\[ \int \frac {c+d x^2}{(e x)^{7/2} \left (a+b x^2\right )^{7/4}} \, dx \]
Optimal antiderivative \[ -\frac {2 c}{5 a e \left (e x \right )^{\frac {5}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}-\frac {2 \left (-5 a d +8 b c \right )}{15 a^{2} e^{3} \left (b \,x^{2}+a \right )^{\frac {3}{4}} \sqrt {e x}}+\frac {8 \left (-5 a d +8 b c \right ) \left (b \,x^{2}+a \right )^{\frac {1}{4}}}{15 a^{3} e^{3} \sqrt {e x}} \]
command
integrate((d*x**2+c)/(e*x)**(7/2)/(b*x**2+a)**(7/4),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ c \left (- \frac {3 a^{3} b^{\frac {17}{4}} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{32 a^{5} b^{4} e^{\frac {7}{2}} x^{2} \Gamma \left (\frac {7}{4}\right ) + 64 a^{4} b^{5} e^{\frac {7}{2}} x^{4} \Gamma \left (\frac {7}{4}\right ) + 32 a^{3} b^{6} e^{\frac {7}{2}} x^{6} \Gamma \left (\frac {7}{4}\right )} + \frac {21 a^{2} b^{\frac {21}{4}} x^{2} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{32 a^{5} b^{4} e^{\frac {7}{2}} x^{2} \Gamma \left (\frac {7}{4}\right ) + 64 a^{4} b^{5} e^{\frac {7}{2}} x^{4} \Gamma \left (\frac {7}{4}\right ) + 32 a^{3} b^{6} e^{\frac {7}{2}} x^{6} \Gamma \left (\frac {7}{4}\right )} + \frac {56 a b^{\frac {25}{4}} x^{4} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{32 a^{5} b^{4} e^{\frac {7}{2}} x^{2} \Gamma \left (\frac {7}{4}\right ) + 64 a^{4} b^{5} e^{\frac {7}{2}} x^{4} \Gamma \left (\frac {7}{4}\right ) + 32 a^{3} b^{6} e^{\frac {7}{2}} x^{6} \Gamma \left (\frac {7}{4}\right )} + \frac {32 b^{\frac {29}{4}} x^{6} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {5}{4}\right )}{32 a^{5} b^{4} e^{\frac {7}{2}} x^{2} \Gamma \left (\frac {7}{4}\right ) + 64 a^{4} b^{5} e^{\frac {7}{2}} x^{4} \Gamma \left (\frac {7}{4}\right ) + 32 a^{3} b^{6} e^{\frac {7}{2}} x^{6} \Gamma \left (\frac {7}{4}\right )}\right ) + d \left (\frac {3 \Gamma \left (- \frac {1}{4}\right )}{8 a b^{\frac {3}{4}} e^{\frac {7}{2}} x^{2} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )} + \frac {\sqrt [4]{b} \Gamma \left (- \frac {1}{4}\right )}{2 a^{2} e^{\frac {7}{2}} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )}\right ) \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________