\[ -4 a^2 y(x)-3 a y(x)^2-b-3 y(x) y'(x)+y''(x)=0 \] ✓ Mathematica : cpu = 7.19286 (sec), leaf count = 3227
\[\left \{\left \{y(x)\to -\frac {2 \left ((-1)^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}+1} 2^{-\frac {3 \left (a^{3/2} \sqrt {4 a^3-3 b}-2 a^3\right )}{4 a^3}+\frac {3 \sqrt {4 a^6-3 a^3 b}}{4 a^3}+1} 3^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} a^{-\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{2 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1} b^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-1} \left (\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}\right ) e^{-2 a x} I_{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right ) c_1{}^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} \Gamma \left (\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1\right ) \left (\frac {e^{-2 a x}}{b}\right )^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-1}+(-1)^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}+1} 2^{-\frac {3 \left (a^{3/2} \sqrt {4 a^3-3 b}-2 a^3\right )}{4 a^3}+\frac {3 \sqrt {4 a^6-3 a^3 b}}{4 a^3}-\frac {3}{2}} 3^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}+\frac {1}{2}} a^{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}-\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{2 a^3}} b^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-\frac {1}{2}} e^{-2 a x} \left (I_{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}-1}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right )+I_{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right )\right ) c_1{}^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}+\frac {1}{2}} \Gamma \left (\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1\right ) \left (\frac {e^{-2 a x}}{b}\right )^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-\frac {1}{2}}+c_2 \left ((-1)^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+1} 2^{-\frac {3 \left (-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}\right )}{4 a^3}-\frac {3 \sqrt {4 a^6-3 a^3 b}}{4 a^3}-\frac {3}{2}} 3^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}+\frac {1}{2}} b^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-\frac {1}{2}} e^{-2 a x} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-\frac {1}{2}} \left (I_{-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}-1}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right )+I_{1-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right )\right ) c_1{}^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}+\frac {1}{2}} \Gamma \left (1-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}\right ) a^{-\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{2 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}+(-1)^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+1} 2^{-\frac {3 \left (-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}\right )}{4 a^3}-\frac {3 \sqrt {4 a^6-3 a^3 b}}{4 a^3}+1} 3^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} b^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-1} \left (\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}\right ) e^{-2 a x} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}-1} I_{-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right ) c_1{}^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} \Gamma \left (1-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}\right ) a^{-\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{2 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1}\right )\right )}{3 \left ((-1)^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}} 2^{-\frac {3 \left (-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}\right )}{4 a^3}-\frac {3 \sqrt {4 a^6-3 a^3 b}}{4 a^3}} 3^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} b^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} I_{-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right ) c_1{}^{\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{4 a^3}+\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} c_2 \Gamma \left (1-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}\right ) a^{-\frac {-2 a^3-\sqrt {4 a^3-3 b} a^{3/2}}{2 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}+(-1)^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}} 2^{\frac {3 \sqrt {4 a^6-3 a^3 b}}{4 a^3}-\frac {3 \left (a^{3/2} \sqrt {4 a^3-3 b}-2 a^3\right )}{4 a^3}} 3^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} b^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} \left (\frac {e^{-2 a x}}{b}\right )^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} I_{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {b} \sqrt {\frac {e^{-2 a x}}{b}} \sqrt {c_1}}{a}\right ) c_1{}^{\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{4 a^3}-\frac {\sqrt {4 a^6-3 a^3 b}}{4 a^3}} \Gamma \left (\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}+1\right ) a^{\frac {\sqrt {4 a^6-3 a^3 b}}{2 a^3}-\frac {a^{3/2} \sqrt {4 a^3-3 b}-2 a^3}{2 a^3}}\right )}\right \}\right \}\] ✓ Maple : cpu = 0.946 (sec), leaf count = 783
\[ \left \{ \int ^{y \left ( x \right ) }\!-6\,{{a}^{2} \left ( -12\,{\it \_a}\,{a}^{3}-9\,{{\it \_a}}^{2}{a}^{2}+ \left ( {\it RootOf} \left ( 2\,{{\sl K}_{1/2\,{\frac {4\,{a}^{3}-3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (-1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )}{\it \_C1}\,{a}^{2}+3\,{{\sl K}_{1/2\,{\frac {4\,{a}^{3}-3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (-1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )}{\it \_C1}\,a{\it \_a}+2\,{{\sl I}_{-1/2\,{\frac {4\,{a}^{3}-3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (-1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )}{a}^{2}+3\,{{\sl I}_{-1/2\,{\frac {4\,{a}^{3}-3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (-1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )}a{\it \_a}+{{\sl K}_{1/2\,{\frac {4\,{a}^{3}-3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (-1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )}\sqrt {4\,{a}^{4}-3\,ab}{\it \_C1}+{\it \_Z}\,{{\sl K}_{1/2\,{\frac {-4\,{a}^{3}+2\,\sqrt {4\,{a}^{4}-3\,ab}a+3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (-1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )}{\it \_C1}+{{\sl I}_{-1/2\,{\frac {4\,{a}^{3}-3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (-1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )}\sqrt {4\,{a}^{4}-3\,ab}-{\it \_Z}\,{{\sl I}_{1/2\,{\frac {-4\,{a}^{3}+2\,\sqrt {4\,{a}^{4}-3\,ab}a+3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (-1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )} \right ) \right ) ^{2}-3\,ab \right ) ^{-1}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-6\,{{a}^{2} \left ( -12\,{\it \_a}\,{a}^{3}-9\,{{\it \_a}}^{2}{a}^{2}+ \left ( {\it RootOf} \left ( 2\,{{\sl K}_{1/2\,{\frac {4\,{a}^{3}-3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )}{\it \_C1}\,{a}^{2}+3\,{{\sl K}_{1/2\,{\frac {4\,{a}^{3}-3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )}{\it \_C1}\,a{\it \_a}+2\,{{\sl I}_{-1/2\,{\frac {4\,{a}^{3}-3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )}{a}^{2}+3\,{{\sl I}_{-1/2\,{\frac {4\,{a}^{3}-3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )}a{\it \_a}+{{\sl K}_{1/2\,{\frac {4\,{a}^{3}-3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )}\sqrt {4\,{a}^{4}-3\,ab}{\it \_C1}+{\it \_Z}\,{{\sl K}_{1/2\,{\frac {-4\,{a}^{3}+2\,\sqrt {4\,{a}^{4}-3\,ab}a+3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )}{\it \_C1}+{{\sl I}_{-1/2\,{\frac {4\,{a}^{3}-3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )}\sqrt {4\,{a}^{4}-3\,ab}-{\it \_Z}\,{{\sl I}_{1/2\,{\frac {-4\,{a}^{3}+2\,\sqrt {4\,{a}^{4}-3\,ab}a+3\,b}{\sqrt {4\,{a}^{4}-3\,ab}a}}}\left (1/2\,{\frac {{\it \_Z}}{{a}^{2}}}\right )} \right ) \right ) ^{2}-3\,ab \right ) ^{-1}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \]