Integrand size = 40, antiderivative size = 105 \[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx=\frac {3 \left (-1+x^5\right )^{2/3} \left (-4-5 x^3+4 x^5\right )}{10 x^5}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1+x^5}}\right )-\log \left (x+\sqrt [3]{-1+x^5}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^5}+\left (-1+x^5\right )^{2/3}\right ) \] Output:
3/10*(x^5-1)^(2/3)*(4*x^5-5*x^3-4)/x^5-3^(1/2)*arctan(3^(1/2)*x/(-x+2*(x^5 -1)^(1/3)))-ln(x+(x^5-1)^(1/3))+1/2*ln(x^2-x*(x^5-1)^(1/3)+(x^5-1)^(2/3))
Time = 2.63 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx=\frac {3 \left (-1+x^5\right )^{2/3} \left (-4-5 x^3+4 x^5\right )}{10 x^5}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{-1+x^5}}\right )-\log \left (x+\sqrt [3]{-1+x^5}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1+x^5}+\left (-1+x^5\right )^{2/3}\right ) \] Input:
Integrate[((-1 + x^5)^(2/3)*(3 + 2*x^5)*(-2 + x^3 + 2*x^5))/(x^6*(-1 + x^3 + x^5)),x]
Output:
(3*(-1 + x^5)^(2/3)*(-4 - 5*x^3 + 4*x^5))/(10*x^5) + Sqrt[3]*ArcTan[(Sqrt[ 3]*x)/(x - 2*(-1 + x^5)^(1/3))] - Log[x + (-1 + x^5)^(1/3)] + Log[x^2 - x* (-1 + x^5)^(1/3) + (-1 + x^5)^(2/3)]/2
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.
| \(\displaystyle \int \frac {\left (x^5-1\right )^{2/3} \left (2 x^5+3\right ) \left (2 x^5+x^3-2\right )}{x^6 \left (x^5+x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (x^5-1\right )^{2/3}}{x}+\frac {6 \left (x^5-1\right )^{2/3}}{x^6}+\frac {3 \left (x^5-1\right )^{2/3}}{x^3}+\frac {\left (x^5-1\right )^{2/3} \left (-5 x^2-3\right )}{x^5+x^3-1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \int \frac {\left (x^5-1\right )^{2/3}}{x^5+x^3-1}dx-5 \int \frac {x^2 \left (x^5-1\right )^{2/3}}{x^5+x^3-1}dx-\frac {3 \left (x^5-1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {2}{5},\frac {3}{5},x^5\right )}{2 x^2 \left (1-x^5\right )^{2/3}}-\frac {6 \left (x^5-1\right )^{2/3}}{5 x^5}+\frac {6}{5} \left (x^5-1\right )^{2/3}\) |
Input:
Int[((-1 + x^5)^(2/3)*(3 + 2*x^5)*(-2 + x^3 + 2*x^5))/(x^6*(-1 + x^3 + x^5 )),x]
Output:
$Aborted
Time = 11.74 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.98
| method | result | size |
| pseudoelliptic | \(\frac {-10 \ln \left (\frac {x +\left (x^{5}-1\right )^{\frac {1}{3}}}{x}\right ) x^{5}+\left (12 x^{5}-15 x^{3}-12\right ) \left (x^{5}-1\right )^{\frac {2}{3}}+5 x^{5} \left (-2 \sqrt {3}\, \arctan \left (\frac {\left (x -2 \left (x^{5}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right )+\ln \left (\frac {x^{2}-x \left (x^{5}-1\right )^{\frac {1}{3}}+\left (x^{5}-1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{10 x^{5}}\) | \(103\) |
| risch | \(\frac {-\frac {3}{2} x^{8}+\frac {3}{2} x^{3}-\frac {12}{5} x^{5}+\frac {6}{5}+\frac {6}{5} x^{10}}{x^{5} \left (x^{5}-1\right )^{\frac {1}{3}}}-\ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-2 x^{5}-3 \left (x^{5}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-3 \left (x^{5}-1\right )^{\frac {2}{3}} x +2 x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2}{x^{5}+x^{3}-1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+x^{5}+3 \left (x^{5}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}-1\right )^{\frac {2}{3}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{x^{5}+x^{3}-1}\right )\) | \(260\) |
| trager | \(\text {Expression too large to display}\) | \(601\) |
Input:
int((x^5-1)^(2/3)*(2*x^5+3)*(2*x^5+x^3-2)/x^6/(x^5+x^3-1),x,method=_RETURN VERBOSE)
Output:
1/10*(-10*ln((x+(x^5-1)^(1/3))/x)*x^5+(12*x^5-15*x^3-12)*(x^5-1)^(2/3)+5*x ^5*(-2*3^(1/2)*arctan(1/3*(x-2*(x^5-1)^(1/3))*3^(1/2)/x)+ln((x^2-x*(x^5-1) ^(1/3)+(x^5-1)^(2/3))/x^2)))/x^5
Time = 4.37 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.37 \[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx=-\frac {10 \, \sqrt {3} x^{5} \arctan \left (\frac {1092 \, \sqrt {3} {\left (x^{5} - 1\right )}^{\frac {1}{3}} x^{2} + 2002 \, \sqrt {3} {\left (x^{5} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (121 \, x^{5} + 576 \, x^{3} - 121\right )}}{3 \, {\left (1331 \, x^{5} - 216 \, x^{3} - 1331\right )}}\right ) + 5 \, x^{5} \log \left (\frac {x^{5} + x^{3} + 3 \, {\left (x^{5} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{5} - 1\right )}^{\frac {2}{3}} x - 1}{x^{5} + x^{3} - 1}\right ) - 3 \, {\left (4 \, x^{5} - 5 \, x^{3} - 4\right )} {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{10 \, x^{5}} \] Input:
integrate((x^5-1)^(2/3)*(2*x^5+3)*(2*x^5+x^3-2)/x^6/(x^5+x^3-1),x, algorit hm="fricas")
Output:
-1/10*(10*sqrt(3)*x^5*arctan(1/3*(1092*sqrt(3)*(x^5 - 1)^(1/3)*x^2 + 2002* sqrt(3)*(x^5 - 1)^(2/3)*x + sqrt(3)*(121*x^5 + 576*x^3 - 121))/(1331*x^5 - 216*x^3 - 1331)) + 5*x^5*log((x^5 + x^3 + 3*(x^5 - 1)^(1/3)*x^2 + 3*(x^5 - 1)^(2/3)*x - 1)/(x^5 + x^3 - 1)) - 3*(4*x^5 - 5*x^3 - 4)*(x^5 - 1)^(2/3) )/x^5
\[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )\right )^{\frac {2}{3}} \cdot \left (2 x^{5} + 3\right ) \left (2 x^{5} + x^{3} - 2\right )}{x^{6} \left (x^{5} + x^{3} - 1\right )}\, dx \] Input:
integrate((x**5-1)**(2/3)*(2*x**5+3)*(2*x**5+x**3-2)/x**6/(x**5+x**3-1),x)
Output:
Integral(((x - 1)*(x**4 + x**3 + x**2 + x + 1))**(2/3)*(2*x**5 + 3)*(2*x** 5 + x**3 - 2)/(x**6*(x**5 + x**3 - 1)), x)
\[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{5} + x^{3} - 2\right )} {\left (2 \, x^{5} + 3\right )} {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{{\left (x^{5} + x^{3} - 1\right )} x^{6}} \,d x } \] Input:
integrate((x^5-1)^(2/3)*(2*x^5+3)*(2*x^5+x^3-2)/x^6/(x^5+x^3-1),x, algorit hm="maxima")
Output:
integrate((2*x^5 + x^3 - 2)*(2*x^5 + 3)*(x^5 - 1)^(2/3)/((x^5 + x^3 - 1)*x ^6), x)
\[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{5} + x^{3} - 2\right )} {\left (2 \, x^{5} + 3\right )} {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{{\left (x^{5} + x^{3} - 1\right )} x^{6}} \,d x } \] Input:
integrate((x^5-1)^(2/3)*(2*x^5+3)*(2*x^5+x^3-2)/x^6/(x^5+x^3-1),x, algorit hm="giac")
Output:
integrate((2*x^5 + x^3 - 2)*(2*x^5 + 3)*(x^5 - 1)^(2/3)/((x^5 + x^3 - 1)*x ^6), x)
Timed out. \[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx=\int \frac {{\left (x^5-1\right )}^{2/3}\,\left (2\,x^5+3\right )\,\left (2\,x^5+x^3-2\right )}{x^6\,\left (x^5+x^3-1\right )} \,d x \] Input:
int(((x^5 - 1)^(2/3)*(2*x^5 + 3)*(x^3 + 2*x^5 - 2))/(x^6*(x^3 + x^5 - 1)), x)
Output:
int(((x^5 - 1)^(2/3)*(2*x^5 + 3)*(x^3 + 2*x^5 - 2))/(x^6*(x^3 + x^5 - 1)), x)
\[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right ) \left (-2+x^3+2 x^5\right )}{x^6 \left (-1+x^3+x^5\right )} \, dx=\frac {12 \left (x^{5}-1\right )^{\frac {2}{3}} x^{5}-15 \left (x^{5}-1\right )^{\frac {2}{3}} x^{3}-30 \left (x^{5}-1\right )^{\frac {2}{3}} x -12 \left (x^{5}-1\right )^{\frac {2}{3}}-120 \left (\int \frac {\left (x^{5}-1\right )^{\frac {2}{3}}}{x^{15}+x^{13}-2 x^{10}-x^{8}+x^{5}}d x \right ) x^{5}+120 \left (\int \frac {\left (x^{5}-1\right )^{\frac {2}{3}}}{x^{12}+x^{10}-2 x^{7}-x^{5}+x^{2}}d x \right ) x^{5}+170 \left (\int \frac {\left (x^{5}-1\right )^{\frac {2}{3}}}{x^{10}+x^{8}-2 x^{5}-x^{3}+1}d x \right ) x^{5}-20 \left (\int \frac {\left (x^{5}-1\right )^{\frac {2}{3}} x^{3}}{x^{10}+x^{8}-2 x^{5}-x^{3}+1}d x \right ) x^{5}}{10 x^{5}} \] Input:
int((x^5-1)^(2/3)*(2*x^5+3)*(2*x^5+x^3-2)/x^6/(x^5+x^3-1),x)
Output:
(12*(x**5 - 1)**(2/3)*x**5 - 15*(x**5 - 1)**(2/3)*x**3 - 30*(x**5 - 1)**(2 /3)*x - 12*(x**5 - 1)**(2/3) - 120*int((x**5 - 1)**(2/3)/(x**15 + x**13 - 2*x**10 - x**8 + x**5),x)*x**5 + 120*int((x**5 - 1)**(2/3)/(x**12 + x**10 - 2*x**7 - x**5 + x**2),x)*x**5 + 170*int((x**5 - 1)**(2/3)/(x**10 + x**8 - 2*x**5 - x**3 + 1),x)*x**5 - 20*int(((x**5 - 1)**(2/3)*x**3)/(x**10 + x* *8 - 2*x**5 - x**3 + 1),x)*x**5)/(10*x**5)