\[ \int \frac {(e x)^{5/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{3/4}} \, dx \]
Optimal antiderivative \[ \frac {\left (-7 a d +8 b c \right ) e \left (e x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {1}{4}}}{16 b^{2}}+\frac {d \left (e x \right )^{\frac {7}{2}} \left (b \,x^{2}+a \right )^{\frac {1}{4}}}{4 b e}+\frac {3 a \left (-7 a d +8 b c \right ) e^{\frac {5}{2}} \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {e}}\right )}{32 b^{\frac {11}{4}}}-\frac {3 a \left (-7 a d +8 b c \right ) e^{\frac {5}{2}} \arctanh \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {e}}\right )}{32 b^{\frac {11}{4}}} \]
command
integrate((e*x)^(5/2)*(d*x^2+c)/(b*x^2+a)^(3/4),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ \frac {1}{64} \, {\left (d {\left (\frac {4 \, {\left (\frac {11 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b}{\sqrt {x}} - \frac {7 \, {\left (b x^{2} + a\right )}^{\frac {5}{4}} a^{2}}{x^{\frac {5}{2}}}\right )}}{b^{4} - \frac {2 \, {\left (b x^{2} + a\right )} b^{3}}{x^{2}} + \frac {{\left (b x^{2} + a\right )}^{2} b^{2}}{x^{4}}} + \frac {21 \, {\left (\frac {2 \, a^{2} \arctan \left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} \sqrt {x}}\right )}{b^{\frac {3}{4}}} - \frac {a^{2} \log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}{b^{\frac {1}{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}\right )}{b^{\frac {3}{4}}}\right )}}{b^{2}}\right )} - 8 \, c {\left (\frac {3 \, {\left (\frac {2 \, a \arctan \left (\frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} \sqrt {x}}\right )}{b^{\frac {3}{4}}} - \frac {a \log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}{b^{\frac {1}{4}} + \frac {{\left (b x^{2} + a\right )}^{\frac {1}{4}}}{\sqrt {x}}}\right )}{b^{\frac {3}{4}}}\right )}}{b} + \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}} a}{{\left (b^{2} - \frac {{\left (b x^{2} + a\right )} b}{x^{2}}\right )} \sqrt {x}}\right )}\right )} e^{\frac {5}{2}} \]
Maxima 5.44 via sagemath 9.3 output
\[ \int \frac {{\left (d x^{2} + c\right )} \left (e x\right )^{\frac {5}{2}}}{{\left (b x^{2} + a\right )}^{\frac {3}{4}}}\,{d x} \]________________________________________________________________________________________