\[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{5/4}} \, dx \]
Optimal antiderivative \[ \frac {d \left (e x \right )^{\frac {5}{2}}}{2 b e \left (b \,x^{2}+a \right )^{\frac {1}{4}}}+\frac {\left (-5 a d +4 b c \right ) e^{\frac {3}{2}} \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {e}}\right )}{4 b^{\frac {9}{4}}}+\frac {\left (-5 a d +4 b c \right ) e^{\frac {3}{2}} \arctanh \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {e}}\right )}{4 b^{\frac {9}{4}}}-\frac {\left (-5 a d +4 b c \right ) e \sqrt {e x}}{2 b^{2} \left (b \,x^{2}+a \right )^{\frac {1}{4}}} \]
command
integrate((e*x)^(3/2)*(d*x^2+c)/(b*x^2+a)^(5/4),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {1}{8} \, {\left (d {\left (\frac {4 \, {\left (4 \, a b - \frac {5 \, {\left (b x^{2} + a\right )} a}{x^{2}}\right )}}{\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}} b^{3}}{\sqrt {x}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{4}} b^{2}}{x^{\frac {5}{2}}}} + \frac {5 \, a {\left (\frac {2 \, \arctan \left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} \sqrt {x}}\right )}{b^{\frac {1}{4}}} + \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}{b^{\frac {1}{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}\right )}{b^{\frac {1}{4}}}\right )}}{b^{2}}\right )} - 4 \, c {\left (\frac {\frac {2 \, \arctan \left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} \sqrt {x}}\right )}{b^{\frac {1}{4}}} + \frac {\log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}{b^{\frac {1}{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}\right )}{b^{\frac {1}{4}}}}{b} + \frac {4 \, \sqrt {x}}{{\left (b x^{2} + a\right )}^{\frac {1}{4}} b}\right )}\right )} e^{\frac {3}{2}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {3}{2}}}{{\left (b x^{2} + a\right )}^{\frac {5}{4}}}\,{d x} \]________________________________________________________________________________________