Integrand size = 48, antiderivative size = 156 \[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\frac {\sqrt [3]{-2+x^2+2 x^3} \left (12-12 x^2+54 x^3+3 x^4-27 x^5-38 x^6\right )}{4 x^4 \left (-2+x^2+x^3\right )}-\frac {7 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-2+x^2+2 x^3}}\right )}{\sqrt {3}}-\frac {7}{3} \log \left (-x+\sqrt [3]{-2+x^2+2 x^3}\right )+\frac {7}{6} \log \left (x^2+x \sqrt [3]{-2+x^2+2 x^3}+\left (-2+x^2+2 x^3\right )^{2/3}\right ) \]
1/4*(2*x^3+x^2-2)^(1/3)*(-38*x^6-27*x^5+3*x^4+54*x^3-12*x^2+12)/x^4/(x^3+x ^2-2)-7/3*arctan(3^(1/2)*x/(x+2*(2*x^3+x^2-2)^(1/3)))*3^(1/2)-7/3*ln(-x+(2 *x^3+x^2-2)^(1/3))+7/6*ln(x^2+x*(2*x^3+x^2-2)^(1/3)+(2*x^3+x^2-2)^(2/3))
Time = 0.72 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\frac {\sqrt [3]{-2+x^2+2 x^3} \left (12-12 x^2+54 x^3+3 x^4-27 x^5-38 x^6\right )}{4 x^4 \left (-2+x^2+x^3\right )}-\frac {7 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-2+x^2+2 x^3}}\right )}{\sqrt {3}}-\frac {7}{3} \log \left (-x+\sqrt [3]{-2+x^2+2 x^3}\right )+\frac {7}{6} \log \left (x^2+x \sqrt [3]{-2+x^2+2 x^3}+\left (-2+x^2+2 x^3\right )^{2/3}\right ) \]
Integrate[((-6 + x^2)*(-2 + x^2)*(2 - x^2 + x^3)*(-2 + x^2 + 2*x^3)^(1/3)) /(x^5*(-2 + x^2 + x^3)^2),x]
((-2 + x^2 + 2*x^3)^(1/3)*(12 - 12*x^2 + 54*x^3 + 3*x^4 - 27*x^5 - 38*x^6) )/(4*x^4*(-2 + x^2 + x^3)) - (7*ArcTan[(Sqrt[3]*x)/(x + 2*(-2 + x^2 + 2*x^ 3)^(1/3))])/Sqrt[3] - (7*Log[-x + (-2 + x^2 + 2*x^3)^(1/3)])/3 + (7*Log[x^ 2 + x*(-2 + x^2 + 2*x^3)^(1/3) + (-2 + x^2 + 2*x^3)^(2/3)])/6
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.
| \(\displaystyle \int \frac {\left (x^2-6\right ) \left (x^2-2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (x^3+x^2-2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {8 \left (x^2-6\right ) \left (x^2-2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 (x-1) x^5}+\frac {\left (x^2-6\right ) \left (x^2-2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 (x-1)^2 x^5}+\frac {(8 x+19) \left (x^2-6\right ) \left (x^2-2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 x^5 \left (x^2+2 x+2\right )}+\frac {(4 x+7) \left (x^2-6\right ) \left (x^2-2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 x^5 \left (x^2+2 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-x^2\right ) \left (6-x^2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (-x^3-x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {8 \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 (x-1) x^5}+\frac {\left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 (x-1)^2 x^5}+\frac {(8 x+19) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 x^5 \left (x^2+2 x+2\right )}+\frac {(4 x+7) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 x^5 \left (x^2+2 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-x^2\right ) \left (6-x^2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (-x^3-x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {8 \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 (x-1) x^5}+\frac {\left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 (x-1)^2 x^5}+\frac {(8 x+19) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 x^5 \left (x^2+2 x+2\right )}+\frac {(4 x+7) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 x^5 \left (x^2+2 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-x^2\right ) \left (6-x^2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (-x^3-x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {8 \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 (x-1) x^5}+\frac {\left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 (x-1)^2 x^5}+\frac {(8 x+19) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 x^5 \left (x^2+2 x+2\right )}+\frac {(4 x+7) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 x^5 \left (x^2+2 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-x^2\right ) \left (6-x^2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (-x^3-x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {8 \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 (x-1) x^5}+\frac {\left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 (x-1)^2 x^5}+\frac {(8 x+19) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 x^5 \left (x^2+2 x+2\right )}+\frac {(4 x+7) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 x^5 \left (x^2+2 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-x^2\right ) \left (6-x^2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (-x^3-x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {8 \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 (x-1) x^5}+\frac {\left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 (x-1)^2 x^5}+\frac {(8 x+19) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 x^5 \left (x^2+2 x+2\right )}+\frac {(4 x+7) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 x^5 \left (x^2+2 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-x^2\right ) \left (6-x^2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (-x^3-x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {8 \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 (x-1) x^5}+\frac {\left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 (x-1)^2 x^5}+\frac {(8 x+19) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 x^5 \left (x^2+2 x+2\right )}+\frac {(4 x+7) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 x^5 \left (x^2+2 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-x^2\right ) \left (6-x^2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (-x^3-x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {8 \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 (x-1) x^5}+\frac {\left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 (x-1)^2 x^5}+\frac {(8 x+19) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 x^5 \left (x^2+2 x+2\right )}+\frac {(4 x+7) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 x^5 \left (x^2+2 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-x^2\right ) \left (6-x^2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (-x^3-x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {8 \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 (x-1) x^5}+\frac {\left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 (x-1)^2 x^5}+\frac {(8 x+19) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 x^5 \left (x^2+2 x+2\right )}+\frac {(4 x+7) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 x^5 \left (x^2+2 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-x^2\right ) \left (6-x^2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (-x^3-x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {8 \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 (x-1) x^5}+\frac {\left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 (x-1)^2 x^5}+\frac {(8 x+19) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 x^5 \left (x^2+2 x+2\right )}+\frac {(4 x+7) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 x^5 \left (x^2+2 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-x^2\right ) \left (6-x^2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (-x^3-x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {8 \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 (x-1) x^5}+\frac {\left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 (x-1)^2 x^5}+\frac {(8 x+19) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 x^5 \left (x^2+2 x+2\right )}+\frac {(4 x+7) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 x^5 \left (x^2+2 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-x^2\right ) \left (6-x^2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (-x^3-x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {8 \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 (x-1) x^5}+\frac {\left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 (x-1)^2 x^5}+\frac {(8 x+19) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 x^5 \left (x^2+2 x+2\right )}+\frac {(4 x+7) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 x^5 \left (x^2+2 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-x^2\right ) \left (6-x^2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (-x^3-x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {8 \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 (x-1) x^5}+\frac {\left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 (x-1)^2 x^5}+\frac {(8 x+19) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 x^5 \left (x^2+2 x+2\right )}+\frac {(4 x+7) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 x^5 \left (x^2+2 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-x^2\right ) \left (6-x^2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (-x^3-x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {8 \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 (x-1) x^5}+\frac {\left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 (x-1)^2 x^5}+\frac {(8 x+19) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 x^5 \left (x^2+2 x+2\right )}+\frac {(4 x+7) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 x^5 \left (x^2+2 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-x^2\right ) \left (6-x^2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (-x^3-x^2+2\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {8 \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 (x-1) x^5}+\frac {\left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 (x-1)^2 x^5}+\frac {(8 x+19) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{125 x^5 \left (x^2+2 x+2\right )}+\frac {(4 x+7) \left (2-x^2\right ) \left (6-x^2\right ) \sqrt [3]{2 x^3+x^2-2} \left (x^3-x^2+2\right )}{25 x^5 \left (x^2+2 x+2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (2-x^2\right ) \left (6-x^2\right ) \left (x^3-x^2+2\right ) \sqrt [3]{2 x^3+x^2-2}}{x^5 \left (-x^3-x^2+2\right )^2}dx\) |
Int[((-6 + x^2)*(-2 + x^2)*(2 - x^2 + x^3)*(-2 + x^2 + 2*x^3)^(1/3))/(x^5* (-2 + x^2 + x^3)^2),x]
3.22.37.3.1 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]]
Time = 3.16 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.28
| method | result | size |
| pseudoelliptic | \(\frac {\left (114 x^{6}+81 x^{5}-9 x^{4}-162 x^{3}+36 x^{2}-36\right ) \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}-14 x^{4} \left (x^{3}+x^{2}-2\right ) \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}\right )}{3 x}\right )+\ln \left (\frac {x^{2}+x \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}+\left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-x +\left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}}{x}\right )\right )}{12 \left (x^{2}+x \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}+\left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}}\right ) x^{4} \left (x -\left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}\right )}\) | \(199\) |
| trager | \(-\frac {\left (38 x^{6}+27 x^{5}-3 x^{4}-54 x^{3}+12 x^{2}-12\right ) \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}}{4 \left (x^{3}+x^{2}-2\right ) x^{4}}-\frac {7 \ln \left (-\frac {20044800 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{3}+468480 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x -1586400 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}-80179200 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{2}+1292640 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{3}-3305 \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x +2329 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} x^{2}+383520 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}+2490 x^{3}+160358400 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2}+1079 x^{2}-767040 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )-2158}{\left (-1+x \right ) \left (x^{2}+2 x +2\right )}\right )}{3}+1120 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) \ln \left (-\frac {149196441600 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{3}+4089968640 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x +2476821600 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}-596785766400 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{2}-9892153440 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{3}+5160045 \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x -13680813 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} x^{2}-1771233600 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}+16568849 x^{3}+1193571532800 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2}+4810311 x^{2}+3542467200 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )-9620622}{\left (-1+x \right ) \left (x^{2}+2 x +2\right )}\right )\) | \(495\) |
| risch | \(\text {Expression too large to display}\) | \(1291\) |
int((x^2-6)*(x^2-2)*(x^3-x^2+2)*(2*x^3+x^2-2)^(1/3)/x^5/(x^3+x^2-2)^2,x,me thod=_RETURNVERBOSE)
1/12*((114*x^6+81*x^5-9*x^4-162*x^3+36*x^2-36)*(2*x^3+x^2-2)^(1/3)-14*x^4* (x^3+x^2-2)*(2*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(2*x^3+x^2-2)^(1/3)))+ln( (x^2+x*(2*x^3+x^2-2)^(1/3)+(2*x^3+x^2-2)^(2/3))/x^2)-2*ln((-x+(2*x^3+x^2-2 )^(1/3))/x)))/(x^2+x*(2*x^3+x^2-2)^(1/3)+(2*x^3+x^2-2)^(2/3))/x^4/(x-(2*x^ 3+x^2-2)^(1/3))
Time = 1.01 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.36 \[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\frac {28 \, \sqrt {3} {\left (x^{7} + x^{6} - 2 \, x^{4}\right )} \arctan \left (\frac {1078 \, \sqrt {3} {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}} x^{2} + 196 \, \sqrt {3} {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (669 \, x^{3} + 32 \, x^{2} - 64\right )}}{1315 \, x^{3} - 8 \, x^{2} + 16}\right ) - 14 \, {\left (x^{7} + x^{6} - 2 \, x^{4}\right )} \log \left (\frac {x^{3} + 3 \, {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}} x^{2} + x^{2} - 3 \, {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {2}{3}} x - 2}{x^{3} + x^{2} - 2}\right ) - 3 \, {\left (38 \, x^{6} + 27 \, x^{5} - 3 \, x^{4} - 54 \, x^{3} + 12 \, x^{2} - 12\right )} {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}}}{12 \, {\left (x^{7} + x^{6} - 2 \, x^{4}\right )}} \]
integrate((x^2-6)*(x^2-2)*(x^3-x^2+2)*(2*x^3+x^2-2)^(1/3)/x^5/(x^3+x^2-2)^ 2,x, algorithm="fricas")
1/12*(28*sqrt(3)*(x^7 + x^6 - 2*x^4)*arctan((1078*sqrt(3)*(2*x^3 + x^2 - 2 )^(1/3)*x^2 + 196*sqrt(3)*(2*x^3 + x^2 - 2)^(2/3)*x + sqrt(3)*(669*x^3 + 3 2*x^2 - 64))/(1315*x^3 - 8*x^2 + 16)) - 14*(x^7 + x^6 - 2*x^4)*log((x^3 + 3*(2*x^3 + x^2 - 2)^(1/3)*x^2 + x^2 - 3*(2*x^3 + x^2 - 2)^(2/3)*x - 2)/(x^ 3 + x^2 - 2)) - 3*(38*x^6 + 27*x^5 - 3*x^4 - 54*x^3 + 12*x^2 - 12)*(2*x^3 + x^2 - 2)^(1/3))/(x^7 + x^6 - 2*x^4)
\[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\int \frac {\left (x + 1\right ) \left (x^{2} - 6\right ) \left (x^{2} - 2\right ) \left (x^{2} - 2 x + 2\right ) \sqrt [3]{2 x^{3} + x^{2} - 2}}{x^{5} \left (x - 1\right )^{2} \left (x^{2} + 2 x + 2\right )^{2}}\, dx \]
Integral((x + 1)*(x**2 - 6)*(x**2 - 2)*(x**2 - 2*x + 2)*(2*x**3 + x**2 - 2 )**(1/3)/(x**5*(x - 1)**2*(x**2 + 2*x + 2)**2), x)
\[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\int { \frac {{\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}} {\left (x^{3} - x^{2} + 2\right )} {\left (x^{2} - 2\right )} {\left (x^{2} - 6\right )}}{{\left (x^{3} + x^{2} - 2\right )}^{2} x^{5}} \,d x } \]
integrate((x^2-6)*(x^2-2)*(x^3-x^2+2)*(2*x^3+x^2-2)^(1/3)/x^5/(x^3+x^2-2)^ 2,x, algorithm="maxima")
integrate((2*x^3 + x^2 - 2)^(1/3)*(x^3 - x^2 + 2)*(x^2 - 2)*(x^2 - 6)/((x^ 3 + x^2 - 2)^2*x^5), x)
\[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\int { \frac {{\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}} {\left (x^{3} - x^{2} + 2\right )} {\left (x^{2} - 2\right )} {\left (x^{2} - 6\right )}}{{\left (x^{3} + x^{2} - 2\right )}^{2} x^{5}} \,d x } \]
integrate((x^2-6)*(x^2-2)*(x^3-x^2+2)*(2*x^3+x^2-2)^(1/3)/x^5/(x^3+x^2-2)^ 2,x, algorithm="giac")
integrate((2*x^3 + x^2 - 2)^(1/3)*(x^3 - x^2 + 2)*(x^2 - 2)*(x^2 - 6)/((x^ 3 + x^2 - 2)^2*x^5), x)
Timed out. \[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\int \frac {\left (x^2-2\right )\,\left (x^2-6\right )\,\left (x^3-x^2+2\right )\,{\left (2\,x^3+x^2-2\right )}^{1/3}}{x^5\,{\left (x^3+x^2-2\right )}^2} \,d x \]