ODE
\[ -x^3-(1-3 x) x y'(x)+y'(x)^3+(1-3 x) y'(x)^2-1=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Dependent variable missing, Use new variable
Mathematica ✓
cpu = 11.579 (sec), leaf count = 379
\[\left \{\left \{y(x)\to \int _1^x\frac {1}{6} \left (6 K[1]-2^{2/3} \sqrt [3]{-9 K[1]+3 \sqrt {12 K[1]^3-3 K[1]^2+54 K[1]+69}-25}+\frac {2 \sqrt [3]{2} (3 K[1]-1)}{\sqrt [3]{-9 K[1]+3 \sqrt {12 K[1]^3-3 K[1]^2+54 K[1]+69}-25}}-2\right )dK[1]+c_1\right \},\left \{y(x)\to \int _1^x\frac {1}{12} \left (12 K[2]+2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{-9 K[2]+3 \sqrt {12 K[2]^3-3 K[2]^2+54 K[2]+69}-25}-\frac {2 i \sqrt [3]{2} \left (-i+\sqrt {3}\right ) (3 K[2]-1)}{\sqrt [3]{-9 K[2]+3 \sqrt {12 K[2]^3-3 K[2]^2+54 K[2]+69}-25}}-4\right )dK[2]+c_1\right \},\left \{y(x)\to \int _1^x\frac {1}{12} \left (12 K[3]+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-9 K[3]+3 \sqrt {12 K[3]^3-3 K[3]^2+54 K[3]+69}-25}+\frac {2 i \sqrt [3]{2} \left (i+\sqrt {3}\right ) (3 K[3]-1)}{\sqrt [3]{-9 K[3]+3 \sqrt {12 K[3]^3-3 K[3]^2+54 K[3]+69}-25}}-4\right )dK[3]+c_1\right \}\right \}\]
Maple ✓
cpu = 0.217 (sec), leaf count = 497
\[\left [y \left (x \right ) = \int \frac {i \left (\sqrt {3}\, \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {2}{3}}+12 \sqrt {3}\, x -4 \sqrt {3}+i \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {2}{3}}-12 i \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}} x +4 i \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}}-12 i x +4 i\right )}{12 \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}}}d x +\textit {\_C1}, y \left (x \right ) = \int \frac {i \left (i \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {2}{3}}-12 i \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}} x -\sqrt {3}\, \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {2}{3}}+4 i \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}}-12 i x -12 \sqrt {3}\, x +4 i+4 \sqrt {3}\right )}{12 \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}}}d x +\textit {\_C1}, y \left (x \right ) = \int \frac {\left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {2}{3}}+6 x \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}}-2 \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}}-12 x +4}{6 \left (36 x +100+12 \sqrt {3}\, \sqrt {\left (x +1\right ) \left (4 x^{2}-5 x +23\right )}\right )^{\frac {1}{3}}}d x +\textit {\_C1}\right ]\] Mathematica raw input
DSolve[-1 - x^3 - (1 - 3*x)*x*y'[x] + (1 - 3*x)*y'[x]^2 + y'[x]^3 == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1] + Inactive[Integrate][(-2 + 6*K[1] + (2*2^(1/3)*(-1 + 3*K[1]))/(-25 - 9*K[1] + 3*Sqrt[69 + 54*K[1] - 3*K[1]^2 + 12*K[1]^3])^(1/3) - 2^(2/3)*(-25 - 9*K[1] + 3*Sqrt[69 + 54*K[1] - 3*K[1]^2 + 12*K[1]^3])^(1/3))/6, {K[1], 1, x}]}, {y[x] -> C[1] + Inactive[Integrate][(-4 + 12*K[2] - ((2*I)*2^(1/3)*(-I + Sqrt[3])*(-1 + 3*K[2]))/(-25 - 9*K[2] + 3*Sqrt[69 + 54*K[2] - 3*K[2]^2 + 12*K[2]^3])^(1/3) + 2^(2/3)*(1 - I*Sqrt[3])*(-25 - 9*K[2] + 3*Sqrt[69 + 54*K[2] - 3*K[2]^2 + 12*K[2]^3])^(1/3))/12, {K[2], 1, x}]}, {y[x] -> C[1] + Inactive[Integrate][(-4 + 12*K[3] + ((2*I)*2^(1/3)*(I + Sqrt[3])*(-1 + 3*K[3]))/(-25 - 9*K[3] + 3*Sqrt[69 + 54*K[3] - 3*K[3]^2 + 12*K[3]^3])^(1/3) + 2^(2/3)*(1 + I*Sqrt[3])*(-25 - 9*K[3] + 3*Sqrt[69 + 54*K[3] - 3*K[3]^2 + 12*K[3]^3])^(1/3))/12, {K[3], 1, x}]}}
Maple raw input
dsolve(diff(y(x),x)^3+(1-3*x)*diff(y(x),x)^2-x*(1-3*x)*diff(y(x),x)-1-x^3 = 0, y(x))
Maple raw output
[y(x) = Int(1/12*I*(3^(1/2)*(36*x+100+12*3^(1/2)*((x+1)*(4*x^2-5*x+23))^(1/2))^(2/3)+12*3^(1/2)*x-4*3^(1/2)+I*(36*x+100+12*3^(1/2)*((x+1)*(4*x^2-5*x+23))^(1/2))^(2/3)-12*I*(36*x+100+12*3^(1/2)*((x+1)*(4*x^2-5*x+23))^(1/2))^(1/3)*x+4*I*(36*x+100+12*3^(1/2)*((x+1)*(4*x^2-5*x+23))^(1/2))^(1/3)-12*I*x+4*I)/(36*x+100+12*3^(1/2)*((x+1)*(4*x^2-5*x+23))^(1/2))^(1/3),x)+_C1, y(x) = Int(1/12*I*(I*(36*x+100+12*3^(1/2)*((x+1)*(4*x^2-5*x+23))^(1/2))^(2/3)-12*I*(36*x+100+12*3^(1/2)*((x+1)*(4*x^2-5*x+23))^(1/2))^(1/3)*x-3^(1/2)*(36*x+100+12*3^(1/2)*((x+1)*(4*x^2-5*x+23))^(1/2))^(2/3)+4*I*(36*x+100+12*3^(1/2)*((x+1)*(4*x^2-5*x+23))^(1/2))^(1/3)-12*I*x-12*3^(1/2)*x+4*I+4*3^(1/2))/(36*x+100+12*3^(1/2)*((x+1)*(4*x^2-5*x+23))^(1/2))^(1/3),x)+_C1, y(x) = Int(1/6*((36*x+100+12*3^(1/2)*((x+1)*(4*x^2-5*x+23))^(1/2))^(2/3)+6*x*(36*x+100+12*3^(1/2)*((x+1)*(4*x^2-5*x+23))^(1/2))^(1/3)-2*(36*x+100+12*3^(1/2)*((x+1)*(4*x^2-5*x+23))^(1/2))^(1/3)-12*x+4)/(36*x+100+12*3^(1/2)*((x+1)*(4*x^2-5*x+23))^(1/2))^(1/3),x)+_C1]