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