ODE
\[ 4 y'(x)^3+4 y'(x)=x \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Dependent variable missing, Solve for \(x\)
Mathematica ✓
cpu = 2.68298 (sec), leaf count = 828
\[\left \{\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{\sqrt {81 x^2+192}-9 x} \left (9 \sqrt {3} x+\sqrt {27 x^2+64}\right )}{96 \sqrt [6]{3}}+\frac {i \sqrt [6]{3} \left (i+\sqrt {3}\right ) \sqrt {27 x^2+64} \left (-243 x^4+27 \sqrt {81 x^2+192} x^3-576 x^2+32 \sqrt {81 x^2+192} x-512\right )}{256 \sqrt [3]{\sqrt {81 x^2+192}-9 x} \left (-27 x^2+3 \sqrt {81 x^2+192} x-64\right )}+\frac {\left (1-i \sqrt {3}\right ) \left (\sqrt {81 x^2+192}-9 x\right )^{2/3} \left (-729 x^4+81 \sqrt {81 x^2+192} x^3-5184 x^2+480 \sqrt {81 x^2+192} x-3584\right )}{1536 \sqrt [3]{3} \left (-27 x^2+3 \sqrt {81 x^2+192} x-32\right )}+c_1\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{\sqrt {81 x^2+192}-9 x} \left (9 \sqrt {3} x+\sqrt {27 x^2+64}\right )}{96 \sqrt [6]{3}}-\frac {i \sqrt [6]{3} \left (-i+\sqrt {3}\right ) \sqrt {27 x^2+64} \left (-243 x^4+27 \sqrt {81 x^2+192} x^3-576 x^2+32 \sqrt {81 x^2+192} x-512\right )}{256 \sqrt [3]{\sqrt {81 x^2+192}-9 x} \left (-27 x^2+3 \sqrt {81 x^2+192} x-64\right )}+\frac {\left (1+i \sqrt {3}\right ) \left (\sqrt {81 x^2+192}-9 x\right )^{2/3} \left (-729 x^4+81 \sqrt {81 x^2+192} x^3-5184 x^2+480 \sqrt {81 x^2+192} x-3584\right )}{1536 \sqrt [3]{3} \left (-27 x^2+3 \sqrt {81 x^2+192} x-32\right )}+c_1\right \},\left \{y(x)\to \frac {1}{6} \left (-\frac {\sqrt [3]{\sqrt {81 x^2+192}-9 x} \left (9 \sqrt {3} x+\sqrt {27 x^2+64}\right )}{8 \sqrt [6]{3}}+\frac {\left (\sqrt {81 x^2+192}-9 x\right )^{2/3} \left (729 x^4-81 \sqrt {81 x^2+192} x^3+5184 x^2-480 \sqrt {81 x^2+192} x+3584\right )}{128 \sqrt [3]{3} \left (-27 x^2+3 \sqrt {81 x^2+192} x-32\right )}+\frac {3 \sqrt [6]{3} \sqrt {27 x^2+64} \left (-243 x^4+27 \sqrt {81 x^2+192} x^3-576 x^2+32 \sqrt {81 x^2+192} x-512\right )}{64 \sqrt [3]{\sqrt {81 x^2+192}-9 x} \left (-27 x^2+3 \sqrt {81 x^2+192} x-64\right )}\right )+c_1\right \}\right \}\]
Maple ✓
cpu = 0.225 (sec), leaf count = 166
\[ \left \{ y \left ( x \right ) =\int \!{{\frac {i}{12}} \left ( \left ( -\sqrt {3}+i \right ) \left ( 27\,x+3\,\sqrt {81\,{x}^{2}+192} \right ) ^{{\frac {2}{3}}}-12\,i-12\,\sqrt {3} \right ) {\frac {1}{\sqrt [3]{27\,x+3\,\sqrt {81\,{x}^{2}+192}}}}}\,{\rm d}x+{\it \_C1},y \left ( x \right ) =\int \!{{\frac {i}{12}} \left ( \left ( \sqrt {3}+i \right ) \left ( 27\,x+3\,\sqrt {81\,{x}^{2}+192} \right ) ^{{\frac {2}{3}}}-12\,i+12\,\sqrt {3} \right ) {\frac {1}{\sqrt [3]{27\,x+3\,\sqrt {81\,{x}^{2}+192}}}}}\,{\rm d}x+{\it \_C1},y \left ( x \right ) =\int \!{\frac {1}{6} \left ( \left ( 27\,x+3\,\sqrt {81\,{x}^{2}+192} \right ) ^{{\frac {2}{3}}}-12 \right ) {\frac {1}{\sqrt [3]{27\,x+3\,\sqrt {81\,{x}^{2}+192}}}}}\,{\rm d}x+{\it \_C1} \right \} \] Mathematica raw input
DSolve[4*y'[x] + 4*y'[x]^3 == x,y[x],x]
Mathematica raw output
{{y[x] -> ((1 + I*Sqrt[3])*(9*Sqrt[3]*x + Sqrt[64 + 27*x^2])*(-9*x + Sqrt[192 + 81*x^2])^(1/3))/(96*3^(1/6)) + ((I/256)*3^(1/6)*(I + Sqrt[3])*Sqrt[64 + 27*x^2]*(-512 - 576*x^2 - 243*x^4 + 32*x*Sqrt[192 + 81*x^2] + 27*x^3*Sqrt[192 + 81*x^2]))/((-9*x + Sqrt[192 + 81*x^2])^(1/3)*(-64 - 27*x^2 + 3*x*Sqrt[192 + 81*x^2])) + ((1 - I*Sqrt[3])*(-9*x + Sqrt[192 + 81*x^2])^(2/3)*(-3584 - 5184*x^2 - 729*x^4 + 480*x*Sqrt[192 + 81*x^2] + 81*x^3*Sqrt[192 + 81*x^2]))/(1536*3^(1/3)*(-32 - 27*x^2 + 3*x*Sqrt[192 + 81*x^2])) + C[1]}, {y[x] -> ((1 - I*Sqrt[3])*(9*Sqrt[3]*x + Sqrt[64 + 27*x^2])*(-9*x + Sqrt[192 + 81*x^2])^(1/3))/(96*3^(1/6)) - ((I/256)*3^(1/6)*(-I + Sqrt[3])*Sqrt[64 + 27*x^2]*(-512 - 576*x^2 - 243*x^4 + 32*x*Sqrt[192 + 81*x^2] + 27*x^3*Sqrt[192 + 81*x^2]))/((-9*x + Sqrt[192 + 81*x^2])^(1/3)*(-64 - 27*x^2 + 3*x*Sqrt[192 + 81*x^2])) + ((1 + I*Sqrt[3])*(-9*x + Sqrt[192 + 81*x^2])^(2/3)*(-3584 - 5184*x^2 - 729*x^4 + 480*x*Sqrt[192 + 81*x^2] + 81*x^3*Sqrt[192 + 81*x^2]))/(1536*3^(1/3)*(-32 - 27*x^2 + 3*x*Sqrt[192 + 81*x^2])) + C[1]}, {y[x] -> (-((9*Sqrt[3]*x + Sqrt[64 + 27*x^2])*(-9*x + Sqrt[192 + 81*x^2])^(1/3))/(8*3^(1/6)) + ((-9*x + Sqrt[192 + 81*x^2])^(2/3)*(3584 + 5184*x^2 + 729*x^4 - 480*x*Sqrt[192 + 81*x^2] - 81*x^3*Sqrt[192 + 81*x^2]))/(128*3^(1/3)*(-32 - 27*x^2 + 3*x*Sqrt[192 + 81*x^2])) + (3*3^(1/6)*Sqrt[64 + 27*x^2]*(-512 - 576*x^2 - 243*x^4 + 32*x*Sqrt[192 + 81*x^2] + 27*x^3*Sqrt[192 + 81*x^2]))/(64*(-9*x + Sqrt[192 + 81*x^2])^(1/3)*(-64 - 27*x^2 + 3*x*Sqrt[192 + 81*x^2])))/6 + C[1]}}
Maple raw input
dsolve(4*diff(y(x),x)^3+4*diff(y(x),x) = x, y(x),'implicit')
Maple raw output
y(x) = Int(1/12*I/(27*x+3*(81*x^2+192)^(1/2))^(1/3)*((-3^(1/2)+I)*(27*x+3*(81*x^2+192)^(1/2))^(2/3)-12*I-12*3^(1/2)),x)+_C1, y(x) = Int(1/12*I/(27*x+3*(81*x^2+192)^(1/2))^(1/3)*((3^(1/2)+I)*(27*x+3*(81*x^2+192)^(1/2))^(2/3)-12*I+12*3^(1/2)),x)+_C1, y(x) = Int(1/6*((27*x+3*(81*x^2+192)^(1/2))^(2/3)-12)/(27*x+3*(81*x^2+192)^(1/2))^(1/3),x)+_C1