K[1]^2 c^2+6 a^2 K[2]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[1] K[2] c^2+4 a^2 b^3 K[1]^3 c+4 a^2 K[2]^3 c+12 a^2 b K[1] K[2]^2 c-8 a^2 b c_1 K[1] c+12 a^2 b^2 K[1]^2 K[2] c-8 a^2 c_1 K[2] c+a^2 b^4 K[1]^4+a^2 K[2]^4+4 a^2 b K[1] K[2]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[1]^2+6 a^2 b^2 K[1]^2 K[2]^2-4 a^2 c_1 K[2]^2+4 a^2 b^3 K[1]^3 K[2]-8 a^2 b c_1 K[1] K[2]-4\right ) a^2\right )}{2 \left (b^2+1\right ) \left (a^2 c^4+4 a^2 b K[1] c^3+4 a^2 K[2] c^3+6 a^2 b^2 K[1]^2 c^2+6 a^2 K[2]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[1] K[2] c^2+4 a^2 b^3 K[1]^3 c+4 a^2 K[2]^3 c+12 a^2 b K[1] K[2]^2 c-8 a^2 b c_1 K[1] c+12 a^2 b^2 K[1]^2 K[2] c-8 a^2 c_1 K[2] c+a^2 b^4 K[1]^4+a^2 K[2]^4+4 a^2 b K[1] K[2]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[1]^2+6 a^2 b^2 K[1]^2 K[2]^2-4 a^2 c_1 K[2]^2+4 a^2 b^3 K[1]^3 K[2]-8 a^2 b c_1 K[1] K[2]-4\right ) \sqrt {-a^2 \left (c^2+2 b K[1] c+2 K[2] c+b^2 K[1]^2+K[2]^2-2 c_1+2 b K[1] K[2]\right ){}^2 \left (a^2 c^4+4 a^2 b K[1] c^3+4 a^2 K[2] c^3+6 a^2 b^2 K[1]^2 c^2+6 a^2 K[2]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[1] K[2] c^2+4 a^2 b^3 K[1]^3 c+4 a^2 K[2]^3 c+12 a^2 b K[1] K[2]^2 c-8 a^2 b c_1 K[1] c+12 a^2 b^2 K[1]^2 K[2] c-8 a^2 c_1 K[2] c+a^2 b^4 K[1]^4+a^2 K[2]^4+4 a^2 b K[1] K[2]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[1]^2+6 a^2 b^2 K[1]^2 K[2]^2-4 a^2 c_1 K[2]^2+4 a^2 b^3 K[1]^3 K[2]-8 a^2 b c_1 K[1] K[2]-4\right )}}\right )dK[1] b^2+b+\int _1^x\left (\frac {b \left (4 a^2 c^3+12 a^2 b K[1] c^2+12 a^2 K[2] c^2+12 a^2 b^2 K[1]^2 c+12 a^2 K[2]^2 c-8 a^2 c_1 c+24 a^2 b K[1] K[2] c+4 a^2 b^3 K[1]^3+4 a^2 K[2]^3+12 a^2 b K[1] K[2]^2-8 a^2 b c_1 K[1]+12 a^2 b^2 K[1]^2 K[2]-8 a^2 c_1 K[2]\right ) \sqrt {-a^2 \left (c^2+2 b K[1] c+2 K[2] c+b^2 K[1]^2+K[2]^2-2 c_1+2 b K[1] K[2]\right ){}^2 \left (a^2 c^4+4 a^2 b K[1] c^3+4 a^2 K[2] c^3+6 a^2 b^2 K[1]^2 c^2+6 a^2 K[2]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[1] K[2] c^2+4 a^2 b^3 K[1]^3 c+4 a^2 K[2]^3 c+12 a^2 b K[1] K[2]^2 c-8 a^2 b c_1 K[1] c+12 a^2 b^2 K[1]^2 K[2] c-8 a^2 c_1 K[2] c+a^2 b^4 K[1]^4+a^2 K[2]^4+4 a^2 b K[1] K[2]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[1]^2+6 a^2 b^2 K[1]^2 K[2]^2-4 a^2 c_1 K[2]^2+4 a^2 b^3 K[1]^3 K[2]-8 a^2 b c_1 K[1] K[2]-4\right )}}{\left (b^2+1\right ) \left (a^2 c^4+4 a^2 b K[1] c^3+4 a^2 K[2] c^3+6 a^2 b^2 K[1]^2 c^2+6 a^2 K[2]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[1] K[2] c^2+4 a^2 b^3 K[1]^3 c+4 a^2 K[2]^3 c+12 a^2 b K[1] K[2]^2 c-8 a^2 b c_1 K[1] c+12 a^2 b^2 K[1]^2 K[2] c-8 a^2 c_1 K[2] c+a^2 b^4 K[1]^4+a^2 K[2]^4+4 a^2 b K[1] K[2]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[1]^2+6 a^2 b^2 K[1]^2 K[2]^2-4 a^2 c_1 K[2]^2+4 a^2 b^3 K[1]^3 K[2]-8 a^2 b c_1 K[1] K[2]-4\right ){}^2}-\frac {b \left (-\left (c^2+2 b K[1] c+2 K[2] c+b^2 K[1]^2+K[2]^2-2 c_1+2 b K[1] K[2]\right ){}^2 \left (4 a^2 c^3+12 a^2 b K[1] c^2+12 a^2 K[2] c^2+12 a^2 b^2 K[1]^2 c+12 a^2 K[2]^2 c-8 a^2 c_1 c+24 a^2 b K[1] K[2] c+4 a^2 b^3 K[1]^3+4 a^2 K[2]^3+12 a^2 b K[1] K[2]^2-8 a^2 b c_1 K[1]+12 a^2 b^2 K[1]^2 K[2]-8 a^2 c_1 K[2]\right ) a^2-2 (2 c+2 b K[1]+2 K[2]) \left (c^2+2 b K[1] c+2 K[2] c+b^2 K[1]^2+K[2]^2-2 c_1+2 b K[1] K[2]\right ) \left (a^2 c^4+4 a^2 b K[1] c^3+4 a^2 K[2] c^3+6 a^2 b^2 K[1]^2 c^2+6 a^2 K[2]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[1] K[2] c^2+4 a^2 b^3 K[1]^3 c+4 a^2 K[2]^3 c+12 a^2 b K[1] K[2]^2 c-8 a^2 b c_1 K[1] c+12 a^2 b^2 K[1]^2 K[2] c-8 a^2 c_1 K[2] c+a^2 b^4 K[1]^4+a^2 K[2]^4+4 a^2 b K[1] K[2]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[1]^2+6 a^2 b^2 K[1]^2 K[2]^2-4 a^2 c_1 K[2]^2+4 a^2 b^3 K[1]^3 K[2]-8 a^2 b c_1 K[1] K[2]-4\right ) a^2\right )}{2 \left (b^2+1\right ) \left (a^2 c^4+4 a^2 b K[1] c^3+4 a^2 K[2] c^3+6 a^2 b^2 K[1]^2 c^2+6 a^2 K[2]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[1] K[2] c^2+4 a^2 b^3 K[1]^3 c+4 a^2 K[2]^3 c+12 a^2 b K[1] K[2]^2 c-8 a^2 b c_1 K[1] c+12 a^2 b^2 K[1]^2 K[2] c-8 a^2 c_1 K[2] c+a^2 b^4 K[1]^4+a^2 K[2]^4+4 a^2 b K[1] K[2]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[1]^2+6 a^2 b^2 K[1]^2 K[2]^2-4 a^2 c_1 K[2]^2+4 a^2 b^3 K[1]^3 K[2]-8 a^2 b c_1 K[1] K[2]-4\right ) \sqrt {-a^2 \left (c^2+2 b K[1] c+2 K[2] c+b^2 K[1]^2+K[2]^2-2 c_1+2 b K[1] K[2]\right ){}^2 \left (a^2 c^4+4 a^2 b K[1] c^3+4 a^2 K[2] c^3+6 a^2 b^2 K[1]^2 c^2+6 a^2 K[2]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[1] K[2] c^2+4 a^2 b^3 K[1]^3 c+4 a^2 K[2]^3 c+12 a^2 b K[1] K[2]^2 c-8 a^2 b c_1 K[1] c+12 a^2 b^2 K[1]^2 K[2] c-8 a^2 c_1 K[2] c+a^2 b^4 K[1]^4+a^2 K[2]^4+4 a^2 b K[1] K[2]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[1]^2+6 a^2 b^2 K[1]^2 K[2]^2-4 a^2 c_1 K[2]^2+4 a^2 b^3 K[1]^3 K[2]-8 a^2 b c_1 K[1] K[2]-4\right )}}\right )dK[1]}{b^2+1}\right )dK[2]=c_2,y(x)\right ],\text {Solve}\left [\int _1^x\left (\frac {\sqrt {-a^2 \left (c^2+2 b K[3] c+2 y(x) c+b^2 K[3]^2+y(x)^2-2 c_1+2 b K[3] y(x)\right ){}^2 \left (a^2 c^4+4 a^2 b K[3] c^3+4 a^2 y(x) c^3+6 a^2 b^2 K[3]^2 c^2+6 a^2 y(x)^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[3] y(x) c^2+4 a^2 b^3 K[3]^3 c+4 a^2 y(x)^3 c+12 a^2 b K[3] y(x)^2 c-8 a^2 b c_1 K[3] c+12 a^2 b^2 K[3]^2 y(x) c-8 a^2 c_1 y(x) c+a^2 b^4 K[3]^4+a^2 y(x)^4+4 a^2 b K[3] y(x)^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[3]^2+6 a^2 b^2 K[3]^2 y(x)^2-4 a^2 c_1 y(x)^2+4 a^2 b^3 K[3]^3 y(x)-8 a^2 b c_1 K[3] y(x)-4\right )} b}{\left (b^2+1\right ) \left (a^2 c^4+4 a^2 b K[3] c^3+4 a^2 y(x) c^3+6 a^2 b^2 K[3]^2 c^2+6 a^2 y(x)^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[3] y(x) c^2+4 a^2 b^3 K[3]^3 c+4 a^2 y(x)^3 c+12 a^2 b K[3] y(x)^2 c-8 a^2 b c_1 K[3] c+12 a^2 b^2 K[3]^2 y(x) c-8 a^2 c_1 y(x) c+a^2 b^4 K[3]^4+a^2 y(x)^4+4 a^2 b K[3] y(x)^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[3]^2+6 a^2 b^2 K[3]^2 y(x)^2-4 a^2 c_1 y(x)^2+4 a^2 b^3 K[3]^3 y(x)-8 a^2 b c_1 K[3] y(x)-4\right )}+\frac {1}{b^2+1}\right )dK[3]+\int _1^{y(x)}\left (\frac {\sqrt {-a^2 \left (c^2+2 b x c+2 K[4] c+b^2 x^2+K[4]^2-2 c_1+2 b x K[4]\right ){}^2 \left (a^2 c^4+4 a^2 b x c^3+4 a^2 K[4] c^3+6 a^2 b^2 x^2 c^2+6 a^2 K[4]^2 c^2-4 a^2 c_1 c^2+12 a^2 b x K[4] c^2+4 a^2 b^3 x^3 c+4 a^2 K[4]^3 c+12 a^2 b x K[4]^2 c-8 a^2 b x c_1 c+12 a^2 b^2 x^2 K[4] c-8 a^2 c_1 K[4] c+a^2 b^4 x^4+a^2 K[4]^4+4 a^2 b x K[4]^3-4 b^2+4 a^2 c_1{}^2+6 a^2 b^2 x^2 K[4]^2-4 a^2 c_1 K[4]^2-4 a^2 b^2 x^2 c_1+4 a^2 b^3 x^3 K[4]-8 a^2 b x c_1 K[4]-4\right )}}{\left (b^2+1\right ) \left (a^2 c^4+4 a^2 b x c^3+4 a^2 K[4] c^3+6 a^2 b^2 x^2 c^2+6 a^2 K[4]^2 c^2-4 a^2 c_1 c^2+12 a^2 b x K[4] c^2+4 a^2 b^3 x^3 c+4 a^2 K[4]^3 c+12 a^2 b x K[4]^2 c-8 a^2 b x c_1 c+12 a^2 b^2 x^2 K[4] c-8 a^2 c_1 K[4] c+a^2 b^4 x^4+a^2 K[4]^4+4 a^2 b x K[4]^3-4 b^2+4 a^2 c_1{}^2+6 a^2 b^2 x^2 K[4]^2-4 a^2 c_1 K[4]^2-4 a^2 b^2 x^2 c_1+4 a^2 b^3 x^3 K[4]-8 a^2 b x c_1 K[4]-4\right )}-\frac {\int _1^x\left (\frac {b \left (-\left (c^2+2 b K[3] c+2 K[4] c+b^2 K[3]^2+K[4]^2-2 c_1+2 b K[3] K[4]\right ){}^2 \left (4 a^2 c^3+12 a^2 b K[3] c^2+12 a^2 K[4] c^2+12 a^2 b^2 K[3]^2 c+12 a^2 K[4]^2 c-8 a^2 c_1 c+24 a^2 b K[3] K[4] c+4 a^2 b^3 K[3]^3+4 a^2 K[4]^3+12 a^2 b K[3] K[4]^2-8 a^2 b c_1 K[3]+12 a^2 b^2 K[3]^2 K[4]-8 a^2 c_1 K[4]\right ) a^2-2 (2 c+2 b K[3]+2 K[4]) \left (c^2+2 b K[3] c+2 K[4] c+b^2 K[3]^2+K[4]^2-2 c_1+2 b K[3] K[4]\right ) \left (a^2 c^4+4 a^2 b K[3] c^3+4 a^2 K[4] c^3+6 a^2 b^2 K[3]^2 c^2+6 a^2 K[4]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[3] K[4] c^2+4 a^2 b^3 K[3]^3 c+4 a^2 K[4]^3 c+12 a^2 b K[3] K[4]^2 c-8 a^2 b c_1 K[3] c+12 a^2 b^2 K[3]^2 K[4] c-8 a^2 c_1 K[4] c+a^2 b^4 K[3]^4+a^2 K[4]^4+4 a^2 b K[3] K[4]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[3]^2+6 a^2 b^2 K[3]^2 K[4]^2-4 a^2 c_1 K[4]^2+4 a^2 b^3 K[3]^3 K[4]-8 a^2 b c_1 K[3] K[4]-4\right ) a^2\right )}{2 \left (b^2+1\right ) \left (a^2 c^4+4 a^2 b K[3] c^3+4 a^2 K[4] c^3+6 a^2 b^2 K[3]^2 c^2+6 a^2 K[4]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[3] K[4] c^2+4 a^2 b^3 K[3]^3 c+4 a^2 K[4]^3 c+12 a^2 b K[3] K[4]^2 c-8 a^2 b c_1 K[3] c+12 a^2 b^2 K[3]^2 K[4] c-8 a^2 c_1 K[4] c+a^2 b^4 K[3]^4+a^2 K[4]^4+4 a^2 b K[3] K[4]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[3]^2+6 a^2 b^2 K[3]^2 K[4]^2-4 a^2 c_1 K[4]^2+4 a^2 b^3 K[3]^3 K[4]-8 a^2 b c_1 K[3] K[4]-4\right ) \sqrt {-a^2 \left (c^2+2 b K[3] c+2 K[4] c+b^2 K[3]^2+K[4]^2-2 c_1+2 b K[3] K[4]\right ){}^2 \left (a^2 c^4+4 a^2 b K[3] c^3+4 a^2 K[4] c^3+6 a^2 b^2 K[3]^2 c^2+6 a^2 K[4]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[3] K[4] c^2+4 a^2 b^3 K[3]^3 c+4 a^2 K[4]^3 c+12 a^2 b K[3] K[4]^2 c-8 a^2 b c_1 K[3] c+12 a^2 b^2 K[3]^2 K[4] c-8 a^2 c_1 K[4] c+a^2 b^4 K[3]^4+a^2 K[4]^4+4 a^2 b K[3] K[4]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[3]^2+6 a^2 b^2 K[3]^2 K[4]^2-4 a^2 c_1 K[4]^2+4 a^2 b^3 K[3]^3 K[4]-8 a^2 b c_1 K[3] K[4]-4\right )}}-\frac {b \left (4 a^2 c^3+12 a^2 b K[3] c^2+12 a^2 K[4] c^2+12 a^2 b^2 K[3]^2 c+12 a^2 K[4]^2 c-8 a^2 c_1 c+24 a^2 b K[3] K[4] c+4 a^2 b^3 K[3]^3+4 a^2 K[4]^3+12 a^2 b K[3] K[4]^2-8 a^2 b c_1 K[3]+12 a^2 b^2 K[3]^2 K[4]-8 a^2 c_1 K[4]\right ) \sqrt {-a^2 \left (c^2+2 b K[3] c+2 K[4] c+b^2 K[3]^2+K[4]^2-2 c_1+2 b K[3] K[4]\right ){}^2 \left (a^2 c^4+4 a^2 b K[3] c^3+4 a^2 K[4] c^3+6 a^2 b^2 K[3]^2 c^2+6 a^2 K[4]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[3] K[4] c^2+4 a^2 b^3 K[3]^3 c+4 a^2 K[4]^3 c+12 a^2 b K[3] K[4]^2 c-8 a^2 b c_1 K[3] c+12 a^2 b^2 K[3]^2 K[4] c-8 a^2 c_1 K[4] c+a^2 b^4 K[3]^4+a^2 K[4]^4+4 a^2 b K[3] K[4]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[3]^2+6 a^2 b^2 K[3]^2 K[4]^2-4 a^2 c_1 K[4]^2+4 a^2 b^3 K[3]^3 K[4]-8 a^2 b c_1 K[3] K[4]-4\right )}}{\left (b^2+1\right ) \left (a^2 c^4+4 a^2 b K[3] c^3+4 a^2 K[4] c^3+6 a^2 b^2 K[3]^2 c^2+6 a^2 K[4]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[3] K[4] c^2+4 a^2 b^3 K[3]^3 c+4 a^2 K[4]^3 c+12 a^2 b K[3] K[4]^2 c-8 a^2 b c_1 K[3] c+12 a^2 b^2 K[3]^2 K[4] c-8 a^2 c_1 K[4] c+a^2 b^4 K[3]^4+a^2 K[4]^4+4 a^2 b K[3] K[4]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[3]^2+6 a^2 b^2 K[3]^2 K[4]^2-4 a^2 c_1 K[4]^2+4 a^2 b^3 K[3]^3 K[4]-8 a^2 b c_1 K[3] K[4]-4\right ){}^2}\right )dK[3] b^2+b+\int _1^x\left (\frac {b \left (-\left (c^2+2 b K[3] c+2 K[4] c+b^2 K[3]^2+K[4]^2-2 c_1+2 b K[3] K[4]\right ){}^2 \left (4 a^2 c^3+12 a^2 b K[3] c^2+12 a^2 K[4] c^2+12 a^2 b^2 K[3]^2 c+12 a^2 K[4]^2 c-8 a^2 c_1 c+24 a^2 b K[3] K[4] c+4 a^2 b^3 K[3]^3+4 a^2 K[4]^3+12 a^2 b K[3] K[4]^2-8 a^2 b c_1 K[3]+12 a^2 b^2 K[3]^2 K[4]-8 a^2 c_1 K[4]\right ) a^2-2 (2 c+2 b K[3]+2 K[4]) \left (c^2+2 b K[3] c+2 K[4] c+b^2 K[3]^2+K[4]^2-2 c_1+2 b K[3] K[4]\right ) \left (a^2 c^4+4 a^2 b K[3] c^3+4 a^2 K[4] c^3+6 a^2 b^2 K[3]^2 c^2+6 a^2 K[4]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[3] K[4] c^2+4 a^2 b^3 K[3]^3 c+4 a^2 K[4]^3 c+12 a^2 b K[3] K[4]^2 c-8 a^2 b c_1 K[3] c+12 a^2 b^2 K[3]^2 K[4] c-8 a^2 c_1 K[4] c+a^2 b^4 K[3]^4+a^2 K[4]^4+4 a^2 b K[3] K[4]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[3]^2+6 a^2 b^2 K[3]^2 K[4]^2-4 a^2 c_1 K[4]^2+4 a^2 b^3 K[3]^3 K[4]-8 a^2 b c_1 K[3] K[4]-4\right ) a^2\right )}{2 \left (b^2+1\right ) \left (a^2 c^4+4 a^2 b K[3] c^3+4 a^2 K[4] c^3+6 a^2 b^2 K[3]^2 c^2+6 a^2 K[4]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[3] K[4] c^2+4 a^2 b^3 K[3]^3 c+4 a^2 K[4]^3 c+12 a^2 b K[3] K[4]^2 c-8 a^2 b c_1 K[3] c+12 a^2 b^2 K[3]^2 K[4] c-8 a^2 c_1 K[4] c+a^2 b^4 K[3]^4+a^2 K[4]^4+4 a^2 b K[3] K[4]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[3]^2+6 a^2 b^2 K[3]^2 K[4]^2-4 a^2 c_1 K[4]^2+4 a^2 b^3 K[3]^3 K[4]-8 a^2 b c_1 K[3] K[4]-4\right ) \sqrt {-a^2 \left (c^2+2 b K[3] c+2 K[4] c+b^2 K[3]^2+K[4]^2-2 c_1+2 b K[3] K[4]\right ){}^2 \left (a^2 c^4+4 a^2 b K[3] c^3+4 a^2 K[4] c^3+6 a^2 b^2 K[3]^2 c^2+6 a^2 K[4]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[3] K[4] c^2+4 a^2 b^3 K[3]^3 c+4 a^2 K[4]^3 c+12 a^2 b K[3] K[4]^2 c-8 a^2 b c_1 K[3] c+12 a^2 b^2 K[3]^2 K[4] c-8 a^2 c_1 K[4] c+a^2 b^4 K[3]^4+a^2 K[4]^4+4 a^2 b K[3] K[4]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[3]^2+6 a^2 b^2 K[3]^2 K[4]^2-4 a^2 c_1 K[4]^2+4 a^2 b^3 K[3]^3 K[4]-8 a^2 b c_1 K[3] K[4]-4\right )}}-\frac {b \left (4 a^2 c^3+12 a^2 b K[3] c^2+12 a^2 K[4] c^2+12 a^2 b^2 K[3]^2 c+12 a^2 K[4]^2 c-8 a^2 c_1 c+24 a^2 b K[3] K[4] c+4 a^2 b^3 K[3]^3+4 a^2 K[4]^3+12 a^2 b K[3] K[4]^2-8 a^2 b c_1 K[3]+12 a^2 b^2 K[3]^2 K[4]-8 a^2 c_1 K[4]\right ) \sqrt {-a^2 \left (c^2+2 b K[3] c+2 K[4] c+b^2 K[3]^2+K[4]^2-2 c_1+2 b K[3] K[4]\right ){}^2 \left (a^2 c^4+4 a^2 b K[3] c^3+4 a^2 K[4] c^3+6 a^2 b^2 K[3]^2 c^2+6 a^2 K[4]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[3] K[4] c^2+4 a^2 b^3 K[3]^3 c+4 a^2 K[4]^3 c+12 a^2 b K[3] K[4]^2 c-8 a^2 b c_1 K[3] c+12 a^2 b^2 K[3]^2 K[4] c-8 a^2 c_1 K[4] c+a^2 b^4 K[3]^4+a^2 K[4]^4+4 a^2 b K[3] K[4]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[3]^2+6 a^2 b^2 K[3]^2 K[4]^2-4 a^2 c_1 K[4]^2+4 a^2 b^3 K[3]^3 K[4]-8 a^2 b c_1 K[3] K[4]-4\right )}}{\left (b^2+1\right ) \left (a^2 c^4+4 a^2 b K[3] c^3+4 a^2 K[4] c^3+6 a^2 b^2 K[3]^2 c^2+6 a^2 K[4]^2 c^2-4 a^2 c_1 c^2+12 a^2 b K[3] K[4] c^2+4 a^2 b^3 K[3]^3 c+4 a^2 K[4]^3 c+12 a^2 b K[3] K[4]^2 c-8 a^2 b c_1 K[3] c+12 a^2 b^2 K[3]^2 K[4] c-8 a^2 c_1 K[4] c+a^2 b^4 K[3]^4+a^2 K[4]^4+4 a^2 b K[3] K[4]^3-4 b^2+4 a^2 c_1{}^2-4 a^2 b^2 c_1 K[3]^2+6 a^2 b^2 K[3]^2 K[4]^2-4 a^2 c_1 K[4]^2+4 a^2 b^3 K[3]^3 K[4]-8 a^2 b c_1 K[3] K[4]-4\right ){}^2}\right )dK[3]}{b^2+1}\right )dK[4]=c_2,y(x)\right ]\right \}\] ✓ Maple : cpu = 2.336 (sec), leaf count = 771

dsolve(diff(diff(y(x),x),x)-a*(c+b*x+y(x))*(diff(y(x),x)^2+1)^(3/2)=0,y(x))
 

\[y \left (x \right ) = -b x +\RootOf \left (-x +\int _{}^{\textit {\_Z}}\frac {4 a^{2} b^{2} c^{2} \textit {\_f}^{2}+4 a^{2} b^{2} c \,\textit {\_f}^{3}+a^{2} b^{2} \textit {\_f}^{4}-8 c_{1} a^{2} b^{2} c \textit {\_f} -4 c_{1} a^{2} b^{2} \textit {\_f}^{2}+4 c_{1}^{2} a^{2} b^{2}-2 \sqrt {-b^{2} \left (\textit {\_f}^{4} a^{2}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 \textit {\_f}^{2} a^{2} c_{1}-8 \textit {\_f} \,a^{2} c c_{1}+4 a^{2} c_{1}^{2}-4 b^{2}-4\right )}\, a c \textit {\_f} -\sqrt {-b^{2} \left (\textit {\_f}^{4} a^{2}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 \textit {\_f}^{2} a^{2} c_{1}-8 \textit {\_f} \,a^{2} c c_{1}+4 a^{2} c_{1}^{2}-4 b^{2}-4\right )}\, a \,\textit {\_f}^{2}-4 b^{4}+2 \sqrt {-b^{2} \left (\textit {\_f}^{4} a^{2}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 \textit {\_f}^{2} a^{2} c_{1}-8 \textit {\_f} \,a^{2} c c_{1}+4 a^{2} c_{1}^{2}-4 b^{2}-4\right )}\, c_{1} a -4 b^{2}}{\left (\textit {\_f}^{4} a^{2}+4 \textit {\_f}^{3} a^{2} c +4 \textit {\_f}^{2} a^{2} c^{2}-4 \textit {\_f}^{2} a^{2} c_{1}-8 \textit {\_f} \,a^{2} c c_{1}+4 a^{2} c_{1}^{2}-4 b^{2}-4\right ) \left (b^{2}+1\right ) b}d \textit {\_f} +c_{2}\right )\]