\[ \int \frac {(e x)^{3/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-a d +b c \right ) \left (e x \right )^{\frac {5}{2}}}{5 a b e \left (b \,x^{2}+a \right )^{\frac {5}{4}}}+\frac {d \,e^{\frac {3}{2}} \arctan \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {e}}\right )}{b^{\frac {9}{4}}}+\frac {d \,e^{\frac {3}{2}} \arctanh \left (\frac {b^{\frac {1}{4}} \sqrt {e x}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {e}}\right )}{b^{\frac {9}{4}}}-\frac {2 d e \sqrt {e x}}{b^{2} \left (b \,x^{2}+a \right )^{\frac {1}{4}}} \]
command
integrate((e*x)**(3/2)*(d*x**2+c)/(b*x**2+a)**(9/4),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {c e^{\frac {3}{2}} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right )}{2 a^{\frac {9}{4}} \sqrt [4]{1 + \frac {b x^{2}}{a}} \Gamma \left (\frac {9}{4}\right ) + 2 a^{\frac {5}{4}} b x^{2} \sqrt [4]{1 + \frac {b x^{2}}{a}} \Gamma \left (\frac {9}{4}\right )} + \frac {d e^{\frac {3}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {9}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {9}{4}} \Gamma \left (\frac {13}{4}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________