ODE
\[ x \left (4 x^2 y(x)-12 x y(x)^2+5 x+3\right ) y'(x)+y(x) \left (6 x^2 y(x)-8 x y(x)^2+10 x+3\right )=0 \] ODE Classification
[_exact, _rational]
Book solution method
Exact equation
Mathematica ✓
cpu = 0.455555 (sec), leaf count = 660
\[\left \{\left \{y(x)\to -\frac {\sqrt [3]{-16 x^9-360 x^7-216 x^6+432 c_1 x^4+8 \sqrt {-4 x^9 \left (x^3+15 x+9\right )^3+\left (2 x^9+45 x^7+27 x^6-54 c_1 x^4\right ){}^2}}}{12 \sqrt [3]{2} x^2}-\frac {\left (x^3+15 x+9\right ) x}{3\ 2^{2/3} \sqrt [3]{-2 x^9-45 x^7-27 x^6+54 c_1 x^4+3 \sqrt {3} \sqrt {-x^8 \left (25 x^6+(30+8 c_1) x^5+509 x^4+180 (5+c_1) x^3+108 (5+c_1) x^2+108 x-108 c_1{}^2\right )}}}+\frac {x}{6}\right \},\left \{y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-16 x^9-360 x^7-216 x^6+432 c_1 x^4+8 \sqrt {-4 x^9 \left (x^3+15 x+9\right )^3+\left (2 x^9+45 x^7+27 x^6-54 c_1 x^4\right ){}^2}}}{24 \sqrt [3]{2} x^2}+\frac {\left (1+i \sqrt {3}\right ) \left (x^3+15 x+9\right ) x}{6\ 2^{2/3} \sqrt [3]{-2 x^9-45 x^7-27 x^6+54 c_1 x^4+3 \sqrt {3} \sqrt {-x^8 \left (25 x^6+(30+8 c_1) x^5+509 x^4+180 (5+c_1) x^3+108 (5+c_1) x^2+108 x-108 c_1{}^2\right )}}}+\frac {x}{6}\right \},\left \{y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-16 x^9-360 x^7-216 x^6+432 c_1 x^4+8 \sqrt {-4 x^9 \left (x^3+15 x+9\right )^3+\left (2 x^9+45 x^7+27 x^6-54 c_1 x^4\right ){}^2}}}{24 \sqrt [3]{2} x^2}+\frac {\left (1-i \sqrt {3}\right ) \left (x^3+15 x+9\right ) x}{6\ 2^{2/3} \sqrt [3]{-2 x^9-45 x^7-27 x^6+54 c_1 x^4+3 \sqrt {3} \sqrt {-x^8 \left (25 x^6+(30+8 c_1) x^5+509 x^4+180 (5+c_1) x^3+108 (5+c_1) x^2+108 x-108 c_1{}^2\right )}}}+\frac {x}{6}\right \}\right \}\]
Maple ✓
cpu = 0.044 (sec), leaf count = 931
\[\left [y \left (x \right ) = \frac {\left (\left (8 x^{5}+180 x^{3}+12 \sqrt {3}\, \sqrt {8 \textit {\_C1} \,x^{5}-25 x^{6}-30 x^{5}+180 \textit {\_C1} \,x^{3}-509 x^{4}+108 x^{2} \textit {\_C1} -900 x^{3}+108 \textit {\_C1}^{2}-540 x^{2}-108 x}+108 x^{2}+216 \textit {\_C1} \right ) x \right )^{\frac {1}{3}}}{12 x}+\frac {x^{3}+15 x +9}{3 \left (\left (8 x^{5}+180 x^{3}+12 \sqrt {3}\, \sqrt {8 \textit {\_C1} \,x^{5}-25 x^{6}-30 x^{5}+180 \textit {\_C1} \,x^{3}-509 x^{4}+108 x^{2} \textit {\_C1} -900 x^{3}+108 \textit {\_C1}^{2}-540 x^{2}-108 x}+108 x^{2}+216 \textit {\_C1} \right ) x \right )^{\frac {1}{3}}}+\frac {x}{6}, y \left (x \right ) = -\frac {\left (\left (8 x^{5}+180 x^{3}+12 \sqrt {3}\, \sqrt {8 \textit {\_C1} \,x^{5}-25 x^{6}-30 x^{5}+180 \textit {\_C1} \,x^{3}-509 x^{4}+108 x^{2} \textit {\_C1} -900 x^{3}+108 \textit {\_C1}^{2}-540 x^{2}-108 x}+108 x^{2}+216 \textit {\_C1} \right ) x \right )^{\frac {1}{3}}}{24 x}-\frac {x^{3}+15 x +9}{6 \left (\left (8 x^{5}+180 x^{3}+12 \sqrt {3}\, \sqrt {8 \textit {\_C1} \,x^{5}-25 x^{6}-30 x^{5}+180 \textit {\_C1} \,x^{3}-509 x^{4}+108 x^{2} \textit {\_C1} -900 x^{3}+108 \textit {\_C1}^{2}-540 x^{2}-108 x}+108 x^{2}+216 \textit {\_C1} \right ) x \right )^{\frac {1}{3}}}+\frac {x}{6}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (8 x^{5}+180 x^{3}+12 \sqrt {3}\, \sqrt {8 \textit {\_C1} \,x^{5}-25 x^{6}-30 x^{5}+180 \textit {\_C1} \,x^{3}-509 x^{4}+108 x^{2} \textit {\_C1} -900 x^{3}+108 \textit {\_C1}^{2}-540 x^{2}-108 x}+108 x^{2}+216 \textit {\_C1} \right ) x \right )^{\frac {1}{3}}}{12 x}-\frac {x^{3}+15 x +9}{3 \left (\left (8 x^{5}+180 x^{3}+12 \sqrt {3}\, \sqrt {8 \textit {\_C1} \,x^{5}-25 x^{6}-30 x^{5}+180 \textit {\_C1} \,x^{3}-509 x^{4}+108 x^{2} \textit {\_C1} -900 x^{3}+108 \textit {\_C1}^{2}-540 x^{2}-108 x}+108 x^{2}+216 \textit {\_C1} \right ) x \right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {\left (\left (8 x^{5}+180 x^{3}+12 \sqrt {3}\, \sqrt {8 \textit {\_C1} \,x^{5}-25 x^{6}-30 x^{5}+180 \textit {\_C1} \,x^{3}-509 x^{4}+108 x^{2} \textit {\_C1} -900 x^{3}+108 \textit {\_C1}^{2}-540 x^{2}-108 x}+108 x^{2}+216 \textit {\_C1} \right ) x \right )^{\frac {1}{3}}}{24 x}-\frac {x^{3}+15 x +9}{6 \left (\left (8 x^{5}+180 x^{3}+12 \sqrt {3}\, \sqrt {8 \textit {\_C1} \,x^{5}-25 x^{6}-30 x^{5}+180 \textit {\_C1} \,x^{3}-509 x^{4}+108 x^{2} \textit {\_C1} -900 x^{3}+108 \textit {\_C1}^{2}-540 x^{2}-108 x}+108 x^{2}+216 \textit {\_C1} \right ) x \right )^{\frac {1}{3}}}+\frac {x}{6}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (8 x^{5}+180 x^{3}+12 \sqrt {3}\, \sqrt {8 \textit {\_C1} \,x^{5}-25 x^{6}-30 x^{5}+180 \textit {\_C1} \,x^{3}-509 x^{4}+108 x^{2} \textit {\_C1} -900 x^{3}+108 \textit {\_C1}^{2}-540 x^{2}-108 x}+108 x^{2}+216 \textit {\_C1} \right ) x \right )^{\frac {1}{3}}}{12 x}-\frac {x^{3}+15 x +9}{3 \left (\left (8 x^{5}+180 x^{3}+12 \sqrt {3}\, \sqrt {8 \textit {\_C1} \,x^{5}-25 x^{6}-30 x^{5}+180 \textit {\_C1} \,x^{3}-509 x^{4}+108 x^{2} \textit {\_C1} -900 x^{3}+108 \textit {\_C1}^{2}-540 x^{2}-108 x}+108 x^{2}+216 \textit {\_C1} \right ) x \right )^{\frac {1}{3}}}\right )}{2}\right ]\] Mathematica raw input
DSolve[y[x]*(3 + 10*x + 6*x^2*y[x] - 8*x*y[x]^2) + x*(3 + 5*x + 4*x^2*y[x] - 12*x*y[x]^2)*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> x/6 - (x*(9 + 15*x + x^3))/(3*2^(2/3)*(-27*x^6 - 45*x^7 - 2*x^9 + 54*x^4*C[1] + 3*Sqrt[3]*Sqrt[-(x^8*(108*x + 509*x^4 + 25*x^6 - 108*C[1]^2 + 108*x^2*(5 + C[1]) + 180*x^3*(5 + C[1]) + x^5*(30 + 8*C[1])))])^(1/3)) - (-216*x^6 - 360*x^7 - 16*x^9 + 432*x^4*C[1] + 8*Sqrt[-4*x^9*(9 + 15*x + x^3)^3 + (27*x^6 + 45*x^7 + 2*x^9 - 54*x^4*C[1])^2])^(1/3)/(12*2^(1/3)*x^2)}, {y[x] -> x/6 + ((1 + I*Sqrt[3])*x*(9 + 15*x + x^3))/(6*2^(2/3)*(-27*x^6 - 45*x^7 - 2*x^9 + 54*x^4*C[1] + 3*Sqrt[3]*Sqrt[-(x^8*(108*x + 509*x^4 + 25*x^6 - 108*C[1]^2 + 108*x^2*(5 + C[1]) + 180*x^3*(5 + C[1]) + x^5*(30 + 8*C[1])))])^(1/3)) + ((1 - I*Sqrt[3])*(-216*x^6 - 360*x^7 - 16*x^9 + 432*x^4*C[1] + 8*Sqrt[-4*x^9*(9 + 15*x + x^3)^3 + (27*x^6 + 45*x^7 + 2*x^9 - 54*x^4*C[1])^2])^(1/3))/(24*2^(1/3)*x^2)}, {y[x] -> x/6 + ((1 - I*Sqrt[3])*x*(9 + 15*x + x^3))/(6*2^(2/3)*(-27*x^6 - 45*x^7 - 2*x^9 + 54*x^4*C[1] + 3*Sqrt[3]*Sqrt[-(x^8*(108*x + 509*x^4 + 25*x^6 - 108*C[1]^2 + 108*x^2*(5 + C[1]) + 180*x^3*(5 + C[1]) + x^5*(30 + 8*C[1])))])^(1/3)) + ((1 + I*Sqrt[3])*(-216*x^6 - 360*x^7 - 16*x^9 + 432*x^4*C[1] + 8*Sqrt[-4*x^9*(9 + 15*x + x^3)^3 + (27*x^6 + 45*x^7 + 2*x^9 - 54*x^4*C[1])^2])^(1/3))/(24*2^(1/3)*x^2)}}
Maple raw input
dsolve(x*(3+5*x-12*x*y(x)^2+4*x^2*y(x))*diff(y(x),x)+(3+10*x-8*x*y(x)^2+6*x^2*y(x))*y(x) = 0, y(x))
Maple raw output
[y(x) = 1/12/x*((8*x^5+180*x^3+12*3^(1/2)*(8*_C1*x^5-25*x^6-30*x^5+180*_C1*x^3-509*x^4+108*_C1*x^2-900*x^3+108*_C1^2-540*x^2-108*x)^(1/2)+108*x^2+216*_C1)*x)^(1/3)+1/3*(x^3+15*x+9)/((8*x^5+180*x^3+12*3^(1/2)*(8*_C1*x^5-25*x^6-30*x^5+180*_C1*x^3-509*x^4+108*_C1*x^2-900*x^3+108*_C1^2-540*x^2-108*x)^(1/2)+108*x^2+216*_C1)*x)^(1/3)+1/6*x, y(x) = -1/24/x*((8*x^5+180*x^3+12*3^(1/2)*(8*_C1*x^5-25*x^6-30*x^5+180*_C1*x^3-509*x^4+108*_C1*x^2-900*x^3+108*_C1^2-540*x^2-108*x)^(1/2)+108*x^2+216*_C1)*x)^(1/3)-1/6*(x^3+15*x+9)/((8*x^5+180*x^3+12*3^(1/2)*(8*_C1*x^5-25*x^6-30*x^5+180*_C1*x^3-509*x^4+108*_C1*x^2-900*x^3+108*_C1^2-540*x^2-108*x)^(1/2)+108*x^2+216*_C1)*x)^(1/3)+1/6*x-1/2*I*3^(1/2)*(1/12/x*((8*x^5+180*x^3+12*3^(1/2)*(8*_C1*x^5-25*x^6-30*x^5+180*_C1*x^3-509*x^4+108*_C1*x^2-900*x^3+108*_C1^2-540*x^2-108*x)^(1/2)+108*x^2+216*_C1)*x)^(1/3)-1/3*(x^3+15*x+9)/((8*x^5+180*x^3+12*3^(1/2)*(8*_C1*x^5-25*x^6-30*x^5+180*_C1*x^3-509*x^4+108*_C1*x^2-900*x^3+108*_C1^2-540*x^2-108*x)^(1/2)+108*x^2+216*_C1)*x)^(1/3)), y(x) = -1/24/x*((8*x^5+180*x^3+12*3^(1/2)*(8*_C1*x^5-25*x^6-30*x^5+180*_C1*x^3-509*x^4+108*_C1*x^2-900*x^3+108*_C1^2-540*x^2-108*x)^(1/2)+108*x^2+216*_C1)*x)^(1/3)-1/6*(x^3+15*x+9)/((8*x^5+180*x^3+12*3^(1/2)*(8*_C1*x^5-25*x^6-30*x^5+180*_C1*x^3-509*x^4+108*_C1*x^2-900*x^3+108*_C1^2-540*x^2-108*x)^(1/2)+108*x^2+216*_C1)*x)^(1/3)+1/6*x+1/2*I*3^(1/2)*(1/12/x*((8*x^5+180*x^3+12*3^(1/2)*(8*_C1*x^5-25*x^6-30*x^5+180*_C1*x^3-509*x^4+108*_C1*x^2-900*x^3+108*_C1^2-540*x^2-108*x)^(1/2)+108*x^2+216*_C1)*x)^(1/3)-1/3*(x^3+15*x+9)/((8*x^5+180*x^3+12*3^(1/2)*(8*_C1*x^5-25*x^6-30*x^5+180*_C1*x^3-509*x^4+108*_C1*x^2-900*x^3+108*_C1^2-540*x^2-108*x)^(1/2)+108*x^2+216*_C1)*x)^(1/3))]