\[ \int \frac {2+3 x}{\left (2^{2/3}+x\right ) \sqrt {-1-x^3}} \, dx \]
Optimal antiderivative \[ \frac {2 \left (2-3 \,2^{\frac {2}{3}}\right ) \arctanh \left (\frac {\left (1+2^{\frac {1}{3}} x \right ) \sqrt {3}}{\sqrt {-x^{3}-1}}\right ) \sqrt {3}}{9}+\frac {2 \left (3+2 \,2^{\frac {1}{3}}\right ) \left (1+x \right ) \EllipticF \left (\frac {1+x +\sqrt {3}}{1+x -\sqrt {3}}, 2 i-i \sqrt {3}\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{2}-x +1}{\left (1+x -\sqrt {3}\right )^{2}}}\, 3^{\frac {3}{4}}}{9 \sqrt {-x^{3}-1}\, \sqrt {\frac {-1-x}{\left (1+x -\sqrt {3}\right )^{2}}}} \]
command
integrate((2+3*x)/(2^(2/3)+x)/(-x^3-1)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {1}{18} \, \sqrt {3} \sqrt {-12 \cdot 2^{\frac {2}{3}} + 18 \cdot 2^{\frac {1}{3}} + 4} \log \left (\frac {25 \, x^{18} - 36000 \, x^{15} + 435000 \, x^{12} + 526400 \, x^{9} - 259200 \, x^{6} - 384000 \, x^{3} + 2 \, \sqrt {3} {\left (6 \, x^{16} - 34 \, x^{15} + 1134 \, x^{14} - 1860 \, x^{13} + 2116 \, x^{12} - 23976 \, x^{11} + 13992 \, x^{10} - 5056 \, x^{9} + 15936 \, x^{7} - 10816 \, x^{6} + 41472 \, x^{5} - 1536 \, x^{4} - 5120 \, x^{3} + 20736 \, x^{2} + 3 \cdot 2^{\frac {2}{3}} {\left (3 \, x^{16} - 17 \, x^{15} + 42 \, x^{14} - 930 \, x^{13} + 1058 \, x^{12} - 888 \, x^{11} + 6996 \, x^{10} - 2528 \, x^{9} + 7968 \, x^{7} - 5408 \, x^{6} + 1536 \, x^{5} - 768 \, x^{4} - 2560 \, x^{3} + 768 \, x^{2} - 1536 \, x - 512\right )} + 2^{\frac {1}{3}} {\left (2 \, x^{16} - 153 \, x^{15} + 378 \, x^{14} - 620 \, x^{13} + 9522 \, x^{12} - 7992 \, x^{11} + 4664 \, x^{10} - 22752 \, x^{9} + 5312 \, x^{7} - 48672 \, x^{6} + 13824 \, x^{5} - 512 \, x^{4} - 23040 \, x^{3} + 6912 \, x^{2} - 1024 \, x - 4608\right )} - 3072 \, x - 1024\right )} \sqrt {-x^{3} - 1} \sqrt {-12 \cdot 2^{\frac {2}{3}} + 18 \cdot 2^{\frac {1}{3}} + 4} - 600 \cdot 2^{\frac {2}{3}} {\left (x^{17} - 121 \, x^{14} + 478 \, x^{11} + 1144 \, x^{8} + 608 \, x^{5} + 64 \, x^{2}\right )} + 1200 \cdot 2^{\frac {1}{3}} {\left (5 \, x^{16} - 176 \, x^{13} + 83 \, x^{10} + 680 \, x^{7} + 544 \, x^{4} + 128 \, x\right )} - 51200}{x^{18} + 24 \, x^{15} + 240 \, x^{12} + 1280 \, x^{9} + 3840 \, x^{6} + 6144 \, x^{3} + 4096}\right ) \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (-\frac {{\left (3 \, x^{3} + 2 \, x^{2} - 2^{\frac {2}{3}} {\left (3 \, x^{2} + 2 \, x\right )} + 2 \cdot 2^{\frac {1}{3}} {\left (3 \, x + 2\right )}\right )} \sqrt {-x^{3} - 1}}{x^{6} + 5 \, x^{3} + 4}, x\right ) \]