\[ \int \frac {x}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {6 x}{a \sqrt {b \,x^{\frac {2}{3}}+a x}}+\frac {8 \sqrt {b \,x^{\frac {2}{3}}+a x}}{a^{2}}-\frac {16 b \sqrt {b \,x^{\frac {2}{3}}+a x}}{a^{3} x^{\frac {1}{3}}} \]
command
integrate(x/(b*x^(2/3)+a*x)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {{\left (98304 \, a^{3} b^{10} + 81920 \, a^{3} b^{9} - 30720 \, a^{3} b^{8} + 1456 \, a^{9} + 6144 \, {\left (16 \, a^{6} - 3 \, a^{3}\right )} b^{7} + 6784 \, {\left (8 \, a^{6} + a^{3}\right )} b^{6} - 192 \, {\left (236 \, a^{6} + a^{3}\right )} b^{5} + 24 \, {\left (1536 \, a^{9} - 2512 \, a^{6} - 3 \, a^{3}\right )} b^{4} + 32 \, {\left (576 \, a^{9} + 379 \, a^{6}\right )} b^{3} - 12 \, {\left (2304 \, a^{9} + 61 \, a^{6}\right )} b^{2} - 3 \, {\left (10112 \, a^{9} + 15 \, a^{6}\right )} b\right )} x^{2} + {\left (98304 \, b^{13} + 81920 \, b^{12} + 6144 \, {\left (16 \, a^{3} - 3\right )} b^{10} - 30720 \, b^{11} + 6784 \, {\left (8 \, a^{3} + 1\right )} b^{9} + 1456 \, a^{6} b^{3} - 192 \, {\left (236 \, a^{3} + 1\right )} b^{8} + 24 \, {\left (1536 \, a^{6} - 2512 \, a^{3} - 3\right )} b^{7} + 32 \, {\left (576 \, a^{6} + 379 \, a^{3}\right )} b^{6} - 12 \, {\left (2304 \, a^{6} + 61 \, a^{3}\right )} b^{5} - 3 \, {\left (10112 \, a^{6} + 15 \, a^{3}\right )} b^{4}\right )} x + 2 \, {\left ({\left (4096 \, a^{4} b^{9} + 6144 \, a^{4} b^{8} + 768 \, a^{4} b^{7} - 4096 \, a^{10} - 144 \, a^{7} b^{2} + 216 \, a^{7} b - 27 \, a^{7} + 256 \, {\left (16 \, a^{7} - 7 \, a^{4}\right )} b^{6} + 48 \, {\left (128 \, a^{7} - 3 \, a^{4}\right )} b^{5} + 24 \, {\left (32 \, a^{7} + 9 \, a^{4}\right )} b^{4} - {\left (5888 \, a^{7} + 27 \, a^{4}\right )} b^{3}\right )} x^{2} - 3 \, {\left (4096 \, a^{2} b^{11} + 6144 \, a^{2} b^{10} + 768 \, a^{2} b^{9} - 144 \, a^{5} b^{4} + 256 \, {\left (16 \, a^{5} - 7 \, a^{2}\right )} b^{8} + 216 \, a^{5} b^{3} + 48 \, {\left (128 \, a^{5} - 3 \, a^{2}\right )} b^{7} + 24 \, {\left (32 \, a^{5} + 9 \, a^{2}\right )} b^{6} - {\left (5888 \, a^{5} + 27 \, a^{2}\right )} b^{5} - {\left (4096 \, a^{8} + 27 \, a^{5}\right )} b^{2}\right )} x^{\frac {4}{3}} + 4 \, {\left (4096 \, a b^{12} + 6144 \, a b^{11} + 768 \, a b^{10} + 256 \, {\left (16 \, a^{4} - 7 \, a\right )} b^{9} - 144 \, a^{4} b^{5} + 48 \, {\left (128 \, a^{4} - 3 \, a\right )} b^{8} + 216 \, a^{4} b^{4} + 24 \, {\left (32 \, a^{4} + 9 \, a\right )} b^{7} - {\left (5888 \, a^{4} + 27 \, a\right )} b^{6} - {\left (4096 \, a^{7} + 27 \, a^{4}\right )} b^{3}\right )} x - {\left (32768 \, b^{13} + 49152 \, b^{12} + 2048 \, {\left (16 \, a^{3} - 7\right )} b^{10} + 6144 \, b^{11} + 384 \, {\left (128 \, a^{3} - 3\right )} b^{9} - 1152 \, a^{3} b^{6} + 192 \, {\left (32 \, a^{3} + 9\right )} b^{8} + 1728 \, a^{3} b^{5} - 8 \, {\left (5888 \, a^{3} + 27\right )} b^{7} - 8 \, {\left (4096 \, a^{6} + 27 \, a^{3}\right )} b^{4} + 5 \, {\left (4096 \, a^{3} b^{10} + 6144 \, a^{3} b^{9} + 768 \, a^{3} b^{8} - 144 \, a^{6} b^{3} + 216 \, a^{6} b^{2} + 256 \, {\left (16 \, a^{6} - 7 \, a^{3}\right )} b^{7} + 48 \, {\left (128 \, a^{6} - 3 \, a^{3}\right )} b^{6} + 24 \, {\left (32 \, a^{6} + 9 \, a^{3}\right )} b^{5} - {\left (5888 \, a^{6} + 27 \, a^{3}\right )} b^{4} - {\left (4096 \, a^{9} + 27 \, a^{6}\right )} b\right )} x\right )} x^{\frac {2}{3}}\right )} \sqrt {a x + b x^{\frac {2}{3}}}}{{\left (4096 \, a^{6} b^{9} + 6144 \, a^{6} b^{8} + 768 \, a^{6} b^{7} - 4096 \, a^{12} - 144 \, a^{9} b^{2} + 216 \, a^{9} b - 27 \, a^{9} + 256 \, {\left (16 \, a^{9} - 7 \, a^{6}\right )} b^{6} + 48 \, {\left (128 \, a^{9} - 3 \, a^{6}\right )} b^{5} + 24 \, {\left (32 \, a^{9} + 9 \, a^{6}\right )} b^{4} - {\left (5888 \, a^{9} + 27 \, a^{6}\right )} b^{3}\right )} x^{2} + {\left (4096 \, a^{3} b^{12} + 6144 \, a^{3} b^{11} + 768 \, a^{3} b^{10} - 144 \, a^{6} b^{5} + 216 \, a^{6} b^{4} + 256 \, {\left (16 \, a^{6} - 7 \, a^{3}\right )} b^{9} + 48 \, {\left (128 \, a^{6} - 3 \, a^{3}\right )} b^{8} + 24 \, {\left (32 \, a^{6} + 9 \, a^{3}\right )} b^{7} - {\left (5888 \, a^{6} + 27 \, a^{3}\right )} b^{6} - {\left (4096 \, a^{9} + 27 \, a^{6}\right )} b^{3}\right )} x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]