\[ \int \frac {x^2}{\sqrt {b x^{2/3}+a x}} \, dx \]
Optimal antiderivative \[ \frac {2048 b^{6} \sqrt {b \,x^{\frac {2}{3}}+a x}}{2145 a^{7}}-\frac {4096 b^{7} \sqrt {b \,x^{\frac {2}{3}}+a x}}{2145 a^{8} x^{\frac {1}{3}}}-\frac {512 b^{5} x^{\frac {1}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{715 a^{6}}+\frac {256 b^{4} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{429 a^{5}}-\frac {224 b^{3} x \sqrt {b \,x^{\frac {2}{3}}+a x}}{429 a^{4}}+\frac {336 b^{2} x^{\frac {4}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{715 a^{3}}-\frac {28 b \,x^{\frac {5}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{65 a^{2}}+\frac {2 x^{2} \sqrt {b \,x^{\frac {2}{3}}+a x}}{5 a} \]
command
integrate(x^2/(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 {{\left (51539607552 \, b^{13} + 10737418240 \, b^{12} + 50331648 \, {\left (64 \, a^{3} - 3\right )} b^{10} - 1006632960 \, b^{11} - 262144 \, {\left (11264 \, a^{3} - 53\right )} b^{9} + 4996992 \, a^{9} - 98304 \, {\left (5504 \, a^{3} + 1\right )} b^{8} + 3072 \, {\left (3194880 \, a^{6} - 114688 \, a^{3} - 3\right )} b^{7} + 114688 \, {\left (18816 \, a^{6} + 103 \, a^{3}\right )} b^{6} - 12288 \, {\left (48816 \, a^{6} + 23 \, a^{3}\right )} b^{5} + 192 \, {\left (21626880 \, a^{9} + 495872 \, a^{6} + 15 \, a^{3}\right )} b^{4} + 256 \, {\left (10690560 \, a^{9} - 24073 \, a^{6}\right )} b^{3} + 3744 \, {\left (133120 \, a^{9} + 49 \, a^{6}\right )} b^{2} - 297 \, {\left (450560 \, a^{9} + 7 \, a^{6}\right )} b\right )} x + 2 \, {\left (429 \, {\left (16777216 \, a^{7} b^{6} + 6291456 \, a^{7} b^{5} + 196608 \, a^{7} b^{4} - 262144 \, a^{10} - 114688 \, a^{7} b^{3} - 2304 \, a^{7} b^{2} + 864 \, a^{7} b - 27 \, a^{7}\right )} x^{3} - 560 \, {\left (16777216 \, a^{4} b^{9} + 6291456 \, a^{4} b^{8} + 196608 \, a^{4} b^{7} - 114688 \, a^{4} b^{6} - 2304 \, a^{4} b^{5} + 864 \, a^{4} b^{4} - {\left (262144 \, a^{7} + 27 \, a^{4}\right )} b^{3}\right )} x^{2} + 1024 \, {\left (16777216 \, a b^{12} + 6291456 \, a b^{11} + 196608 \, a b^{10} - 114688 \, a b^{9} - 2304 \, a b^{8} + 864 \, a b^{7} - {\left (262144 \, a^{4} + 27 \, a\right )} b^{6}\right )} x - 2 \, {\left (17179869184 \, b^{13} + 6442450944 \, b^{12} + 201326592 \, b^{11} - 117440512 \, b^{10} - 2359296 \, b^{9} - 1024 \, {\left (262144 \, a^{3} + 27\right )} b^{7} + 884736 \, b^{8} + 231 \, {\left (16777216 \, a^{6} b^{7} + 6291456 \, a^{6} b^{6} + 196608 \, a^{6} b^{5} - 114688 \, a^{6} b^{4} - 2304 \, a^{6} b^{3} + 864 \, a^{6} b^{2} - {\left (262144 \, a^{9} + 27 \, a^{6}\right )} b\right )} x^{2} - 320 \, {\left (16777216 \, a^{3} b^{10} + 6291456 \, a^{3} b^{9} + 196608 \, a^{3} b^{8} - 114688 \, a^{3} b^{7} - 2304 \, a^{3} b^{6} + 864 \, a^{3} b^{5} - {\left (262144 \, a^{6} + 27 \, a^{3}\right )} b^{4}\right )} x\right )} x^{\frac {2}{3}} + 24 \, {\left (21 \, {\left (16777216 \, a^{5} b^{8} + 6291456 \, a^{5} b^{7} + 196608 \, a^{5} b^{6} - 114688 \, a^{5} b^{5} - 2304 \, a^{5} b^{4} + 864 \, a^{5} b^{3} - {\left (262144 \, a^{8} + 27 \, a^{5}\right )} b^{2}\right )} x^{2} - 32 \, {\left (16777216 \, a^{2} b^{11} + 6291456 \, a^{2} b^{10} + 196608 \, a^{2} b^{9} - 114688 \, a^{2} b^{8} - 2304 \, a^{2} b^{7} + 864 \, a^{2} b^{6} - {\left (262144 \, a^{5} + 27 \, a^{2}\right )} b^{5}\right )} x\right )} x^{\frac {1}{3}}\right )} \sqrt {a x + b x^{\frac {2}{3}}}}{2145 \, {\left (16777216 \, a^{8} b^{6} + 6291456 \, a^{8} b^{5} + 196608 \, a^{8} b^{4} - 262144 \, a^{11} - 114688 \, a^{8} b^{3} - 2304 \, a^{8} b^{2} + 864 \, a^{8} b - 27 \, a^{8}\right )} x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]