\[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{7/4}} \, dx \]
Optimal antiderivative \[ -\frac {2 c}{5 a e \left (e x \right )^{\frac {5}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}-\frac {2 \left (-5 a d +8 b c \right )}{15 a^{2} e^{3} \left (b \,x^{2}+a \right )^{\frac {3}{4}} \sqrt {e x}}+\frac {8 \left (-5 a d +8 b c \right ) \left (b \,x^{2}+a \right )^{\frac {1}{4}}}{15 a^{3} e^{3} \sqrt {e x}} \]
command
integrate((d*x^2+c)/(e*x)^(7/2)/(b*x^2+a)^(7/4),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ -\frac {2}{15} \, {\left (5 \, d {\left (\frac {b x^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} a^{2}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}}}{a^{2} \sqrt {x}}\right )} - c {\left (\frac {5 \, b^{2} x^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} a^{3}} + \frac {3 \, {\left (\frac {10 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}} b}{\sqrt {x}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{4}}}{x^{\frac {5}{2}}}\right )}}{a^{3}}\right )}\right )} e^{\left (-\frac {7}{2}\right )} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \left (e x\right )^{\frac {7}{2}}}\,{d x} \]________________________________________________________________________________________