\[ \int \frac {c+d x^2}{(e x)^{15/2} \left (a+b x^2\right )^{3/4}} \, dx \]
Optimal antiderivative \[ -\frac {2 c \left (b \,x^{2}+a \right )^{\frac {1}{4}}}{13 a e \left (e x \right )^{\frac {13}{2}}}+\frac {2 \left (-13 a d +12 b c \right ) \left (b \,x^{2}+a \right )^{\frac {1}{4}}}{13 a^{2} e^{3} \left (e x \right )^{\frac {9}{2}}}-\frac {16 \left (-13 a d +12 b c \right ) \left (b \,x^{2}+a \right )^{\frac {5}{4}}}{65 a^{3} e^{3} \left (e x \right )^{\frac {9}{2}}}+\frac {64 \left (-13 a d +12 b c \right ) \left (b \,x^{2}+a \right )^{\frac {9}{4}}}{585 a^{4} e^{3} \left (e x \right )^{\frac {9}{2}}} \]
command
integrate((d*x^2+c)/(e*x)^(15/2)/(b*x^2+a)^(3/4),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ -\frac {2}{585} \, {\left (\frac {13 \, {\left (\frac {45 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}} b^{2}}{\sqrt {x}} - \frac {18 \, {\left (b x^{2} + a\right )}^{\frac {5}{4}} b}{x^{\frac {5}{2}}} + \frac {5 \, {\left (b x^{2} + a\right )}^{\frac {9}{4}}}{x^{\frac {9}{2}}}\right )} d}{a^{3}} - \frac {3 \, {\left (\frac {195 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}} b^{3}}{\sqrt {x}} - \frac {117 \, {\left (b x^{2} + a\right )}^{\frac {5}{4}} b^{2}}{x^{\frac {5}{2}}} + \frac {65 \, {\left (b x^{2} + a\right )}^{\frac {9}{4}} b}{x^{\frac {9}{2}}} - \frac {15 \, {\left (b x^{2} + a\right )}^{\frac {13}{4}}}{x^{\frac {13}{2}}}\right )} c}{a^{4}}\right )} e^{\left (-\frac {15}{2}\right )} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {15}{2}}}\,{d x} \]________________________________________________________________________________________