4.22.12 $4y^{\prime }(x{)}^3+4y^{\prime }(x)=x$

ODE
$$4y^{\prime }(x{)}^3+4y^{\prime }(x)=x$$ ODE Classification

[_quadrature]

Book solution method
Missing Variables ODE, Dependent variable missing, Solve for $x$

Mathematica ✓
cpu = 2.68298 (sec), leaf count = 828

$$\{\{y(x)\to \frac{(1+i\sqrt{3})\sqrt[3]{\sqrt{81x^2+192}-9x}(9\sqrt{3}x+\sqrt{27x^2+64})}{96\sqrt[6]{3}}+\frac{i\sqrt[6]{3}(i+\sqrt{3})\sqrt{27x^2+64}(-243x^4+27\sqrt{81x^2+192}{x}^3-576x^2+32\sqrt{81x^2+192}x-512)}{256\sqrt[3]{\sqrt{81x^2+192}-9x}(-27x^2+3\sqrt{81x^2+192}x-64)}+\frac{(1-i\sqrt{3}){(\sqrt{81x^2+192}-9x)}^{2/3}(-729x^4+81\sqrt{81x^2+192}{x}^3-5184x^2+480\sqrt{81x^2+192}x-3584)}{1536\sqrt[3]{3}(-27x^2+3\sqrt{81x^2+192}x-32)}+c_1\},\{y(x)\to \frac{(1-i\sqrt{3})\sqrt[3]{\sqrt{81x^2+192}-9x}(9\sqrt{3}x+\sqrt{27x^2+64})}{96\sqrt[6]{3}}-\frac{i\sqrt[6]{3}(-i+\sqrt{3})\sqrt{27x^2+64}(-243x^4+27\sqrt{81x^2+192}{x}^3-576x^2+32\sqrt{81x^2+192}x-512)}{256\sqrt[3]{\sqrt{81x^2+192}-9x}(-27x^2+3\sqrt{81x^2+192}x-64)}+\frac{(1+i\sqrt{3}){(\sqrt{81x^2+192}-9x)}^{2/3}(-729x^4+81\sqrt{81x^2+192}{x}^3-5184x^2+480\sqrt{81x^2+192}x-3584)}{1536\sqrt[3]{3}(-27x^2+3\sqrt{81x^2+192}x-32)}+c_1\},\{y(x)\to \frac{1}{6}(-\frac{\sqrt[3]{\sqrt{81x^2+192}-9x}(9\sqrt{3}x+\sqrt{27x^2+64})}{8\sqrt[6]{3}}+\frac{{(\sqrt{81x^2+192}-9x)}^{2/3}(729x^4-81\sqrt{81x^2+192}{x}^3+5184x^2-480\sqrt{81x^2+192}x+3584)}{128\sqrt[3]{3}(-27x^2+3\sqrt{81x^2+192}x-32)}+\frac{3\sqrt[6]{3}\sqrt{27x^2+64}(-243x^4+27\sqrt{81x^2+192}{x}^3-576x^2+32\sqrt{81x^2+192}x-512)}{64\sqrt[3]{\sqrt{81x^2+192}-9x}(-27x^2+3\sqrt{81x^2+192}x-64)})+c_1\}\}$$

Maple ✓
cpu = 0.225 (sec), leaf count = 166

$$\{y\left(x\right)=\int \frac{i}{12}((-\sqrt{3}+i){(27x+3\sqrt{81x^2+192})}^{\frac{2}{3}}-12i-12\sqrt{3})\frac{1}{\sqrt[3]{27x+3\sqrt{81x^2+192}}}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1},y\left(x\right)=\int \frac{i}{12}((\sqrt{3}+i){(27x+3\sqrt{81x^2+192})}^{\frac{2}{3}}-12i+12\sqrt{3})\frac{1}{\sqrt[3]{27x+3\sqrt{81x^2+192}}}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1},y\left(x\right)=\int \frac{1}{6}({(27x+3\sqrt{81x^2+192})}^{\frac{2}{3}}-12)\frac{1}{\sqrt[3]{27x+3\sqrt{81x^2+192}}}\mathrm{d}x+\mathit{\_}\mathit{C}\mathit{1}\}$$ 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[19
2 + 81*x^2] + 27*x^3*Sqrt[192 + 81*x^2]))/((-9*x + Sqrt[192 + 81*x^2])^(1/3)*(-6
4 - 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 - 4
80*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[19
2 + 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+1
92)^(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+1
92)^(1/2))^(1/3),x)+_C1