\[ \int \frac {(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{7/4}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (-a d +b c \right ) \left (e x \right )^{\frac {9}{2}}}{3 a b e \left (b \,x^{2}+a \right )^{\frac {3}{4}}}-\frac {\left (-3 a d +2 b c \right ) e \left (e x \right )^{\frac {5}{2}} \left (b \,x^{2}+a \right )^{\frac {1}{4}}}{3 a \,b^{2}}+\frac {5 \left (-3 a d +2 b c \right ) e^{2} \left (1+\frac {a}{b \,x^{2}}\right )^{\frac {3}{4}} \left (e x \right )^{\frac {3}{2}} \sqrt {2}\, \sqrt {\frac {x \sqrt {b}+\sqrt {a}\, \sqrt {\frac {b \,x^{2}+a}{a}}}{\sqrt {a}\, \sqrt {\frac {b \,x^{2}+a}{a}}}}\, \EllipticF \left (\sin \left (\frac {\mathrm {arccot}\left (\frac {x \sqrt {b}}{\sqrt {a}}\right )}{2}\right ), \sqrt {2}\right ) \sqrt {a}}{12 \cos \left (\frac {\mathrm {arccot}\left (\frac {x \sqrt {b}}{\sqrt {a}}\right )}{2}\right ) b^{\frac {5}{2}} \left (b \,x^{2}+a \right )^{\frac {3}{4}}}+\frac {5 \left (-3 a d +2 b c \right ) e^{3} \left (b \,x^{2}+a \right )^{\frac {1}{4}} \sqrt {e x}}{6 b^{3}} \]
command
integrate((e*x)**(7/2)*(d*x**2+c)/(b*x**2+a)**(7/4),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \frac {c e^{\frac {7}{2}} x^{\frac {9}{2}} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {7}{4}} \Gamma \left (\frac {13}{4}\right )} + \frac {d e^{\frac {7}{2}} x^{\frac {13}{2}} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {7}{4}} \Gamma \left (\frac {17}{4}\right )} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________