\[ \boxed { {x}^{4}{\it d4y} \left ( x \right ) + \left ( 6-4\,a-4\,c \right ) {x}^{3}{\frac {{\rm d}^{3}}{{\rm d}{x}^{3}}}y \left ( x \right ) + \left ( -2\,{\nu }^{2}{c}^{2}+2\,{a}^{2}+4\, \left ( a+c-1 \right ) ^{2}+4\, \left ( a-1 \right ) \left ( c-1 \right ) -1 \right ) {x}^{2}{\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) + \left ( 2\,{\nu }^{2}{c}^{2}-2\,{a}^{2}- \left ( 2\,a-1 \right ) \left ( 2\,c-1 \right ) \right ) \left ( 2\,a+2\,c-1 \right ) x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( \left ( -{\nu }^{2}{c}^{2}+{a}^{2} \right ) \left ( -{\nu }^{2}{c}^{2}+{a}^{2}+4\,ac+4\,{c}^{2} \right ) -{b}^{4}{c}^{4}{x}^{4\,c} \right ) y \left ( x \right ) =0} \]
Mathematica: cpu = 0.154020 (sec), leaf count = 470 \[ \left \{\left \{y(x)\to c_1 \Gamma (1-\nu ) (-1)^{\frac {a-c \nu }{4 c}} 2^{-\frac {2 (a-c \nu )}{c}-\nu -1} b^{\frac {a-c \nu }{c}+\nu } \left (x^{4 c}\right )^{\frac {a-c \nu }{4 c}+\frac {\nu }{4}} \left (J_{-\nu }\left (b \sqrt [4]{x^{4 c}}\right )+I_{-\nu }\left (b \sqrt [4]{x^{4 c}}\right )\right )+c_2 \Gamma (2-\nu ) (-1)^{\frac {a-c \nu +2 c}{4 c}} 2^{-\frac {2 (a-c \nu +2 c)}{c}-\nu +1} b^{\frac {a-c \nu +2 c}{c}+\nu -2} \left (x^{4 c}\right )^{\frac {a-c \nu +2 c}{4 c}+\frac {\nu -2}{4}} \left (I_{-\nu }\left (b \sqrt [4]{x^{4 c}}\right )-J_{-\nu }\left (b \sqrt [4]{x^{4 c}}\right )\right )+c_3 \Gamma (\nu +1) (-1)^{\frac {a+c \nu }{4 c}} 2^{-\frac {2 (a+c \nu )}{c}+\nu -1} b^{\frac {a+c \nu }{c}-\nu } \left (x^{4 c}\right )^{\frac {a+c \nu }{4 c}-\frac {\nu }{4}} \left (J_{\nu }\left (b \sqrt [4]{x^{4 c}}\right )+I_{\nu }\left (b \sqrt [4]{x^{4 c}}\right )\right )+c_4 \Gamma (\nu +2) (-1)^{\frac {a+c \nu +2 c}{4 c}} 2^{-\frac {2 (a+c \nu +2 c)}{c}+\nu +1} b^{\frac {a+c \nu +2 c}{c}-\nu -2} \left (x^{4 c}\right )^{\frac {a+c \nu +2 c}{4 c}+\frac {1}{4} (-\nu -2)} \left (I_{\nu }\left (b \sqrt [4]{x^{4 c}}\right )-J_{\nu }\left (b \sqrt [4]{x^{4 c}}\right )\right )\right \}\right \} \]
Maple: cpu = 0.047 (sec), leaf count = 57 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{x}^{a}{{\sl J}_{\nu }\left ({x }^{c}b\right )}+{\it \_C2}\,{x}^{a}{{\sl Y}_{\nu }\left ({x}^{c}b\right )} +{\it \_C3}\,{x}^{a}{{\sl J}_{\nu }\left (ib{x}^{c}\right )}+{\it \_C4}\, {x}^{a}{{\sl Y}_{\nu }\left (ib{x}^{c}\right )} \right \} \]