\[ \int \frac {x^3}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {6 x^{3}}{a \sqrt {b \,x^{\frac {2}{3}}+a x}}+\frac {32768 b^{6} \sqrt {b \,x^{\frac {2}{3}}+a x}}{2145 a^{8}}-\frac {65536 b^{7} \sqrt {b \,x^{\frac {2}{3}}+a x}}{2145 a^{9} x^{\frac {1}{3}}}-\frac {8192 b^{5} x^{\frac {1}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{715 a^{7}}+\frac {4096 b^{4} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{429 a^{6}}-\frac {3584 b^{3} x \sqrt {b \,x^{\frac {2}{3}}+a x}}{429 a^{5}}+\frac {5376 b^{2} x^{\frac {4}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{715 a^{4}}-\frac {448 b \,x^{\frac {5}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{65 a^{3}}+\frac {32 x^{2} \sqrt {b \,x^{\frac {2}{3}}+a x}}{5 a^{2}} \]
command
integrate(x^3/(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
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]