\[ y(x) \left (\left (a^2-c^2 \nu ^2\right ) \left (a^2+4 a c-c^2 \nu ^2+4 c^2\right )-b^4 c^4 x^{4 c}\right )+x^2 \left (2 a^2+4 (a+c-1)^2+4 (a-1) (c-1)-2 c^2 \nu ^2-1\right ) y''(x)+x (2 a+2 c-1) \left (-2 a^2-(2 a-1) (2 c-1)+2 c^2 \nu ^2\right ) y'(x)+x^3 (-4 a-4 c+6) y^{(3)}(x)+x^4 y^{(4)}(x)=0 \] ✓ Mathematica : cpu = 0.0989139 (sec), leaf count = 470
DSolve[((a^2 - c^2*nu^2)*(a^2 + 4*a*c + 4*c^2 - c^2*nu^2) - b^4*c^4*x^(4*c))*y[x] + (-1 + 2*a + 2*c)*(-2*a^2 - (-1 + 2*a)*(-1 + 2*c) + 2*c^2*nu^2)*x*Derivative[1][y][x] + (-1 + 2*a^2 + 4*(-1 + a)*(-1 + c) + 4*(-1 + a + c)^2 - 2*c^2*nu^2)*x^2*Derivative[2][y][x] + (6 - 4*a - 4*c)*x^3*Derivative[3][y][x] + x^4*Derivative[4][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 \Gamma (1-\nu ) (-1)^{\frac {a-c \nu }{4 c}} 2^{-\frac {2 (a-c \nu )}{c}-\nu -1} b^{\frac {a-c \nu }{c}+\nu } \left (x^{4 c}\right )^{\frac {a-c \nu }{4 c}+\frac {\nu }{4}} \left (J_{-\nu }\left (b \sqrt [4]{x^{4 c}}\right )+I_{-\nu }\left (b \sqrt [4]{x^{4 c}}\right )\right )+c_2 \Gamma (2-\nu ) (-1)^{\frac {a-c \nu +2 c}{4 c}} 2^{-\frac {2 (a-c \nu +2 c)}{c}-\nu +1} b^{\frac {a-c \nu +2 c}{c}+\nu -2} \left (x^{4 c}\right )^{\frac {a-c \nu +2 c}{4 c}+\frac {\nu -2}{4}} \left (I_{-\nu }\left (b \sqrt [4]{x^{4 c}}\right )-J_{-\nu }\left (b \sqrt [4]{x^{4 c}}\right )\right )+c_3 \Gamma (\nu +1) (-1)^{\frac {a+c \nu }{4 c}} 2^{-\frac {2 (a+c \nu )}{c}+\nu -1} b^{\frac {a+c \nu }{c}-\nu } \left (x^{4 c}\right )^{\frac {a+c \nu }{4 c}-\frac {\nu }{4}} \left (J_{\nu }\left (b \sqrt [4]{x^{4 c}}\right )+I_{\nu }\left (b \sqrt [4]{x^{4 c}}\right )\right )+c_4 \Gamma (\nu +2) (-1)^{\frac {a+c \nu +2 c}{4 c}} 2^{-\frac {2 (a+c \nu +2 c)}{c}+\nu +1} b^{\frac {a+c \nu +2 c}{c}-\nu -2} \left (x^{4 c}\right )^{\frac {a+c \nu +2 c}{4 c}+\frac {1}{4} (-\nu -2)} \left (I_{\nu }\left (b \sqrt [4]{x^{4 c}}\right )-J_{\nu }\left (b \sqrt [4]{x^{4 c}}\right )\right )\right \}\right \}\] ✓ Maple : cpu = 0.094 (sec), leaf count = 49
dsolve(x^4*diff(diff(diff(diff(y(x),x),x),x),x)+(6-4*a-4*c)*x^3*diff(diff(diff(y(x),x),x),x)+(-2*nu^2*c^2+2*a^2+4*(a+c-1)^2+4*(a-1)*(c-1)-1)*x^2*diff(diff(y(x),x),x)+(2*nu^2*c^2-2*a^2-(2*a-1)*(2*c-1))*(2*a+2*c-1)*x*diff(y(x),x)+((-c^2*nu^2+a^2)*(-c^2*nu^2+a^2+4*a*c+4*c^2)-b^4*c^4*x^(4*c))*y(x)=0,y(x))
\[y \left (x \right ) = x^{a} \left (\BesselY \left (\nu , i b \,x^{c}\right ) c_{4}+\BesselY \left (\nu , b \,x^{c}\right ) c_{2}+\BesselJ \left (\nu , i b \,x^{c}\right ) c_{3}+\BesselJ \left (\nu , b \,x^{c}\right ) c_{1}\right )\]