\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{7/2}} \, dx \]
Optimal antiderivative \[ \frac {11 \left (3+5 x \right )^{\frac {3}{2}}}{21 \left (1-2 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {5}{2}}}-\frac {7738 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{252105}+\frac {9206 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{84035}+\frac {99 \sqrt {3+5 x}}{49 \left (2+3 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}-\frac {1432 \sqrt {1-2 x}\, \sqrt {3+5 x}}{1715 \left (2+3 x \right )^{\frac {5}{2}}}-\frac {4437 \sqrt {1-2 x}\, \sqrt {3+5 x}}{12005 \left (2+3 x \right )^{\frac {3}{2}}}-\frac {27618 \sqrt {1-2 x}\, \sqrt {3+5 x}}{84035 \sqrt {2+3 x}} \]
command
integrate((3+5*x)^(5/2)/(1-2*x)^(5/2)/(2+3*x)^(7/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left (1491372 \, x^{4} + 1056186 \, x^{3} - 718167 \, x^{2} - 640441 \, x - 88623\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{252105 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{648 \, x^{7} + 756 \, x^{6} - 378 \, x^{5} - 609 \, x^{4} + 56 \, x^{3} + 168 \, x^{2} - 16}, x\right ) \]