\[ a y(x)+y''(x)+y(x) y'(x)-y(x)^3=0 \] ✓ Mathematica : cpu = 7.99959 (sec), leaf count = 990
DSolve[a*y[x] - y[x]^3 + y[x]*Derivative[1][y][x] + Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{\frac {e^{6 c_1} \left (a-K[1]^2\right )^2}{2 \sqrt [3]{e^{18 c_1} K[1]^6-3 a e^{18 c_1} K[1]^4+3 a^2 e^{18 c_1} K[1]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[1]^6+3 a e^{30 c_1} K[1]^4-3 a^2 e^{30 c_1} K[1]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}+\frac {1}{2} \left (a-K[1]^2\right )+\frac {1}{2} e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[1]^6-3 a e^{18 c_1} K[1]^4+3 a^2 e^{18 c_1} K[1]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[1]^6+3 a e^{30 c_1} K[1]^4-3 a^2 e^{30 c_1} K[1]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}dK[1]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1+i \sqrt {3}\right ) e^{6 c_1} \left (a-K[2]^2\right )^2}{4 \sqrt [3]{e^{18 c_1} K[2]^6-3 a e^{18 c_1} K[2]^4+3 a^2 e^{18 c_1} K[2]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[2]^6+3 a e^{30 c_1} K[2]^4-3 a^2 e^{30 c_1} K[2]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}+\frac {1}{2} \left (a-K[2]^2\right )-\frac {1}{4} \left (1-i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[2]^6-3 a e^{18 c_1} K[2]^4+3 a^2 e^{18 c_1} K[2]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[2]^6+3 a e^{30 c_1} K[2]^4-3 a^2 e^{30 c_1} K[2]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}dK[2]\& \right ][x+c_2]\right \},\left \{y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{-\frac {\left (1-i \sqrt {3}\right ) e^{6 c_1} \left (a-K[3]^2\right )^2}{4 \sqrt [3]{e^{18 c_1} K[3]^6-3 a e^{18 c_1} K[3]^4+3 a^2 e^{18 c_1} K[3]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[3]^6+3 a e^{30 c_1} K[3]^4-3 a^2 e^{30 c_1} K[3]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}+\frac {1}{2} \left (a-K[3]^2\right )-\frac {1}{4} \left (1+i \sqrt {3}\right ) e^{-6 c_1} \sqrt [3]{e^{18 c_1} K[3]^6-3 a e^{18 c_1} K[3]^4+3 a^2 e^{18 c_1} K[3]^2-2 e^{12 c_1}-a^3 e^{18 c_1}+2 \sqrt {-e^{30 c_1} K[3]^6+3 a e^{30 c_1} K[3]^4-3 a^2 e^{30 c_1} K[3]^2+e^{24 c_1}+a^3 e^{30 c_1}}}}dK[3]\& \right ][x+c_2]\right \}\right \}\] ✓ Maple : cpu = 2.071 (sec), leaf count = 108
dsolve(diff(diff(y(x),x),x)+y(x)*diff(y(x),x)-y(x)^3+a*y(x)=0,y(x))
\[\int _{}^{y \left (x \right )}\frac {4 \RootOf \left (\left (-4 \textit {\_a}^{6}+12 a \,\textit {\_a}^{4}-12 \textit {\_a}^{2} a^{2}+4 a^{3}+320 c_{1}\right ) \textit {\_Z}^{9}+\left (-189 \textit {\_a}^{6}+567 a \,\textit {\_a}^{4}-567 \textit {\_a}^{2} a^{2}+189 a^{3}+15120 c_{1}\right ) \textit {\_Z}^{6}+238140 c_{1} \textit {\_Z}^{3}+1250235 c_{1}\right )^{3}+63}{-63 \textit {\_a}^{2}+63 a}d \textit {\_a} -x -c_{2} = 0\]