\[ \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {-a-b x^3}} \, dx \]
Optimal antiderivative \[ \frac {2 \arctan \left (\frac {\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )^{2}}{3 a^{\frac {1}{6}} \sqrt {-b \,x^{3}-a}}\right )}{3 a^{\frac {1}{6}} b^{\frac {1}{3}}} \]
command
integrate((a^(1/3)+b^(1/3)*x)/(2*a^(1/3)-b^(1/3)*x)/(-b*x^3-a)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \left [\frac {1}{6} \, a^{\frac {1}{3}} \sqrt {-\frac {1}{a b^{\frac {2}{3}}}} \log \left (\frac {b^{6} x^{18} + 7800 \, a b^{5} x^{15} + 535272 \, a^{2} b^{4} x^{12} + 5147264 \, a^{3} b^{3} x^{9} + 10516992 \, a^{4} b^{2} x^{6} + 5922816 \, a^{5} b x^{3} + 557056 \, a^{6} + 144 \, {\left (7 \, b^{5} x^{16} + 1169 \, a b^{4} x^{13} + 20266 \, a^{2} b^{3} x^{10} + 66976 \, a^{3} b^{2} x^{7} + 58112 \, a^{4} b x^{4} + 10240 \, a^{5} x\right )} a^{\frac {2}{3}} b^{\frac {1}{3}} + 72 \, {\left (b^{5} x^{17} + 581 \, a b^{4} x^{14} + 19108 \, a^{2} b^{3} x^{11} + 106336 \, a^{3} b^{2} x^{8} + 137984 \, a^{4} b x^{5} + 50176 \, a^{5} x^{2}\right )} a^{\frac {1}{3}} b^{\frac {2}{3}} + 12 \, {\left ({\left (b^{5} x^{16} + 1568 \, a b^{4} x^{13} + 72520 \, a^{2} b^{3} x^{10} + 498304 \, a^{3} b^{2} x^{7} + 625664 \, a^{4} b x^{4} + 139264 \, a^{5} x\right )} \sqrt {-b x^{3} - a} a^{\frac {2}{3}} b^{\frac {2}{3}} + 6 \, {\left (41 \, a b^{5} x^{14} + 4268 \, a^{2} b^{4} x^{11} + 52896 \, a^{3} b^{3} x^{8} + 116480 \, a^{4} b^{2} x^{5} + 48128 \, a^{5} b x^{2}\right )} \sqrt {-b x^{3} - a} a^{\frac {1}{3}} + {\left (25 \, a b^{5} x^{15} + 7202 \, a^{2} b^{4} x^{12} + 167392 \, a^{3} b^{3} x^{9} + 647296 \, a^{4} b^{2} x^{6} + 468992 \, a^{5} b x^{3} + 40960 \, a^{6}\right )} \sqrt {-b x^{3} - a} b^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{a b^{\frac {2}{3}}}}}{b^{6} x^{18} - 48 \, a b^{5} x^{15} + 960 \, a^{2} b^{4} x^{12} - 10240 \, a^{3} b^{3} x^{9} + 61440 \, a^{4} b^{2} x^{6} - 196608 \, a^{5} b x^{3} + 262144 \, a^{6}}\right ), -\frac {1}{3} \, a^{\frac {1}{3}} \sqrt {\frac {1}{a b^{\frac {2}{3}}}} \arctan \left (\frac {{\left ({\left (11 \, b x^{4} - 16 \, a x\right )} \sqrt {-b x^{3} - a} a^{\frac {2}{3}} b^{\frac {2}{3}} + {\left (b^{2} x^{5} + 28 \, a b x^{2}\right )} \sqrt {-b x^{3} - a} a^{\frac {1}{3}} - {\left (17 \, a b x^{3} - 10 \, a^{2}\right )} \sqrt {-b x^{3} - a} b^{\frac {1}{3}}\right )} \sqrt {\frac {1}{a b^{\frac {2}{3}}}}}{6 \, {\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}}\right )\right ] \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]