\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{13/2} \sqrt {3+5 x}} \, dx \]
Optimal antiderivative \[ -\frac {23204503328 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{6417873}-\frac {697995152 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{6417873}+\frac {14 \left (1-2 x \right )^{\frac {3}{2}} \sqrt {3+5 x}}{33 \left (2+3 x \right )^{\frac {11}{2}}}+\frac {4508 \sqrt {1-2 x}\, \sqrt {3+5 x}}{891 \left (2+3 x \right )^{\frac {9}{2}}}+\frac {171004 \sqrt {1-2 x}\, \sqrt {3+5 x}}{6237 \left (2+3 x \right )^{\frac {7}{2}}}+\frac {7173272 \sqrt {1-2 x}\, \sqrt {3+5 x}}{43659 \left (2+3 x \right )^{\frac {5}{2}}}+\frac {333930952 \sqrt {1-2 x}\, \sqrt {3+5 x}}{305613 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {23204503328 \sqrt {1-2 x}\, \sqrt {3+5 x}}{2139291 \sqrt {2+3 x}} \]
command
integrate((1-2*x)^(5/2)/(2+3*x)^(13/2)/(3+5*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (2819347154352 \, x^{5} + 9492493272732 \, x^{4} + 12787628716260 \, x^{3} + 8615827181322 \, x^{2} + 2903435279352 \, x + 391506734113\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2139291 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384}, x\right ) \]