Integrand size = 34, antiderivative size = 100 \[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\frac {3 \sqrt [3]{1+2 x^7}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+2 x^7}}\right )-\log \left (x+\sqrt [3]{1+2 x^7}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+2 x^7}+\left (1+2 x^7\right )^{2/3}\right ) \]
3*(2*x^7+1)^(1/3)/x+3^(1/2)*arctan(3^(1/2)*x/(-x+2*(2*x^7+1)^(1/3)))-ln(x+ (2*x^7+1)^(1/3))+1/2*ln(x^2-x*(2*x^7+1)^(1/3)+(2*x^7+1)^(2/3))
Time = 9.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\frac {3 \sqrt [3]{1+2 x^7}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+2 x^7}}\right )-\log \left (x+\sqrt [3]{1+2 x^7}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+2 x^7}+\left (1+2 x^7\right )^{2/3}\right ) \]
(3*(1 + 2*x^7)^(1/3))/x + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + 2*x^7)^( 1/3))] - Log[x + (1 + 2*x^7)^(1/3)] + Log[x^2 - x*(1 + 2*x^7)^(1/3) + (1 + 2*x^7)^(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.
| \(\displaystyle \int \frac {\sqrt [3]{2 x^7+1} \left (8 x^7-3\right )}{x^2 \left (2 x^7+x^3+1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x \left (14 x^4+3\right ) \sqrt [3]{2 x^7+1}}{2 x^7+x^3+1}-\frac {3 \sqrt [3]{2 x^7+1}}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \int \frac {x \sqrt [3]{2 x^7+1}}{2 x^7+x^3+1}dx+14 \int \frac {x^5 \sqrt [3]{2 x^7+1}}{2 x^7+x^3+1}dx+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{7},\frac {6}{7},-2 x^7\right )}{x}\) |
3.14.99.3.1 Defintions of rubi rules used
Time = 20.93 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.15
| method | result | size |
| pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-2^{\frac {1}{3}} \left (8 x^{7}+4\right )^{\frac {1}{3}}+x \right )}{3 x}\right ) x -2 \ln \left (\frac {2^{\frac {2}{3}} x +\left (8 x^{7}+4\right )^{\frac {1}{3}}}{x}\right ) x +\ln \left (\frac {-2^{\frac {2}{3}} \left (8 x^{7}+4\right )^{\frac {1}{3}} x +2 \,2^{\frac {1}{3}} x^{2}+\left (8 x^{7}+4\right )^{\frac {2}{3}}}{x^{2}}\right ) x +3 \,2^{\frac {1}{3}} \left (8 x^{7}+4\right )^{\frac {1}{3}}}{2 x}\) | \(115\) |
| risch | \(\frac {3 \left (2 x^{7}+1\right )^{\frac {1}{3}}}{x}+\frac {\left (-\ln \left (-\frac {-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{14}-4 x^{14}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{10}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{10}-6 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x^{8}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-4 x^{7}+3 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {2}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-3 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{\left (2 x^{7}+1\right ) \left (2 x^{7}+x^{3}+1\right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{14}-4 x^{14}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{10}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x^{8}+2 x^{10}-6 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x^{8}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{7}-4 x^{7}+3 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {2}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x +x^{3}-3 \left (4 x^{14}+4 x^{7}+1\right )^{\frac {1}{3}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1}{\left (2 x^{7}+1\right ) \left (2 x^{7}+x^{3}+1\right )}\right )\right ) {\left (\left (2 x^{7}+1\right )^{2}\right )}^{\frac {1}{3}}}{\left (2 x^{7}+1\right )^{\frac {2}{3}}}\) | \(498\) |
| trager | \(\text {Expression too large to display}\) | \(620\) |
1/2*(2*3^(1/2)*arctan(1/3*3^(1/2)*(-2^(1/3)*(8*x^7+4)^(1/3)+x)/x)*x-2*ln(( 2^(2/3)*x+(8*x^7+4)^(1/3))/x)*x+ln((-2^(2/3)*(8*x^7+4)^(1/3)*x+2*2^(1/3)*x ^2+(8*x^7+4)^(2/3))/x^2)*x+3*2^(1/3)*(8*x^7+4)^(1/3))/x
Time = 8.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (\frac {8377128467638 \, \sqrt {3} {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}} x^{2} + 15171948325814 \, \sqrt {3} {\left (2 \, x^{7} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (2102123379894 \, x^{7} + 4448471619035 \, x^{3} + 1051061689947\right )}}{60468559237154 \, x^{7} - 5089335571601 \, x^{3} + 30234279618577}\right ) - x \log \left (\frac {2 \, x^{7} + x^{3} + 3 \, {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (2 \, x^{7} + 1\right )}^{\frac {2}{3}} x + 1}{2 \, x^{7} + x^{3} + 1}\right ) + 6 \, {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}}}{2 \, x} \]
1/2*(2*sqrt(3)*x*arctan((8377128467638*sqrt(3)*(2*x^7 + 1)^(1/3)*x^2 + 151 71948325814*sqrt(3)*(2*x^7 + 1)^(2/3)*x + sqrt(3)*(2102123379894*x^7 + 444 8471619035*x^3 + 1051061689947))/(60468559237154*x^7 - 5089335571601*x^3 + 30234279618577)) - x*log((2*x^7 + x^3 + 3*(2*x^7 + 1)^(1/3)*x^2 + 3*(2*x^ 7 + 1)^(2/3)*x + 1)/(2*x^7 + x^3 + 1)) + 6*(2*x^7 + 1)^(1/3))/x
\[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\int \frac {\sqrt [3]{2 x^{7} + 1} \cdot \left (8 x^{7} - 3\right )}{x^{2} \cdot \left (2 x^{7} + x^{3} + 1\right )}\, dx \]
\[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\int { \frac {{\left (8 \, x^{7} - 3\right )} {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (2 \, x^{7} + x^{3} + 1\right )} x^{2}} \,d x } \]
\[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\int { \frac {{\left (8 \, x^{7} - 3\right )} {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (2 \, x^{7} + x^{3} + 1\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\int \frac {{\left (2\,x^7+1\right )}^{1/3}\,\left (8\,x^7-3\right )}{x^2\,\left (2\,x^7+x^3+1\right )} \,d x \]