\[ \int \frac {x^4}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {6 x^{4}}{a \sqrt {b \,x^{\frac {2}{3}}+a x}}-\frac {524288 b^{9} \sqrt {b \,x^{\frac {2}{3}}+a x}}{29393 a^{11}}+\frac {1048576 b^{10} \sqrt {b \,x^{\frac {2}{3}}+a x}}{29393 a^{12} x^{\frac {1}{3}}}+\frac {393216 b^{8} x^{\frac {1}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{29393 a^{10}}-\frac {327680 b^{7} x^{\frac {2}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{29393 a^{9}}+\frac {40960 b^{6} x \sqrt {b \,x^{\frac {2}{3}}+a x}}{4199 a^{8}}-\frac {36864 b^{5} x^{\frac {4}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{4199 a^{7}}+\frac {33792 b^{4} x^{\frac {5}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{4199 a^{6}}-\frac {16896 b^{3} x^{2} \sqrt {b \,x^{\frac {2}{3}}+a x}}{2261 a^{5}}+\frac {15840 b^{2} x^{\frac {7}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{2261 a^{4}}-\frac {880 b \,x^{\frac {8}{3}} \sqrt {b \,x^{\frac {2}{3}}+a x}}{133 a^{3}}+\frac {44 x^{3} \sqrt {b \,x^{\frac {2}{3}}+a x}}{7 a^{2}} \]
command
integrate(x^4/(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 \[ \text {Timed out} \]