\[ \int \frac {x^3}{\sqrt {b x^{2/3}+a x}} \, dx \]
Optimal antiderivative \[ -\frac {262144 b^{9} \sqrt {b \,x^{\frac {2}{3}}+a x}}{323323 a^{10}}+\frac {524288 b^{10} \sqrt {b \,x^{\frac {2}{3}}+a x}}{323323 a^{11} x^{\frac {1}{3}}}+\frac {196608 b^{8} x^{\frac {1}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{323323 a^{9}}-\frac {163840 b^{7} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{323323 a^{8}}+\frac {20480 b^{6} x \sqrt {b \,x^{\frac {2}{3}}+a x}}{46189 a^{7}}-\frac {18432 b^{5} x^{\frac {4}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{46189 a^{6}}+\frac {1536 b^{4} x^{\frac {5}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{4199 a^{5}}-\frac {768 b^{3} x^{2} \sqrt {b \,x^{\frac {2}{3}}+a x}}{2261 a^{4}}+\frac {720 b^{2} x^{\frac {7}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{2261 a^{3}}-\frac {40 b \,x^{\frac {8}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{133 a^{2}}+\frac {2 x^{3} \sqrt {b \,x^{\frac {2}{3}}+a x}}{7 a} \]
command
integrate(x^3/(b*x^(2/3)+a*x)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {2 \, {\left ({\left (3298534883328 \, b^{16} + 687194767360 \, b^{15} + 3221225472 \, {\left (64 \, a^{3} - 3\right )} b^{13} - 64424509440 \, b^{14} - 16777216 \, {\left (11264 \, a^{3} - 53\right )} b^{12} - 269004736 \, a^{12} - 6291456 \, {\left (5504 \, a^{3} + 1\right )} b^{11} + 196608 \, {\left (3194880 \, a^{6} - 114688 \, a^{3} - 3\right )} b^{10} + 7340032 \, {\left (18816 \, a^{6} + 103 \, a^{3}\right )} b^{9} - 786432 \, {\left (48816 \, a^{6} + 23 \, a^{3}\right )} b^{8} - 12288 \, {\left (45731840 \, a^{9} - 495872 \, a^{6} - 15 \, a^{3}\right )} b^{7} - 114688 \, {\left (1349120 \, a^{9} + 3439 \, a^{6}\right )} b^{6} + 3913728 \, {\left (5600 \, a^{9} + 3 \, a^{6}\right )} b^{5} - 2112 \, {\left (101384192 \, a^{12} + 1958400 \, a^{9} + 63 \, a^{6}\right )} b^{4} - 36608 \, {\left (3784704 \, a^{12} - 8101 \, a^{9}\right )} b^{3} - 109824 \, {\left (226688 \, a^{12} + 85 \, a^{9}\right )} b^{2} + 7293 \, {\left (974848 \, a^{12} + 15 \, a^{9}\right )} b\right )} x - {\left (46189 \, {\left (16777216 \, a^{10} b^{6} + 6291456 \, a^{10} b^{5} + 196608 \, a^{10} b^{4} - 262144 \, a^{13} - 114688 \, a^{10} b^{3} - 2304 \, a^{10} b^{2} + 864 \, a^{10} b - 27 \, a^{10}\right )} x^{4} - 54912 \, {\left (16777216 \, a^{7} b^{9} + 6291456 \, a^{7} b^{8} + 196608 \, a^{7} b^{7} - 114688 \, a^{7} b^{6} - 2304 \, a^{7} b^{5} + 864 \, a^{7} b^{4} - {\left (262144 \, a^{10} + 27 \, a^{7}\right )} b^{3}\right )} x^{3} + 71680 \, {\left (16777216 \, a^{4} b^{12} + 6291456 \, a^{4} b^{11} + 196608 \, a^{4} b^{10} - 114688 \, a^{4} b^{9} - 2304 \, a^{4} b^{8} + 864 \, a^{4} b^{7} - {\left (262144 \, a^{7} + 27 \, a^{4}\right )} b^{6}\right )} x^{2} - 131072 \, {\left (16777216 \, a b^{15} + 6291456 \, a b^{14} + 196608 \, a b^{13} - 114688 \, a b^{12} - 2304 \, a b^{11} + 864 \, a b^{10} - {\left (262144 \, a^{4} + 27 \, a\right )} b^{9}\right )} x + 4 \, {\left (1099511627776 \, b^{16} + 412316860416 \, b^{15} + 12884901888 \, b^{14} - 7516192768 \, b^{13} - 150994944 \, b^{12} - 65536 \, {\left (262144 \, a^{3} + 27\right )} b^{10} + 56623104 \, b^{11} - 12155 \, {\left (16777216 \, a^{9} b^{7} + 6291456 \, a^{9} b^{6} + 196608 \, a^{9} b^{5} - 114688 \, a^{9} b^{4} - 2304 \, a^{9} b^{3} + 864 \, a^{9} b^{2} - {\left (262144 \, a^{12} + 27 \, a^{9}\right )} b\right )} x^{3} + 14784 \, {\left (16777216 \, a^{6} b^{10} + 6291456 \, a^{6} b^{9} + 196608 \, a^{6} b^{8} - 114688 \, a^{6} b^{7} - 2304 \, a^{6} b^{6} + 864 \, a^{6} b^{5} - {\left (262144 \, a^{9} + 27 \, a^{6}\right )} b^{4}\right )} x^{2} - 20480 \, {\left (16777216 \, a^{3} b^{13} + 6291456 \, a^{3} b^{12} + 196608 \, a^{3} b^{11} - 114688 \, a^{3} b^{10} - 2304 \, a^{3} b^{9} + 864 \, a^{3} b^{8} - {\left (262144 \, a^{6} + 27 \, a^{3}\right )} b^{7}\right )} x\right )} x^{\frac {2}{3}} + 24 \, {\left (2145 \, {\left (16777216 \, a^{8} b^{8} + 6291456 \, a^{8} b^{7} + 196608 \, a^{8} b^{6} - 114688 \, a^{8} b^{5} - 2304 \, a^{8} b^{4} + 864 \, a^{8} b^{3} - {\left (262144 \, a^{11} + 27 \, a^{8}\right )} b^{2}\right )} x^{3} - 2688 \, {\left (16777216 \, a^{5} b^{11} + 6291456 \, a^{5} b^{10} + 196608 \, a^{5} b^{9} - 114688 \, a^{5} b^{8} - 2304 \, a^{5} b^{7} + 864 \, a^{5} b^{6} - {\left (262144 \, a^{8} + 27 \, a^{5}\right )} b^{5}\right )} x^{2} + 4096 \, {\left (16777216 \, a^{2} b^{14} + 6291456 \, a^{2} b^{13} + 196608 \, a^{2} b^{12} - 114688 \, a^{2} b^{11} - 2304 \, a^{2} b^{10} + 864 \, a^{2} b^{9} - {\left (262144 \, a^{5} + 27 \, a^{2}\right )} b^{8}\right )} x\right )} x^{\frac {1}{3}}\right )} \sqrt {a x + b x^{\frac {2}{3}}}\right )}}{323323 \, {\left (16777216 \, a^{11} b^{6} + 6291456 \, a^{11} b^{5} + 196608 \, a^{11} b^{4} - 262144 \, a^{14} - 114688 \, a^{11} b^{3} - 2304 \, a^{11} b^{2} + 864 \, a^{11} b - 27 \, a^{11}\right )} x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]