Integrand size = 71, antiderivative size = 423 \[ \int \frac {\left (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 \left (-1-x+x^2+x^4\right )} \, dx=\frac {\left (-1-x+x^2-3 x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^4}+\sqrt [3]{2} 3^{5/6} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2\ 2^{2/3} \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{3^{5/6}}\right )-\frac {5 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {5}{3} \log \left (-1+\sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}\right )-\sqrt [3]{6} \log \left (-3+6^{2/3} \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}\right )-\frac {5}{6} \log \left (1+\sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}+\left (\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}\right )^{2/3}\right )+\frac {\sqrt [3]{3} \log \left (3+6^{2/3} \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}+2 \sqrt [3]{6} \left (\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}\right )^{2/3}\right )}{2^{2/3}} \]
(-3*x^4+x^2-x-1)*((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(1/3)/x^4+2^(1/3)*3^(5/ 6)*arctan(1/3*3^(1/2)+2/3*2^(2/3)*((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(1/3)* 3^(1/6))-5/3*arctan(1/3*3^(1/2)+2/3*((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(1/3 )*3^(1/2))*3^(1/2)+5/3*ln(-1+((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(1/3))-6^(1 /3)*ln(-3+6^(2/3)*((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(1/3))-5/6*ln(1+((2*x^ 4-x^2+x+1)/(3*x^4-x^2+x+1))^(1/3)+((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(2/3)) +1/2*3^(1/3)*ln(3+6^(2/3)*((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(1/3)+2*6^(1/3 )*((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(2/3))*2^(1/3)
\[ \int \frac {\left (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 \left (-1-x+x^2+x^4\right )} \, dx=\int \frac {\left (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 \left (-1-x+x^2+x^4\right )} \, dx \]
Integrate[((-4 - 3*x + 2*x^2)*(1 + x - x^2 + x^4)*((1 + x - x^2 + 2*x^4)/( 1 + x - x^2 + 3*x^4))^(1/3))/(x^5*(-1 - x + x^2 + x^4)),x]
Integrate[((-4 - 3*x + 2*x^2)*(1 + x - x^2 + x^4)*((1 + x - x^2 + 2*x^4)/( 1 + x - x^2 + 3*x^4))^(1/3))/(x^5*(-1 - x + x^2 + x^4)), x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (2 x^2-3 x-4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (x^4+x^2-x-1\right )} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (2 x^2-3 x-4\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}} \left (x^4-x^2+x+1\right )}{5 (x-1) x^5}+\frac {\left (-x^2-2 x-4\right ) \left (2 x^2-3 x-4\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}} \left (x^4-x^2+x+1\right )}{5 x^5 \left (x^3+x^2+2 x+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (-x^4-x^2+x+1\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-2 x^2+3 x+4\right ) \left (x^2+2 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 x^5 \left (x^3+x^2+2 x+1\right )}-\frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 (x-1) x^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (-x^4-x^2+x+1\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-2 x^2+3 x+4\right ) \left (x^2+2 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 x^5 \left (x^3+x^2+2 x+1\right )}-\frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 (x-1) x^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (-x^4-x^2+x+1\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-2 x^2+3 x+4\right ) \left (x^2+2 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 x^5 \left (x^3+x^2+2 x+1\right )}-\frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 (x-1) x^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (-x^4-x^2+x+1\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-2 x^2+3 x+4\right ) \left (x^2+2 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 x^5 \left (x^3+x^2+2 x+1\right )}-\frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 (x-1) x^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (-x^4-x^2+x+1\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-2 x^2+3 x+4\right ) \left (x^2+2 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 x^5 \left (x^3+x^2+2 x+1\right )}-\frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 (x-1) x^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (-x^4-x^2+x+1\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-2 x^2+3 x+4\right ) \left (x^2+2 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 x^5 \left (x^3+x^2+2 x+1\right )}-\frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 (x-1) x^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (-x^4-x^2+x+1\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-2 x^2+3 x+4\right ) \left (x^2+2 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 x^5 \left (x^3+x^2+2 x+1\right )}-\frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 (x-1) x^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (-x^4-x^2+x+1\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-2 x^2+3 x+4\right ) \left (x^2+2 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 x^5 \left (x^3+x^2+2 x+1\right )}-\frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 (x-1) x^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (-x^4-x^2+x+1\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-2 x^2+3 x+4\right ) \left (x^2+2 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 x^5 \left (x^3+x^2+2 x+1\right )}-\frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 (x-1) x^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (-x^4-x^2+x+1\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-2 x^2+3 x+4\right ) \left (x^2+2 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 x^5 \left (x^3+x^2+2 x+1\right )}-\frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 (x-1) x^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (-x^4-x^2+x+1\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-2 x^2+3 x+4\right ) \left (x^2+2 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 x^5 \left (x^3+x^2+2 x+1\right )}-\frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 (x-1) x^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (-x^4-x^2+x+1\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-2 x^2+3 x+4\right ) \left (x^2+2 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 x^5 \left (x^3+x^2+2 x+1\right )}-\frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 (x-1) x^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (-x^4-x^2+x+1\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-2 x^2+3 x+4\right ) \left (x^2+2 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 x^5 \left (x^3+x^2+2 x+1\right )}-\frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 (x-1) x^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (-x^4-x^2+x+1\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {\left (-2 x^2+3 x+4\right ) \left (x^2+2 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 x^5 \left (x^3+x^2+2 x+1\right )}-\frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{5 (x-1) x^5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (-2 x^2+3 x+4\right ) \left (x^4-x^2+x+1\right ) \sqrt [3]{\frac {2 x^4-x^2+x+1}{3 x^4-x^2+x+1}}}{x^5 \left (-x^4-x^2+x+1\right )}dx\) |
Int[((-4 - 3*x + 2*x^2)*(1 + x - x^2 + x^4)*((1 + x - x^2 + 2*x^4)/(1 + x - x^2 + 3*x^4))^(1/3))/(x^5*(-1 - x + x^2 + x^4)),x]
3.31.20.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
\[\int \frac {\left (2 x^{2}-3 x -4\right ) \left (x^{4}-x^{2}+x +1\right ) \left (\frac {2 x^{4}-x^{2}+x +1}{3 x^{4}-x^{2}+x +1}\right )^{\frac {1}{3}}}{x^{5} \left (x^{4}+x^{2}-x -1\right )}d x\]
int((2*x^2-3*x-4)*(x^4-x^2+x+1)*((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(1/3)/x^ 5/(x^4+x^2-x-1),x)
Leaf count of result is larger than twice the leaf count of optimal. 960 vs. \(2 (379) = 758\).
Time = 38.80 (sec) , antiderivative size = 960, normalized size of antiderivative = 2.27 \[ \int \frac {\left (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 \left (-1-x+x^2+x^4\right )} \, dx=\text {Too large to display} \]
integrate((2*x^2-3*x-4)*(x^4-x^2+x+1)*((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(1 /3)/x^5/(x^4+x^2-x-1),x, algorithm="fricas")
-1/6*(2*sqrt(3)*(-6)^(1/3)*x^4*arctan(1/3*(6*sqrt(3)*(-6)^(2/3)*(1947*x^12 - 2263*x^10 + 2263*x^9 + 3128*x^8 - 1730*x^7 - 974*x^6 + 2057*x^5 + 865*x ^4 - 545*x^3 + 327*x + 109)*((2*x^4 - x^2 + x + 1)/(3*x^4 - x^2 + x + 1))^ (1/3) + 24*sqrt(3)*(-6)^(1/3)*(39*x^12 + 11*x^10 - 11*x^9 - 34*x^8 + 46*x^ 7 + 28*x^6 - 61*x^5 - 23*x^4 + 25*x^3 - 15*x - 5)*((2*x^4 - x^2 + x + 1)/( 3*x^4 - x^2 + x + 1))^(2/3) + sqrt(3)*(16199*x^12 - 20631*x^10 + 20631*x^9 + 29268*x^8 - 17274*x^7 - 9826*x^6 + 20841*x^5 + 8637*x^4 - 5945*x^3 + 35 67*x + 1189))/(17497*x^12 - 20409*x^10 + 20409*x^9 + 28188*x^8 - 15558*x^7 - 8750*x^6 + 18471*x^5 + 7779*x^4 - 4855*x^3 + 2913*x + 971)) - 10*sqrt(3 )*x^4*arctan((26407150*sqrt(3)*(3*x^4 - x^2 + x + 1)*((2*x^4 - x^2 + x + 1 )/(3*x^4 - x^2 + x + 1))^(2/3) + 15172108*sqrt(3)*(3*x^4 - x^2 + x + 1)*(( 2*x^4 - x^2 + x + 1)/(3*x^4 - x^2 + x + 1))^(1/3) + sqrt(3)*(47470762*x^4 - 20789629*x^2 + 20789629*x + 20789629))/(29760814*x^4 - 16852563*x^2 + 16 852563*x + 16852563)) + (-6)^(1/3)*x^4*log((12*(-6)^(2/3)*(39*x^8 - 28*x^6 + 28*x^5 + 33*x^4 - 10*x^3 - 5*x^2 + 10*x + 5)*((2*x^4 - x^2 + x + 1)/(3* x^4 - x^2 + x + 1))^(2/3) - (-6)^(1/3)*(649*x^8 - 538*x^6 + 538*x^5 + 647* x^4 - 218*x^3 - 109*x^2 + 218*x + 109) + 18*(75*x^8 - 58*x^6 + 58*x^5 + 69 *x^4 - 22*x^3 - 11*x^2 + 22*x + 11)*((2*x^4 - x^2 + x + 1)/(3*x^4 - x^2 + x + 1))^(1/3))/(x^8 + 2*x^6 - 2*x^5 - x^4 - 2*x^3 - x^2 + 2*x + 1)) - 2*(- 6)^(1/3)*x^4*log(((-6)^(2/3)*(x^4 + x^2 - x - 1) + 18*(-6)^(1/3)*(3*x^4...
Timed out. \[ \int \frac {\left (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 \left (-1-x+x^2+x^4\right )} \, dx=\text {Timed out} \]
integrate((2*x**2-3*x-4)*(x**4-x**2+x+1)*((2*x**4-x**2+x+1)/(3*x**4-x**2+x +1))**(1/3)/x**5/(x**4+x**2-x-1),x)
\[ \int \frac {\left (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 \left (-1-x+x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - x^{2} + x + 1\right )} {\left (2 \, x^{2} - 3 \, x - 4\right )} \left (\frac {2 \, x^{4} - x^{2} + x + 1}{3 \, x^{4} - x^{2} + x + 1}\right )^{\frac {1}{3}}}{{\left (x^{4} + x^{2} - x - 1\right )} x^{5}} \,d x } \]
integrate((2*x^2-3*x-4)*(x^4-x^2+x+1)*((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(1 /3)/x^5/(x^4+x^2-x-1),x, algorithm="maxima")
integrate((x^4 - x^2 + x + 1)*(2*x^2 - 3*x - 4)*((2*x^4 - x^2 + x + 1)/(3* x^4 - x^2 + x + 1))^(1/3)/((x^4 + x^2 - x - 1)*x^5), x)
\[ \int \frac {\left (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 \left (-1-x+x^2+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - x^{2} + x + 1\right )} {\left (2 \, x^{2} - 3 \, x - 4\right )} \left (\frac {2 \, x^{4} - x^{2} + x + 1}{3 \, x^{4} - x^{2} + x + 1}\right )^{\frac {1}{3}}}{{\left (x^{4} + x^{2} - x - 1\right )} x^{5}} \,d x } \]
integrate((2*x^2-3*x-4)*(x^4-x^2+x+1)*((2*x^4-x^2+x+1)/(3*x^4-x^2+x+1))^(1 /3)/x^5/(x^4+x^2-x-1),x, algorithm="giac")
integrate((x^4 - x^2 + x + 1)*(2*x^2 - 3*x - 4)*((2*x^4 - x^2 + x + 1)/(3* x^4 - x^2 + x + 1))^(1/3)/((x^4 + x^2 - x - 1)*x^5), x)
Timed out. \[ \int \frac {\left (-4-3 x+2 x^2\right ) \left (1+x-x^2+x^4\right ) \sqrt [3]{\frac {1+x-x^2+2 x^4}{1+x-x^2+3 x^4}}}{x^5 \left (-1-x+x^2+x^4\right )} \, dx=\int \frac {{\left (\frac {2\,x^4-x^2+x+1}{3\,x^4-x^2+x+1}\right )}^{1/3}\,\left (-2\,x^2+3\,x+4\right )\,\left (x^4-x^2+x+1\right )}{x^5\,\left (-x^4-x^2+x+1\right )} \,d x \]
int((((x - x^2 + 2*x^4 + 1)/(x - x^2 + 3*x^4 + 1))^(1/3)*(3*x - 2*x^2 + 4) *(x - x^2 + x^4 + 1))/(x^5*(x - x^2 - x^4 + 1)),x)