\[ \int \frac {c+d x^2}{(e x)^{5/2} \left (a+b x^2\right )^{9/4}} \, dx \]
Optimal antiderivative \[ -\frac {2 c}{3 a e \left (e x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {5}{4}}}-\frac {2 \left (-3 a d +8 b c \right ) \sqrt {e x}}{15 a^{2} e^{3} \left (b \,x^{2}+a \right )^{\frac {5}{4}}}-\frac {8 \left (-3 a d +8 b c \right ) \sqrt {e x}}{15 a^{3} e^{3} \left (b \,x^{2}+a \right )^{\frac {1}{4}}} \]
command
integrate((d*x**2+c)/(e*x)**(5/2)/(b*x**2+a)**(9/4),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ c \left (\frac {5 a^{2} b^{\frac {19}{4}} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{32 a^{5} b^{4} e^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right ) + 64 a^{4} b^{5} e^{\frac {5}{2}} x^{2} \Gamma \left (\frac {9}{4}\right ) + 32 a^{3} b^{6} e^{\frac {5}{2}} x^{4} \Gamma \left (\frac {9}{4}\right )} + \frac {40 a b^{\frac {23}{4}} x^{2} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{32 a^{5} b^{4} e^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right ) + 64 a^{4} b^{5} e^{\frac {5}{2}} x^{2} \Gamma \left (\frac {9}{4}\right ) + 32 a^{3} b^{6} e^{\frac {5}{2}} x^{4} \Gamma \left (\frac {9}{4}\right )} + \frac {32 b^{\frac {27}{4}} x^{4} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{32 a^{5} b^{4} e^{\frac {5}{2}} \Gamma \left (\frac {9}{4}\right ) + 64 a^{4} b^{5} e^{\frac {5}{2}} x^{2} \Gamma \left (\frac {9}{4}\right ) + 32 a^{3} b^{6} e^{\frac {5}{2}} x^{4} \Gamma \left (\frac {9}{4}\right )}\right ) + d \left (\frac {5 a \Gamma \left (\frac {1}{4}\right )}{8 a^{3} \sqrt [4]{b} e^{\frac {5}{2}} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {9}{4}\right ) + 8 a^{2} b^{\frac {5}{4}} e^{\frac {5}{2}} x^{2} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {9}{4}\right )} + \frac {4 b x^{2} \Gamma \left (\frac {1}{4}\right )}{8 a^{3} \sqrt [4]{b} e^{\frac {5}{2}} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {9}{4}\right ) + 8 a^{2} b^{\frac {5}{4}} e^{\frac {5}{2}} x^{2} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (\frac {9}{4}\right )}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________