ODE No. 489

3.489 ODE No. 489

${(y (x))}^{2} {(\frac{d}{d x} y (x))}^{2} + 2 x y (x) \frac{d}{d x} y (x) + a {(y (x))}^{2} + b x + c = 0$

Mathematica: cpu = 0 (sec), leaf count = 0 $Hanged$

Maple: cpu = 2.621 (sec), leaf count = 5525 ${y (x) = - \frac{1}{2 a (a + 1)} \sqrt{a (16 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{5} x^{2} - 16 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{4} b x + 64 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{4} x^{2} + 4 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{3} b^{2} - 48 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{3} b x + 96 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{3} x^{2} + 8 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{2} b^{2} - 48 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{2} b x + 64 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{2} x^{2} + 4 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a b^{2} - 16 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a b x + 16 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a x^{2} - 4 a^{2} b x - 4 a^{2} c - 4 a b x - 8 a c - b^{2} - 4 c)}, y (x) = \frac{1}{2 a (a + 1)} \sqrt{a (16 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{5} x^{2} - 16 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{4} b x + 64 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{4} x^{2} + 4 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{3} b^{2} - 48 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{3} b x + 96 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{3} x^{2} + 8 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{2} b^{2} - 48 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{2} b x + 64 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a^{2} x^{2} + 4 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a b^{2} - 16 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a b x + 16 R o o t O f (- b \ln (2 a x - b + 2 x) + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a a + 2_C 1 a + 2 \int^{_Z} 1 / 4 \frac{b}{(4_a a^{3} + 12_a a^{2} + 12_a a + a^{2} + 4_a + 3 a + 2)_a} (- 4_a a^{2} + \sqrt{- e^{4 \frac{a + 1}{b}} (4_a a^{3} + 8_a a^{2} + 4_a a - 1)} e^{- 2 \frac{a + 1}{b}} - 8_a a - 4_a - 1) d_a + 2_C 1) a x^{2} - 4 a^{2} b x - 4 a^{2} c - 4 a b x - 8 a c - b^{2} - 4 c)}}$

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