ODE
\[ y'(x)^3=f(x) \left (a+b y(x)+c y(x)^2\right ) \] ODE Classification
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
Book solution method
No Missing Variables ODE, Solve for \(y'\)
Mathematica ✓
cpu = 1.80384 (sec), leaf count = 473
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\frac {3 \left (2 \text {$\#$1} c-\sqrt {b^2-4 a c}+b\right ) \sqrt [3]{\frac {2 \text {$\#$1} c+\sqrt {b^2-4 a c}+b}{\sqrt {b^2-4 a c}}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};\frac {-b-2 c \text {$\#$1}+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{4 \sqrt [3]{2} c \sqrt [3]{\text {$\#$1} (\text {$\#$1} c+b)+a}}\& \right ]\left [\int _1^x \sqrt [3]{f(K[1])} \, dK[1]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {3 \left (2 \text {$\#$1} c-\sqrt {b^2-4 a c}+b\right ) \sqrt [3]{\frac {2 \text {$\#$1} c+\sqrt {b^2-4 a c}+b}{\sqrt {b^2-4 a c}}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};\frac {-b-2 c \text {$\#$1}+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{4 \sqrt [3]{2} c \sqrt [3]{\text {$\#$1} (\text {$\#$1} c+b)+a}}\& \right ]\left [\int _1^x -\sqrt [3]{-1} \sqrt [3]{f(K[2])} \, dK[2]+c_1\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {3 \left (2 \text {$\#$1} c-\sqrt {b^2-4 a c}+b\right ) \sqrt [3]{\frac {2 \text {$\#$1} c+\sqrt {b^2-4 a c}+b}{\sqrt {b^2-4 a c}}} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};\frac {-b-2 c \text {$\#$1}+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )}{4 \sqrt [3]{2} c \sqrt [3]{\text {$\#$1} (\text {$\#$1} c+b)+a}}\& \right ]\left [\int _1^x (-1)^{2/3} \sqrt [3]{f(K[3])} \, dK[3]+c_1\right ]\right \}\right \}\]
Maple ✓
cpu = 0.537 (sec), leaf count = 191
\[ \left \{ \int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt [3]{{{\it \_a}}^{2}c+{\it \_a}\,b+a}}}{d{\it \_a}}+\int ^{x}\!-{1\sqrt [3]{ \left ( a+by \left ( x \right ) +c \left ( y \left ( x \right ) \right ) ^{2} \right ) f \left ( {\it \_a} \right ) }{\frac {1}{\sqrt [3]{a+by \left ( x \right ) +c \left ( y \left ( x \right ) \right ) ^{2}}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt [3]{{{\it \_a}}^{2}c+{\it \_a}\,b+a}}}{d{\it \_a}}+\int ^{x}\!-{\frac {i\sqrt {3}-1}{2}\sqrt [3]{ \left ( a+by \left ( x \right ) +c \left ( y \left ( x \right ) \right ) ^{2} \right ) f \left ( {\it \_a} \right ) }{\frac {1}{\sqrt [3]{a+by \left ( x \right ) +c \left ( y \left ( x \right ) \right ) ^{2}}}}}{d{\it \_a}}+{\it \_C1}=0,\int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt [3]{{{\it \_a}}^{2}c+{\it \_a}\,b+a}}}{d{\it \_a}}+\int ^{x}\!{\frac {i\sqrt {3}+1}{2}\sqrt [3]{ \left ( a+by \left ( x \right ) +c \left ( y \left ( x \right ) \right ) ^{2} \right ) f \left ( {\it \_a} \right ) }{\frac {1}{\sqrt [3]{a+by \left ( x \right ) +c \left ( y \left ( x \right ) \right ) ^{2}}}}}{d{\it \_a}}+{\it \_C1}=0 \right \} \] Mathematica raw input
DSolve[y'[x]^3 == f[x]*(a + b*y[x] + c*y[x]^2),y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[(3*Hypergeometric2F1[1/3, 2/3, 5/3, (-b + Sqrt[b^2 - 4*a*c] - 2*c*#1)/(2*Sqrt[b^2 - 4*a*c])]*(b - Sqrt[b^2 - 4*a*c] + 2*c*#1)*((b + Sqrt[b^2 - 4*a*c] + 2*c*#1)/Sqrt[b^2 - 4*a*c])^(1/3))/(4*2^(1/3)*c*(a + #1*(b + c*#1))^(1/3)) & ][C[1] + Integrate[f[K[1]]^(1/3), {K[1], 1, x}]]}, {y[x] -> InverseFunction[(3*Hypergeometric2F1[1/3, 2/3, 5/3, (-b + Sqrt[b^2 - 4*a*c] - 2*c*#1)/(2*Sqrt[b^2 - 4*a*c])]*(b - Sqrt[b^2 - 4*a*c] + 2*c*#1)*((b + Sqrt[b^2 - 4*a*c] + 2*c*#1)/Sqrt[b^2 - 4*a*c])^(1/3))/(4*2^(1/3)*c*(a + #1*(b + c*#1))^(1/3)) & ][C[1] + Integrate[-((-1)^(1/3)*f[K[2]]^(1/3)), {K[2], 1, x}]]}, {y[x] -> InverseFunction[(3*Hypergeometric2F1[1/3, 2/3, 5/3, (-b + Sqrt[b^2 - 4*a*c] - 2*c*#1)/(2*Sqrt[b^2 - 4*a*c])]*(b - Sqrt[b^2 - 4*a*c] + 2*c*#1)*((b + Sqrt[b^2 - 4*a*c] + 2*c*#1)/Sqrt[b^2 - 4*a*c])^(1/3))/(4*2^(1/3)*c*(a + #1*(b + c*#1))^(1/3)) & ][C[1] + Integrate[(-1)^(2/3)*f[K[3]]^(1/3), {K[3], 1, x}]]}}
Maple raw input
dsolve(diff(y(x),x)^3 = (a+b*y(x)+c*y(x)^2)*f(x), y(x),'implicit')
Maple raw output
Intat(1/(_a^2*c+_a*b+a)^(1/3),_a = y(x))+Intat(-((a+b*y(x)+c*y(x)^2)*f(_a))^(1/3)/(a+b*y(x)+c*y(x)^2)^(1/3),_a = x)+_C1 = 0, Intat(1/(_a^2*c+_a*b+a)^(1/3),_a = y(x))+Intat(1/2*((a+b*y(x)+c*y(x)^2)*f(_a))^(1/3)*(I*3^(1/2)+1)/(a+b*y(x)+c*y(x)^2)^(1/3),_a = x)+_C1 = 0, Intat(1/(_a^2*c+_a*b+a)^(1/3),_a = y(x))+Intat(-1/2*((a+b*y(x)+c*y(x)^2)*f(_a))^(1/3)*(I*3^(1/2)-1)/(a+b*y(x)+c*y(x)^2)^(1/3),_a = x)+_C1 = 0