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