\[ \int \frac {(2+3 x)^{5/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx \]
Optimal antiderivative \[ \frac {\left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{3 \left (1-2 x \right )^{\frac {3}{2}}}-\frac {12601 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{420}-\frac {69819 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{70}-\frac {170 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{33 \sqrt {1-2 x}}-\frac {28283 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{462}-\frac {1355 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}\, \sqrt {2+3 x}}{154}-\frac {12601 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}{28} \]
command
integrate((2+3*x)^(5/2)*(3+5*x)^(5/2)/(1-2*x)^(5/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {{\left (2700 \, x^{4} + 12960 \, x^{3} + 36606 \, x^{2} - 175958 \, x + 66663\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{84 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}, x\right ) \]