\[ \int \frac {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (51 x^2+70 x^3+x^3 \log \left (\frac {4}{x^2}\right )\right )}{48+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (48 x^2+72 x^3\right )} \, dx \]
Optimal antiderivative \[ x -\frac {x}{4 \left (3 x +{\mathrm e}^{\frac {x \ln \left (\frac {4}{x^{2}}\right )}{3}} x^{2}+2\right )} \]
command
Integrate[(42 + 144*x + 108*x^2 + 3*2^(2 + (4*x)/3)*(x^(-2))^((2*x)/3)*x^4 + 2^((2*x)/3)*(x^(-2))^(x/3)*(51*x^2 + 70*x^3 + x^3*Log[4/x^2]))/(48 + 144*x + 108*x^2 + 3*2^(2 + (4*x)/3)*(x^(-2))^((2*x)/3)*x^4 + 2^((2*x)/3)*(x^(-2))^(x/3)*(48*x^2 + 72*x^3)),x]
Mathematica 13.1 output
\[ \int \frac {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (51 x^2+70 x^3+x^3 \log \left (\frac {4}{x^2}\right )\right )}{48+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (48 x^2+72 x^3\right )} \, dx \]
Mathematica 12.3 output
\[ \frac {1}{4} x \left (4-\frac {1}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x}\right ) \]