Integrand size = 34, antiderivative size = 100 \[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\frac {3 \sqrt [3]{1+2 x^7}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+2 x^7}}\right )-\log \left (x+\sqrt [3]{1+2 x^7}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+2 x^7}+\left (1+2 x^7\right )^{2/3}\right ) \]
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\[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 \sqrt [3]{1+2 x^7}}{x^2}+\frac {x \left (3+14 x^4\right ) \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7}\right ) \, dx \\ & = -\left (3 \int \frac {\sqrt [3]{1+2 x^7}}{x^2} \, dx\right )+\int \frac {x \left (3+14 x^4\right ) \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7} \, dx \\ & = \frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{7},\frac {6}{7},-2 x^7\right )}{x}+\int \left (\frac {3 x \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7}+\frac {14 x^5 \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7}\right ) \, dx \\ & = \frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{7},\frac {6}{7},-2 x^7\right )}{x}+3 \int \frac {x \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7} \, dx+14 \int \frac {x^5 \sqrt [3]{1+2 x^7}}{1+x^3+2 x^7} \, dx \\ \end{align*}
Time = 9.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\frac {3 \sqrt [3]{1+2 x^7}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+2 x^7}}\right )-\log \left (x+\sqrt [3]{1+2 x^7}\right )+\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+2 x^7}+\left (1+2 x^7\right )^{2/3}\right ) \]
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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 | \(\text {Expression too large to display}\) | \(620\) |
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Time = 8.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (\frac {8377128467638 \, \sqrt {3} {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}} x^{2} + 15171948325814 \, \sqrt {3} {\left (2 \, x^{7} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (2102123379894 \, x^{7} + 4448471619035 \, x^{3} + 1051061689947\right )}}{60468559237154 \, x^{7} - 5089335571601 \, x^{3} + 30234279618577}\right ) - x \log \left (\frac {2 \, x^{7} + x^{3} + 3 \, {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (2 \, x^{7} + 1\right )}^{\frac {2}{3}} x + 1}{2 \, x^{7} + x^{3} + 1}\right ) + 6 \, {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}}}{2 \, x} \]
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\[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\int \frac {\sqrt [3]{2 x^{7} + 1} \cdot \left (8 x^{7} - 3\right )}{x^{2} \cdot \left (2 x^{7} + x^{3} + 1\right )}\, dx \]
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\[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\int { \frac {{\left (8 \, x^{7} - 3\right )} {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (2 \, x^{7} + x^{3} + 1\right )} x^{2}} \,d x } \]
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\[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\int { \frac {{\left (8 \, x^{7} - 3\right )} {\left (2 \, x^{7} + 1\right )}^{\frac {1}{3}}}{{\left (2 \, x^{7} + x^{3} + 1\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{1+2 x^7} \left (-3+8 x^7\right )}{x^2 \left (1+x^3+2 x^7\right )} \, dx=\int \frac {{\left (2\,x^7+1\right )}^{1/3}\,\left (8\,x^7-3\right )}{x^2\,\left (2\,x^7+x^3+1\right )} \,d x \]
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