\[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (2+3 x \right )^{\frac {7}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{3 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {722133 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{38500}-\frac {6547351 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{10500}-\frac {56 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}}}{11 \sqrt {1-2 x}}-\frac {1341 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{154}-\frac {140289 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{3850}-\frac {2166399 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{7700} \]
command
integrate((2+3*x)^(7/2)*(3+5*x)^(3/2)/(1-2*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (40500 \, x^{4} + 198180 \, x^{3} + 567906 \, x^{2} - 2751916 \, x + 1041609\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2100 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}, x\right ) \]