\[ \int \frac {(b+a x) \left (-a q+b p x^2\right )}{\left (q+p x^3\right )^{2/3} \left (b^3 c+d q+3 a b^2 c x+3 a^2 b c x^2+\left (a^3 c+d p\right ) x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {\arctan \left (\frac {\sqrt {3}\, b \,c^{\frac {1}{3}}+\sqrt {3}\, a \,c^{\frac {1}{3}} x}{b \,c^{\frac {1}{3}}+a \,c^{\frac {1}{3}} x -2 d^{\frac {1}{3}} \left (p \,x^{3}+q \right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3 c^{\frac {2}{3}} d^{\frac {1}{3}}}+\frac {\ln \left (b^{2} c^{\frac {1}{3}}+a b \,c^{\frac {1}{3}} x +b \,d^{\frac {1}{3}} \left (p \,x^{3}+q \right )^{\frac {1}{3}}\right )}{3 c^{\frac {2}{3}} d^{\frac {1}{3}}}-\frac {\ln \left (b^{4} c^{\frac {2}{3}}+2 a \,b^{3} c^{\frac {2}{3}} x +a^{2} b^{2} c^{\frac {2}{3}} x^{2}+\left (-b^{3} c^{\frac {1}{3}} d^{\frac {1}{3}}-a \,b^{2} c^{\frac {1}{3}} d^{\frac {1}{3}} x \right ) \left (p \,x^{3}+q \right )^{\frac {1}{3}}+b^{2} d^{\frac {2}{3}} \left (p \,x^{3}+q \right )^{\frac {2}{3}}\right )}{6 c^{\frac {2}{3}} d^{\frac {1}{3}}} \]
command
Integrate[((b + a*x)*(-(a*q) + b*p*x^2))/((q + p*x^3)^(2/3)*(b^3*c + d*q + 3*a*b^2*c*x + 3*a^2*b*c*x^2 + (a^3*c + d*p)*x^3)),x]
Mathematica 13.1 output
\[ \frac {2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{c} (b+a x)}{b \sqrt [3]{c}+a \sqrt [3]{c} x-2 \sqrt [3]{d} \sqrt [3]{q+p x^3}}\right )+2 \log \left (b \left (b \sqrt [3]{c}+a \sqrt [3]{c} x+\sqrt [3]{d} \sqrt [3]{q+p x^3}\right )\right )-\log \left (b^2 \left (b^2 c^{2/3}+2 a b c^{2/3} x+a^2 c^{2/3} x^2-b \sqrt [3]{c} \sqrt [3]{d} \sqrt [3]{q+p x^3}-a \sqrt [3]{c} \sqrt [3]{d} x \sqrt [3]{q+p x^3}+d^{2/3} \left (q+p x^3\right )^{2/3}\right )\right )}{6 c^{2/3} \sqrt [3]{d}} \]
Mathematica 12.3 output
\[ \int \frac {(b+a x) \left (-a q+b p x^2\right )}{\left (q+p x^3\right )^{2/3} \left (b^3 c+d q+3 a b^2 c x+3 a^2 b c x^2+\left (a^3 c+d p\right ) x^3\right )} \, dx \]________________________________________________________________________________________