\[ x^2 \left (a y(x)^2+y'(x)\right )+b x^{\alpha }+c=0 \] ✓ Mathematica : cpu = 0.250037 (sec), leaf count = 1787
DSolve[c + b*x^alpha + x^2*(a*y[x]^2 + Derivative[1][y][x]) == 0,y[x],x]
\[\left \{\left \{y(x)\to \frac {a^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} \alpha ^{-\frac {i \sqrt {4 a c-1} \alpha +\alpha }{\alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}+1} b^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} \left (\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}\right ) \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}-1} J_{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \Gamma \left (\frac {\sqrt {1-4 a c}}{\alpha }+1\right ) x^{\alpha -1}+\frac {1}{2} a^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}+\frac {1}{2}} \alpha ^{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}-\frac {i \sqrt {4 a c-1} \alpha +\alpha }{\alpha ^2}} b^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}+\frac {1}{2}} \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}-\frac {1}{2}} \left (J_{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right )-J_{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}+1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right )\right ) \Gamma \left (\frac {\sqrt {1-4 a c}}{\alpha }+1\right ) x^{\alpha -1}+c_1 \left (a^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} \alpha ^{-\frac {\alpha -i \alpha \sqrt {4 a c-1}}{\alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}+1} b^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} \left (\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}\right ) \left (x^{\alpha }\right )^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}-1} J_{-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \Gamma \left (1-\frac {\sqrt {1-4 a c}}{\alpha }\right ) x^{\alpha -1}+\frac {1}{2} a^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}+\frac {1}{2}} \alpha ^{-\frac {\alpha -i \alpha \sqrt {4 a c-1}}{\alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}} b^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}+\frac {1}{2}} \left (x^{\alpha }\right )^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}-\frac {1}{2}} \left (J_{-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right )-J_{1-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right )\right ) \Gamma \left (1-\frac {\sqrt {1-4 a c}}{\alpha }\right ) x^{\alpha -1}\right )}{a \left (a^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} b^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} \left (x^{\alpha }\right )^{\frac {\alpha -i \alpha \sqrt {4 a c-1}}{2 \alpha ^2}+\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} J_{-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) c_1 \Gamma \left (1-\frac {\sqrt {1-4 a c}}{\alpha }\right ) \alpha ^{-\frac {\alpha -i \alpha \sqrt {4 a c-1}}{\alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}+a^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} b^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} \left (x^{\alpha }\right )^{\frac {i \sqrt {4 a c-1} \alpha +\alpha }{2 \alpha ^2}-\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{2 \alpha ^2}} J_{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {x^{\alpha }}}{\alpha }\right ) \Gamma \left (\frac {\sqrt {1-4 a c}}{\alpha }+1\right ) \alpha ^{\frac {\sqrt {\alpha ^2-4 a \alpha ^2 c}}{\alpha ^2}-\frac {i \sqrt {4 a c-1} \alpha +\alpha }{\alpha ^2}}\right )}\right \}\right \}\] ✓ Maple : cpu = 0.11 (sec), leaf count = 219
dsolve(x^2*(diff(y(x),x)+a*y(x)^2)+b*x^alpha+c = 0,y(x))
\[y \left (x \right ) = \frac {-2 \left (\BesselY \left (\frac {\sqrt {-4 a c +1}+\alpha }{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1}+\BesselJ \left (\frac {\sqrt {-4 a c +1}+\alpha }{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right ) \sqrt {a b}\, x^{\frac {\alpha }{2}}+\left (\sqrt {-4 a c +1}+1\right ) \left (\BesselY \left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1}+\BesselJ \left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right )}{2 x a \left (\BesselY \left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right ) c_{1}+\BesselJ \left (\frac {\sqrt {-4 a c +1}}{\alpha }, \frac {2 \sqrt {a b}\, x^{\frac {\alpha }{2}}}{\alpha }\right )\right )}\]