Integrand size = 32, antiderivative size = 90 \[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right )}{x^3 \left (-1-x^3+x^5\right )} \, dx=\frac {3 \left (-1+x^5\right )^{2/3}}{2 x^2}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^5}}\right )+\log \left (-x+\sqrt [3]{-1+x^5}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^5}+\left (-1+x^5\right )^{2/3}\right ) \]
3/2*(x^5-1)^(2/3)/x^2-3^(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 = 1.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right )}{x^3 \left (-1-x^3+x^5\right )} \, dx=\frac {3 \left (-1+x^5\right )^{2/3}}{2 x^2}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^5}}\right )+\log \left (-x+\sqrt [3]{-1+x^5}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^5}+\left (-1+x^5\right )^{2/3}\right ) \]
(3*(-1 + x^5)^(2/3))/(2*x^2) - 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.
| \(\displaystyle \int \frac {\left (x^5-1\right )^{2/3} \left (2 x^5+3\right )}{x^3 \left (x^5-x^3-1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (5 x^2-3\right ) \left (x^5-1\right )^{2/3}}{x^5-x^3-1}-\frac {3 \left (x^5-1\right )^{2/3}}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \int \frac {\left (x^5-1\right )^{2/3}}{x^5-x^3-1}dx+5 \int \frac {x^2 \left (x^5-1\right )^{2/3}}{x^5-x^3-1}dx+\frac {3 \left (x^5-1\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {2}{5},\frac {3}{5},x^5\right )}{2 x^2 \left (1-x^5\right )^{2/3}}\) |
3.13.44.3.1 Defintions of rubi rules used
Time = 15.91 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.06
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{5}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}+2 \ln \left (\frac {-x +\left (x^{5}-1\right )^{\frac {1}{3}}}{x}\right ) x^{2}-\ln \left (\frac {x^{2}+x \left (x^{5}-1\right )^{\frac {1}{3}}+\left (x^{5}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+3 \left (x^{5}-1\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(95\) |
| risch | \(\frac {3 \left (x^{5}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-x^{5}-3 \left (x^{5}-1\right )^{\frac {2}{3}} x -3 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{x^{5}-x^{3}-1}\right )-\ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-2 x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 \left (x^{5}-1\right )^{\frac {2}{3}} x -3 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2}{x^{5}-x^{3}-1}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-2 x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 \left (x^{5}-1\right )^{\frac {2}{3}} x -3 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2}{x^{5}-x^{3}-1}\right )\) | \(288\) |
| trager | \(\frac {3 \left (x^{5}-1\right )^{\frac {2}{3}}}{2 x^{2}}-12 \ln \left (\frac {19600128 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{5}-3828888 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{5}-75950496 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-807395 x^{5}+17181936 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}} x -397620 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-17651292 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+33135 \left (x^{5}-1\right )^{\frac {2}{3}} x +1398693 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-1015755 x^{3}-19600128 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+3828888 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+807395}{x^{5}-x^{3}-1}\right ) \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+\ln \left (-\frac {25830425627884197504 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{5}+44491798914989188740 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{5}-100092899308051265328 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}+13811764079612376111 x^{5}-205927896932357002488 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}} x +284838003526717488132 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-129590444982696930036 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}-23736500293893124011 \left (x^{5}-1\right )^{\frac {2}{3}} x +6575842216196707137 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}+2833182375305102792 x^{3}-25830425627884197504 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-44491798914989188740 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-13811764079612376111}{x^{5}-x^{3}-1}\right )-\ln \left (\frac {19600128 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{5}-3828888 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{5}-75950496 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-807395 x^{5}+17181936 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {2}{3}} x -397620 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-17651292 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+33135 \left (x^{5}-1\right )^{\frac {2}{3}} x +1398693 \left (x^{5}-1\right )^{\frac {1}{3}} x^{2}-1015755 x^{3}-19600128 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+3828888 \operatorname {RootOf}\left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+807395}{x^{5}-x^{3}-1}\right )\) | \(594\) |
1/2*(2*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^5-1)^(1/3)))*x^2+2*ln((-x+(x^5 -1)^(1/3))/x)*x^2-ln((x^2+x*(x^5-1)^(1/3)+(x^5-1)^(2/3))/x^2)*x^2+3*(x^5-1 )^(2/3))/x^2
Time = 3.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.50 \[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right )}{x^3 \left (-1-x^3+x^5\right )} \, dx=-\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {67616276 \, \sqrt {3} {\left (x^{5} - 1\right )}^{\frac {1}{3}} x^{2} + 10249526 \, \sqrt {3} {\left (x^{5} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (1423013 \, x^{5} + 37509888 \, x^{3} - 1423013\right )}}{300763 \, x^{5} - 86350888 \, x^{3} - 300763}\right ) - x^{2} \log \left (\frac {x^{5} - x^{3} + 3 \, {\left (x^{5} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{5} - 1\right )}^{\frac {2}{3}} x - 1}{x^{5} - x^{3} - 1}\right ) - 3 \, {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \]
-1/2*(2*sqrt(3)*x^2*arctan((67616276*sqrt(3)*(x^5 - 1)^(1/3)*x^2 + 1024952 6*sqrt(3)*(x^5 - 1)^(2/3)*x + sqrt(3)*(1423013*x^5 + 37509888*x^3 - 142301 3))/(300763*x^5 - 86350888*x^3 - 300763)) - x^2*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*(x^5 - 1)^(2/ 3))/x^2
\[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right )}{x^3 \left (-1-x^3+x^5\right )} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )\right )^{\frac {2}{3}} \cdot \left (2 x^{5} + 3\right )}{x^{3} \left (x^{5} - x^{3} - 1\right )}\, dx \]
Integral(((x - 1)*(x**4 + x**3 + x**2 + x + 1))**(2/3)*(2*x**5 + 3)/(x**3* (x**5 - x**3 - 1)), x)
\[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right )}{x^3 \left (-1-x^3+x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{5} + 3\right )} {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{{\left (x^{5} - x^{3} - 1\right )} x^{3}} \,d x } \]
\[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right )}{x^3 \left (-1-x^3+x^5\right )} \, dx=\int { \frac {{\left (2 \, x^{5} + 3\right )} {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{{\left (x^{5} - x^{3} - 1\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (-1+x^5\right )^{2/3} \left (3+2 x^5\right )}{x^3 \left (-1-x^3+x^5\right )} \, dx=\int -\frac {{\left (x^5-1\right )}^{2/3}\,\left (2\,x^5+3\right )}{x^3\,\left (-x^5+x^3+1\right )} \,d x \]