ODE No. 144

$$x^2(ay(x{)}^2+y^{\prime }(x))+b{x}^{\alpha }+c=0$$ ✓ 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\{\{y(x)\to \frac{a^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{\alpha }^{-\frac{i\sqrt{4ac-1}\alpha +\alpha }{{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}+1}{b}^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}(\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}){\left(x^{\alpha }\right)}^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}-1}{J}_{\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right)\mathrm{\Gamma }(\frac{\sqrt{1-4ac}}{\alpha }+1)x^{\alpha -1}+\frac{1}{2}{a}^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}+\frac{1}{2}}{\alpha }^{\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}-\frac{i\sqrt{4ac-1}\alpha +\alpha }{{\alpha }^2}}{b}^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}+\frac{1}{2}}{\left(x^{\alpha }\right)}^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}-\frac{1}{2}}(J_{\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}-1}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right)-J_{\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}+1}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right))\mathrm{\Gamma }(\frac{\sqrt{1-4ac}}{\alpha }+1)x^{\alpha -1}+c_1(a^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{\alpha }^{-\frac{\alpha -i\alpha \sqrt{4ac-1}}{{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}+1}{b}^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}(\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}){\left(x^{\alpha }\right)}^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}-1}{J}_{-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right)\mathrm{\Gamma }(1-\frac{\sqrt{1-4ac}}{\alpha })x^{\alpha -1}+\frac{1}{2}{a}^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}+\frac{1}{2}}{\alpha }^{-\frac{\alpha -i\alpha \sqrt{4ac-1}}{{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}}{b}^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}+\frac{1}{2}}{\left(x^{\alpha }\right)}^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}-\frac{1}{2}}(J_{-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}-1}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right)-J_{1-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right))\mathrm{\Gamma }(1-\frac{\sqrt{1-4ac}}{\alpha })x^{\alpha -1})}{a(a^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{b}^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{\left(x^{\alpha }\right)}^{\frac{\alpha -i\alpha \sqrt{4ac-1}}{2{\alpha }^2}+\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{J}_{-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right)c_1\mathrm{\Gamma }(1-\frac{\sqrt{1-4ac}}{\alpha }){\alpha }^{-\frac{\alpha -i\alpha \sqrt{4ac-1}}{{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}}+a^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{b}^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{\left(x^{\alpha }\right)}^{\frac{i\sqrt{4ac-1}\alpha +\alpha }{2{\alpha }^2}-\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{2{\alpha }^2}}{J}_{\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}}\left(\frac{2\sqrt{a}\sqrt{b}\sqrt{x^{\alpha }}}{\alpha }\right)\mathrm{\Gamma }(\frac{\sqrt{1-4ac}}{\alpha }+1){\alpha }^{\frac{\sqrt{{\alpha }^2-4a{\alpha }^{2}c}}{{\alpha }^2}-\frac{i\sqrt{4ac-1}\alpha +\alpha }{{\alpha }^2}})}\}\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))
 

$$y\left(x\right)=\frac{-2(\mathrm{BesselY}(\frac{\sqrt{-4ac+1}+\alpha }{\alpha },\frac{2\sqrt{ab}{x}^{\frac{\alpha }{2}}}{\alpha })c_1+\mathrm{BesselJ}(\frac{\sqrt{-4ac+1}+\alpha }{\alpha },\frac{2\sqrt{ab}{x}^{\frac{\alpha }{2}}}{\alpha }))\sqrt{ab}{x}^{\frac{\alpha }{2}}+(\sqrt{-4ac+1}+1)(\mathrm{BesselY}(\frac{\sqrt{-4ac+1}}{\alpha },\frac{2\sqrt{ab}{x}^{\frac{\alpha }{2}}}{\alpha })c_1+\mathrm{BesselJ}(\frac{\sqrt{-4ac+1}}{\alpha },\frac{2\sqrt{ab}{x}^{\frac{\alpha }{2}}}{\alpha }))}{2xa(\mathrm{BesselY}(\frac{\sqrt{-4ac+1}}{\alpha },\frac{2\sqrt{ab}{x}^{\frac{\alpha }{2}}}{\alpha })c_1+\mathrm{BesselJ}(\frac{\sqrt{-4ac+1}}{\alpha },\frac{2\sqrt{ab}{x}^{\frac{\alpha }{2}}}{\alpha }))}$$