\[ \int \frac {c+d x^2}{(e x)^{9/2} \left (a+b x^2\right )^{9/4}} \, dx \]
Optimal antiderivative \[ -\frac {2 c}{7 a e \left (e x \right )^{\frac {7}{2}} \left (b \,x^{2}+a \right )^{\frac {5}{4}}}-\frac {2 \left (-7 a d +12 b c \right )}{35 a^{2} e^{3} \left (e x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {5}{4}}}-\frac {16 \left (-7 a d +12 b c \right )}{35 a^{3} e^{3} \left (e x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {1}{4}}}+\frac {64 \left (-7 a d +12 b c \right ) \left (b \,x^{2}+a \right )^{\frac {3}{4}}}{105 a^{4} e^{3} \left (e x \right )^{\frac {3}{2}}} \]
command
integrate((d*x^2+c)/(e*x)^(9/2)/(b*x^2+a)^(9/4),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {2}{105} \, {\left (7 \, d {\left (\frac {3 \, {\left (b^{2} - \frac {10 \, {\left (b x^{2} + a\right )} b}{x^{2}}\right )} x^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} a^{3}} - \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}}}{a^{3} x^{\frac {3}{2}}}\right )} - 3 \, c {\left (\frac {7 \, {\left (b^{3} - \frac {15 \, {\left (b x^{2} + a\right )} b^{2}}{x^{2}}\right )} x^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} a^{4}} - \frac {5 \, {\left (\frac {7 \, {\left (b x^{2} + a\right )}^{\frac {3}{4}} b}{x^{\frac {3}{2}}} - \frac {{\left (b x^{2} + a\right )}^{\frac {7}{4}}}{x^{\frac {7}{2}}}\right )}}{a^{4}}\right )}\right )} e^{\left (-\frac {9}{2}\right )} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {9}{4}} \left (e x\right )^{\frac {9}{2}}}\,{d x} \]________________________________________________________________________________________