\[ \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{9/4}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (-a d +6 b c \right ) \left (e x \right )^{\frac {3}{2}}}{5 a^{2} e^{3} \left (b \,x^{2}+a \right )^{\frac {5}{4}}}-\frac {2 c}{a e \left (b \,x^{2}+a \right )^{\frac {5}{4}} \sqrt {e x}}+\frac {2 \left (-a d +6 b c \right ) \left (1+\frac {a}{b \,x^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {\frac {x \sqrt {b}+\sqrt {a}\, \sqrt {\frac {b \,x^{2}+a}{a}}}{\sqrt {a}\, \sqrt {\frac {b \,x^{2}+a}{a}}}}\, \EllipticE \left (\sin \left (\frac {\mathrm {arccot}\left (\frac {x \sqrt {b}}{\sqrt {a}}\right )}{2}\right ), \sqrt {2}\right ) \sqrt {e x}}{5 \cos \left (\frac {\mathrm {arccot}\left (\frac {x \sqrt {b}}{\sqrt {a}}\right )}{2}\right ) a^{\frac {5}{2}} e^{2} \left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {b}} \]
command
integrate((d*x**2+c)/(e*x)**(3/2)/(b*x**2+a)**(9/4),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {c \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {9}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {9}{4}} e^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} + \frac {d x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {9}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {9}{4}} e^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________