4.15.5 \(\left (-10 x^2 y(x)^3+3 y(x)^2+2\right ) y'(x)=x \left (5 y(x)^4+1\right )\)

ODE
\[ \left (-10 x^2 y(x)^3+3 y(x)^2+2\right ) y'(x)=x \left (5 y(x)^4+1\right ) \] ODE Classification

[_exact, _rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`]]

Book solution method
Exact equation

Mathematica
cpu = 0.199571 (sec), leaf count = 2097

\[\left \{\left \{y(x)\to -\frac {\sqrt {3} \sqrt {\frac {5 \sqrt [3]{6} \sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}} x^2+\frac {10\ 6^{2/3} \left (5 x^4-10 c_1 x^2-2\right ) x^2}{\sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}}}+3}{x^4}} x^2+\sqrt {3} \sqrt {-\frac {5 \sqrt [3]{6} \sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}} x^4+\frac {10\ 6^{2/3} \left (5 x^4-10 c_1 x^2-2\right ) x^4}{\sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}}}-6 x^2+\frac {6 \sqrt {3} \left (100 x^4+1\right )}{\sqrt {\frac {5 \sqrt [3]{6} \sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}} x^2+\frac {10\ 6^{2/3} \left (5 x^4-10 c_1 x^2-2\right ) x^2}{\sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}}}+3}{x^4}}}}{x^6}} x^2-3}{30 x^2}\right \},\left \{y(x)\to \frac {-\sqrt {3} \sqrt {\frac {5 \sqrt [3]{6} \sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}} x^2+\frac {10\ 6^{2/3} \left (5 x^4-10 c_1 x^2-2\right ) x^2}{\sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}}}+3}{x^4}} x^2+\sqrt {3} \sqrt {-\frac {5 \sqrt [3]{6} \sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}} x^4+\frac {10\ 6^{2/3} \left (5 x^4-10 c_1 x^2-2\right ) x^4}{\sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}}}-6 x^2+\frac {6 \sqrt {3} \left (100 x^4+1\right )}{\sqrt {\frac {5 \sqrt [3]{6} \sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}} x^2+\frac {10\ 6^{2/3} \left (5 x^4-10 c_1 x^2-2\right ) x^2}{\sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}}}+3}{x^4}}}}{x^6}} x^2+3}{30 x^2}\right \},\left \{y(x)\to \frac {\sqrt {3} \sqrt {\frac {5 \sqrt [3]{6} \sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}} x^2+\frac {10\ 6^{2/3} \left (5 x^4-10 c_1 x^2-2\right ) x^2}{\sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}}}+3}{x^4}} x^2-\sqrt {3} \sqrt {\frac {-5 \sqrt [3]{6} \sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}} x^4-\frac {10\ 6^{2/3} \left (5 x^4-10 c_1 x^2-2\right ) x^4}{\sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}}}+6 x^2+\frac {6 \sqrt {3} \left (100 x^4+1\right )}{\sqrt {\frac {5 \sqrt [3]{6} \sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}} x^2+\frac {10\ 6^{2/3} \left (5 x^4-10 c_1 x^2-2\right ) x^2}{\sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}}}+3}{x^4}}}}{x^6}} x^2+3}{30 x^2}\right \},\left \{y(x)\to \frac {\sqrt {3} \sqrt {\frac {5 \sqrt [3]{6} \sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}} x^2+\frac {10\ 6^{2/3} \left (5 x^4-10 c_1 x^2-2\right ) x^2}{\sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}}}+3}{x^4}} x^2+\sqrt {3} \sqrt {\frac {-5 \sqrt [3]{6} \sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}} x^4-\frac {10\ 6^{2/3} \left (5 x^4-10 c_1 x^2-2\right ) x^4}{\sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}}}+6 x^2+\frac {6 \sqrt {3} \left (100 x^4+1\right )}{\sqrt {\frac {5 \sqrt [3]{6} \sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}} x^2+\frac {10\ 6^{2/3} \left (5 x^4-10 c_1 x^2-2\right ) x^2}{\sqrt [3]{189 x^2-18 c_1+\sqrt {3} \sqrt {27 \left (21 x^2-2 c_1\right ){}^2-16 \left (5 x^4-10 c_1 x^2-2\right ){}^3}}}+3}{x^4}}}}{x^6}} x^2+3}{30 x^2}\right \}\right \}\]

Maple
cpu = 0.021 (sec), leaf count = 27

\[ \left \{ -{\frac {5\,{x}^{2} \left (y \relax (x ) \right ) ^{4}}{2}}+ \left (y \relax (x ) \right ) ^{3}-{\frac {{x}^{2}}{2}}+2\,y \relax (x ) +{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[(2 + 3*y[x]^2 - 10*x^2*y[x]^3)*y'[x] == x*(1 + 5*y[x]^4),y[x],x]

Mathematica raw output

{{y[x] -> -(-3 + Sqrt[3]*x^2*Sqrt[(3 + (10*6^(2/3)*x^2*(-2 + 5*x^4 - 10*x^2*C[1]
))/(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16*(-2 + 5*x^4 - 1
0*x^2*C[1])^3])^(1/3) + 5*6^(1/3)*x^2*(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[27*(21*x
^2 - 2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])^3])^(1/3))/x^4] + Sqrt[3]*x^2*Sqr
t[-((-6*x^2 + (10*6^(2/3)*x^4*(-2 + 5*x^4 - 10*x^2*C[1]))/(189*x^2 - 18*C[1] + S
qrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])^3])^(1/3) + 5
*6^(1/3)*x^4*(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16*(-2 +
 5*x^4 - 10*x^2*C[1])^3])^(1/3) + (6*Sqrt[3]*(1 + 100*x^4))/Sqrt[(3 + (10*6^(2/3
)*x^2*(-2 + 5*x^4 - 10*x^2*C[1]))/(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[27*(21*x^2 -
 2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])^3])^(1/3) + 5*6^(1/3)*x^2*(189*x^2 - 
18*C[1] + Sqrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])^3]
)^(1/3))/x^4])/x^6)])/(30*x^2)}, {y[x] -> (3 - Sqrt[3]*x^2*Sqrt[(3 + (10*6^(2/3)
*x^2*(-2 + 5*x^4 - 10*x^2*C[1]))/(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[27*(21*x^2 - 
2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])^3])^(1/3) + 5*6^(1/3)*x^2*(189*x^2 - 1
8*C[1] + Sqrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])^3])
^(1/3))/x^4] + Sqrt[3]*x^2*Sqrt[-((-6*x^2 + (10*6^(2/3)*x^4*(-2 + 5*x^4 - 10*x^2
*C[1]))/(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16*(-2 + 5*x^
4 - 10*x^2*C[1])^3])^(1/3) + 5*6^(1/3)*x^4*(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[27*
(21*x^2 - 2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])^3])^(1/3) + (6*Sqrt[3]*(1 + 
100*x^4))/Sqrt[(3 + (10*6^(2/3)*x^2*(-2 + 5*x^4 - 10*x^2*C[1]))/(189*x^2 - 18*C[
1] + Sqrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])^3])^(1/
3) + 5*6^(1/3)*x^2*(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16
*(-2 + 5*x^4 - 10*x^2*C[1])^3])^(1/3))/x^4])/x^6)])/(30*x^2)}, {y[x] -> (3 + Sqr
t[3]*x^2*Sqrt[(3 + (10*6^(2/3)*x^2*(-2 + 5*x^4 - 10*x^2*C[1]))/(189*x^2 - 18*C[1
] + Sqrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])^3])^(1/3
) + 5*6^(1/3)*x^2*(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16*
(-2 + 5*x^4 - 10*x^2*C[1])^3])^(1/3))/x^4] - Sqrt[3]*x^2*Sqrt[(6*x^2 - (10*6^(2/
3)*x^4*(-2 + 5*x^4 - 10*x^2*C[1]))/(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[27*(21*x^2 
- 2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])^3])^(1/3) - 5*6^(1/3)*x^4*(189*x^2 -
 18*C[1] + Sqrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])^3
])^(1/3) + (6*Sqrt[3]*(1 + 100*x^4))/Sqrt[(3 + (10*6^(2/3)*x^2*(-2 + 5*x^4 - 10*
x^2*C[1]))/(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16*(-2 + 5
*x^4 - 10*x^2*C[1])^3])^(1/3) + 5*6^(1/3)*x^2*(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[
27*(21*x^2 - 2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])^3])^(1/3))/x^4])/x^6])/(3
0*x^2)}, {y[x] -> (3 + Sqrt[3]*x^2*Sqrt[(3 + (10*6^(2/3)*x^2*(-2 + 5*x^4 - 10*x^
2*C[1]))/(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16*(-2 + 5*x
^4 - 10*x^2*C[1])^3])^(1/3) + 5*6^(1/3)*x^2*(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[27
*(21*x^2 - 2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])^3])^(1/3))/x^4] + Sqrt[3]*x
^2*Sqrt[(6*x^2 - (10*6^(2/3)*x^4*(-2 + 5*x^4 - 10*x^2*C[1]))/(189*x^2 - 18*C[1] 
+ Sqrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])^3])^(1/3) 
- 5*6^(1/3)*x^4*(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16*(-
2 + 5*x^4 - 10*x^2*C[1])^3])^(1/3) + (6*Sqrt[3]*(1 + 100*x^4))/Sqrt[(3 + (10*6^(
2/3)*x^2*(-2 + 5*x^4 - 10*x^2*C[1]))/(189*x^2 - 18*C[1] + Sqrt[3]*Sqrt[27*(21*x^
2 - 2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])^3])^(1/3) + 5*6^(1/3)*x^2*(189*x^2
 - 18*C[1] + Sqrt[3]*Sqrt[27*(21*x^2 - 2*C[1])^2 - 16*(-2 + 5*x^4 - 10*x^2*C[1])
^3])^(1/3))/x^4])/x^6])/(30*x^2)}}

Maple raw input

dsolve((2-10*x^2*y(x)^3+3*y(x)^2)*diff(y(x),x) = x*(1+5*y(x)^4), y(x),'implicit')

Maple raw output

-5/2*x^2*y(x)^4+y(x)^3-1/2*x^2+2*y(x)+_C1 = 0