\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (1-2 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {5}{2}}}{39 \left (2+3 x \right )^{\frac {13}{2}}}-\frac {245282464136 \EllipticE \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{61322776515}-\frac {7391549624 \EllipticF \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}, \frac {\sqrt {1155}}{33}\right ) \sqrt {33}}{61322776515}-\frac {20992 \left (3+5 x \right )^{\frac {3}{2}} \sqrt {1-2 x}}{81081 \left (2+3 x \right )^{\frac {9}{2}}}+\frac {362 \left (3+5 x \right )^{\frac {5}{2}} \sqrt {1-2 x}}{1287 \left (2+3 x \right )^{\frac {11}{2}}}-\frac {2174468 \sqrt {1-2 x}\, \sqrt {3+5 x}}{11918907 \left (2+3 x \right )^{\frac {7}{2}}}+\frac {73596464 \sqrt {1-2 x}\, \sqrt {3+5 x}}{417161745 \left (2+3 x \right )^{\frac {5}{2}}}+\frac {3523482724 \sqrt {1-2 x}\, \sqrt {3+5 x}}{2920132215 \left (2+3 x \right )^{\frac {3}{2}}}+\frac {245282464136 \sqrt {1-2 x}\, \sqrt {3+5 x}}{20440925505 \sqrt {2+3 x}} \]
command
integrate((1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {2 \, {\left (89405458177572 \, x^{6} + 360618554767050 \, x^{5} + 606171513555828 \, x^{4} + 543590753927373 \, x^{3} + 274263621177573 \, x^{2} + 73802680969881 \, x + 8272877174903\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{20440925505 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256}, x\right ) \]