\[ \int \left (b x^{2/3}+a x\right )^{3/2} \, dx \]
Optimal antiderivative \[ \frac {2 \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {5}{2}}}{5 a}-\frac {512 b^{5} \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {5}{2}}}{15015 a^{6} x^{\frac {5}{3}}}+\frac {256 b^{4} \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {5}{2}}}{3003 a^{5} x^{\frac {4}{3}}}-\frac {64 b^{3} \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {5}{2}}}{429 a^{4} x}+\frac {32 b^{2} \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {5}{2}}}{143 a^{3} x^{\frac {2}{3}}}-\frac {4 b \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {5}{2}}}{13 a^{2} x^{\frac {1}{3}}} \]
command
integrate((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 {2 \, {\left (4 \, {\left (805306368 \, b^{13} + 167772160 \, b^{12} + 786432 \, {\left (64 \, a^{3} - 3\right )} b^{10} - 15728640 \, b^{11} - 4096 \, {\left (11264 \, a^{3} - 53\right )} b^{9} + 4372368 \, a^{9} - 1536 \, {\left (5504 \, a^{3} + 1\right )} b^{8} - 48 \, {\left (242810880 \, a^{6} + 114688 \, a^{3} + 3\right )} b^{7} - 1792 \, {\left (1353984 \, a^{6} - 103 \, a^{3}\right )} b^{6} + 192 \, {\left (1152384 \, a^{6} - 23 \, a^{3}\right )} b^{5} - 3 \, {\left (3633315840 \, a^{9} - 12027392 \, a^{6} - 15 \, a^{3}\right )} b^{4} - 112 \, {\left (35389440 \, a^{9} + 29281 \, a^{6}\right )} b^{3} - 819 \, {\left (368640 \, a^{9} - 31 \, a^{6}\right )} b^{2} + 693 \, {\left (40960 \, a^{9} + 3 \, a^{6}\right )} b\right )} x + {\left (3003 \, {\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} - 70 \, {\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} + 128 \, {\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 - 16 \, {\left (268435456 \, b^{13} + 100663296 \, b^{12} + 3145728 \, b^{11} - 1835008 \, b^{10} - 36864 \, b^{9} - 16 \, {\left (262144 \, a^{3} + 27\right )} b^{7} + 13824 \, 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} - 5 \, {\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}} + 3 \, {\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}}}\right )}}{15015 \, {\left (16777216 \, a^{6} b^{6} + 6291456 \, a^{6} b^{5} + 196608 \, a^{6} b^{4} - 262144 \, a^{9} - 114688 \, a^{6} b^{3} - 2304 \, a^{6} b^{2} + 864 \, a^{6} b - 27 \, a^{6}\right )} x} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]