\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{5/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (1-2 x \right )^{\frac {5}{2}} \left (2+3 x \right )^{\frac {7}{2}}}{15 \left (3+5 x \right )^{\frac {3}{2}}}-\frac {1065118 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{14765625}-\frac {595387 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{14765625}-\frac {442 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {7}{2}}}{75 \sqrt {3+5 x}}+\frac {373022 \left (2+3 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{196875}+\frac {59662 \left (2+3 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{7875}-\frac {524 \left (2+3 x \right )^{\frac {7}{2}} \sqrt {1-2 x}\, \sqrt {3+5 x}}{225}+\frac {500501 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{984375} \]
command
integrate((1-2*x)^(5/2)*(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (4725000 \, x^{5} + 1327500 \, x^{4} - 5654250 \, x^{3} + 470675 \, x^{2} + 4026600 \, x + 1215489\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{984375 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27}, x\right ) \]