Integrand size = 45, antiderivative size = 150 \[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-2 x+2 x^3-x^8}}\right )}{2^{2/3}}-\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{-2 x+2 x^3-x^8}\right )}{2^{2/3}}+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-2 x+2 x^3-x^8}+\sqrt [3]{2} \left (-2 x+2 x^3-x^8\right )^{2/3}\right )}{2\ 2^{2/3}} \]
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\[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx=\int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{-2 x+2 x^3-x^8} \int \frac {\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7} \left (-4+5 x^7\right )}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}} \\ & = -\frac {\sqrt [3]{-2 x+2 x^3-x^8} \int \frac {\sqrt [3]{x} \left (-4+5 x^7\right )}{\left (-2+2 x^2-x^7\right )^{2/3} \left (2+x^7\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}} \\ & = -\frac {\sqrt [3]{-2 x+2 x^3-x^8} \int \left (\frac {5 \sqrt [3]{x}}{\left (-2+2 x^2-x^7\right )^{2/3}}-\frac {14 \sqrt [3]{x}}{\left (-2+2 x^2-x^7\right )^{2/3} \left (2+x^7\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}} \\ & = -\frac {\left (5 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}+\frac {\left (14 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-2+2 x^2-x^7\right )^{2/3} \left (2+x^7\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}} \\ & = \frac {\left (14 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \left (-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}-x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}+\sqrt [7]{-1} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}-(-1)^{2/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}+(-1)^{3/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}-(-1)^{4/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}+(-1)^{5/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}-\frac {\sqrt [3]{x}}{7\ 2^{6/7} \left (-\sqrt [7]{2}-(-1)^{6/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}} \\ & = -\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}-x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}+\sqrt [7]{-1} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}-(-1)^{2/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}+(-1)^{3/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}-(-1)^{4/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}+(-1)^{5/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (\sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [7]{2}-(-1)^{6/7} x\right ) \left (-2+2 x^2-x^7\right )^{2/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}} \\ & = -\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}-x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}+\sqrt [7]{-1} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}-(-1)^{2/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}+(-1)^{3/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}-(-1)^{4/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}+(-1)^{5/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [7]{2}-(-1)^{6/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}} \\ & = -\frac {\left (15 \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-2+2 x^6-x^{21}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (-\frac {1}{\left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {\sqrt [7]{2}}{\left (-\sqrt [7]{2}-x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{6/7}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {(-1)^{6/7} \sqrt [7]{2}}{\left (-\sqrt [7]{2}+\sqrt [7]{-1} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (\frac {(-1)^{5/7}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {(-1)^{5/7} \sqrt [7]{2}}{\left (-\sqrt [7]{2}-(-1)^{2/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{4/7}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {(-1)^{4/7} \sqrt [7]{2}}{\left (-\sqrt [7]{2}+(-1)^{3/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (\frac {(-1)^{3/7}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {(-1)^{3/7} \sqrt [7]{2}}{\left (-\sqrt [7]{2}-(-1)^{4/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{2/7}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}-\frac {(-1)^{2/7} \sqrt [7]{2}}{\left (-\sqrt [7]{2}+(-1)^{5/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}}-\frac {\left (3 \sqrt [7]{2} \sqrt [3]{-2 x+2 x^3-x^8}\right ) \text {Subst}\left (\int \left (\frac {\sqrt [7]{-1}}{\left (-2+2 x^6-x^{21}\right )^{2/3}}+\frac {\sqrt [7]{-2}}{\left (-\sqrt [7]{2}-(-1)^{6/7} x^3\right ) \left (-2+2 x^6-x^{21}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-2+2 x^2-x^7}} \\ & = \text {Too large to display} \\ \end{align*}
Time = 3.52 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.20 \[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx=\frac {x^{2/3} \left (2-2 x^2+x^7\right )^{2/3} \left (-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2^{2/3} \sqrt [3]{2-2 x^2+x^7}}\right )-2 \log \left (2 x^{2/3}+2^{2/3} \sqrt [3]{2-2 x^2+x^7}\right )+\log \left (-2 x^{4/3}+2^{2/3} x^{2/3} \sqrt [3]{2-2 x^2+x^7}-\sqrt [3]{2} \left (2-2 x^2+x^7\right )^{2/3}\right )\right )}{2\ 2^{2/3} \left (-x \left (2-2 x^2+x^7\right )\right )^{2/3}} \]
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Time = 229.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(\frac {2^{\frac {1}{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} {\left (-x \left (x^{7}-2 x^{2}+2\right )\right )}^{\frac {1}{3}}+x \right )}{3 x}\right )+\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} {\left (-x \left (x^{7}-2 x^{2}+2\right )\right )}^{\frac {1}{3}} x +{\left (-x \left (x^{7}-2 x^{2}+2\right )\right )}^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-2^{\frac {1}{3}} x +{\left (-x \left (x^{7}-2 x^{2}+2\right )\right )}^{\frac {1}{3}}}{x}\right )\right )}{4}\) | \(119\) |
trager | \(\text {Expression too large to display}\) | \(1000\) |
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Leaf count of result is larger than twice the leaf count of optimal. 390 vs. \(2 (122) = 244\).
Time = 5.09 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.60 \[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx=\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (-\frac {4^{\frac {1}{6}} \sqrt {3} {\left (6 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{15} - 18 \, x^{10} + 4 \, x^{8} + 36 \, x^{5} - 36 \, x^{3} + 4 \, x\right )} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}} - 12 \, \left (-1\right )^{\frac {1}{3}} {\left (x^{14} - 6 \, x^{9} + 4 \, x^{7} - 12 \, x^{2} + 4\right )} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} {\left (x^{21} - 36 \, x^{16} + 6 \, x^{14} + 180 \, x^{11} - 144 \, x^{9} + 12 \, x^{7} - 216 \, x^{6} + 360 \, x^{4} - 144 \, x^{2} + 8\right )}\right )}}{6 \, {\left (x^{21} + 6 \, x^{14} - 108 \, x^{11} + 12 \, x^{7} + 216 \, x^{6} - 216 \, x^{4} + 8\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {2}{3}} {\left (x^{7} - 6 \, x^{2} + 2\right )} + 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{14} - 18 \, x^{9} + 4 \, x^{7} + 36 \, x^{4} - 36 \, x^{2} + 4\right )} + 24 \, {\left (x^{8} - 3 \, x^{3} + 2 \, x\right )} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}}}{x^{14} + 4 \, x^{7} + 4}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}} x + 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{7} + 2\right )} + 6 \, {\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {2}{3}}}{x^{7} + 2}\right ) \]
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Timed out. \[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx=\int { \frac {{\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} - 4\right )}}{{\left (x^{7} - 2 \, x^{2} + 2\right )} {\left (x^{7} + 2\right )}} \,d x } \]
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\[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx=\int { \frac {{\left (-x^{8} + 2 \, x^{3} - 2 \, x\right )}^{\frac {1}{3}} {\left (5 \, x^{7} - 4\right )}}{{\left (x^{7} - 2 \, x^{2} + 2\right )} {\left (x^{7} + 2\right )}} \,d x } \]
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Timed out. \[ \int \frac {\left (-4+5 x^7\right ) \sqrt [3]{-2 x+2 x^3-x^8}}{\left (2+x^7\right ) \left (2-2 x^2+x^7\right )} \, dx=\int \frac {\left (5\,x^7-4\right )\,{\left (-x^8+2\,x^3-2\,x\right )}^{1/3}}{\left (x^7+2\right )\,\left (x^7-2\,x^2+2\right )} \,d x \]
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