Integrand size = 62, antiderivative size = 124 \[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+2 x^3-x^4-x^7}}\right )-\log \left (-x+\sqrt [3]{x+2 x^3-x^4-x^7}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{x+2 x^3-x^4-x^7}+\left (x+2 x^3-x^4-x^7\right )^{2/3}\right ) \]
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\[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=\int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{x+2 x^3-x^4-x^7} \int \frac {\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6} \left (2+x^3+4 x^6\right )}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}} \\ & = -\frac {\sqrt [3]{x+2 x^3-x^4-x^7} \int \frac {\sqrt [3]{x} \left (2+x^3+4 x^6\right )}{\left (1+2 x^2-x^3-x^6\right )^{2/3} \left (-1-x^2+x^3+x^6\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}} \\ & = -\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3 \left (2+x^9+4 x^{18}\right )}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3} \left (-1-x^6+x^9+x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}} \\ & = -\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \left (\frac {4 x^3}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3}}+\frac {x^3 \left (6+4 x^6-3 x^9\right )}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3} \left (-1-x^6+x^9+x^{18}\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}} \\ & = -\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3 \left (6+4 x^6-3 x^9\right )}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3} \left (-1-x^6+x^9+x^{18}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (12 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}} \\ & = -\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \left (\frac {1}{3 (-1+x) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}+\frac {-2-x}{3 \left (1+x+x^2\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}+\frac {1-4 x^3-2 x^6-5 x^9-x^{12}}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (12 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}} \\ & = -\frac {\sqrt [3]{x+2 x^3-x^4-x^7} \text {Subst}\left (\int \frac {1}{(-1+x) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\sqrt [3]{x+2 x^3-x^4-x^7} \text {Subst}\left (\int \frac {-2-x}{\left (1+x+x^2\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {1-4 x^3-2 x^6-5 x^9-x^{12}}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (12 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}} \\ & = -\frac {\sqrt [3]{x+2 x^3-x^4-x^7} \text {Subst}\left (\int \frac {1}{(-1+x) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\sqrt [3]{x+2 x^3-x^4-x^7} \text {Subst}\left (\int \left (\frac {-1+i \sqrt {3}}{\left (1-i \sqrt {3}+2 x\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}+\frac {-1-i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \left (\frac {1}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}-\frac {4 x^3}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}-\frac {2 x^6}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}-\frac {5 x^9}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}-\frac {x^{12}}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (12 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}} \\ & = -\frac {\sqrt [3]{x+2 x^3-x^4-x^7} \text {Subst}\left (\int \frac {1}{(-1+x) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {1}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}+\frac {\left (3 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^{12}}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}+\frac {\left (6 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^6}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (12 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}+\frac {\left (12 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}+\frac {\left (15 \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {x^9}{\left (1+x^3+2 x^6+x^9+x^{12}+x^{15}\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}}-\frac {\left (\left (-1+i \sqrt {3}\right ) \sqrt [3]{x+2 x^3-x^4-x^7}\right ) \text {Subst}\left (\int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \left (1+2 x^6-x^9-x^{18}\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{1+2 x^2-x^3-x^6}} \\ \end{align*}
Time = 1.27 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.36 \[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=-\frac {x^{2/3} \left (-1-2 x^2+x^3+x^6\right )^{2/3} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}-2 \sqrt [3]{-1-2 x^2+x^3+x^6}}\right )+2 \log \left (x^{2/3}+\sqrt [3]{-1-2 x^2+x^3+x^6}\right )-\log \left (x^{4/3}-x^{2/3} \sqrt [3]{-1-2 x^2+x^3+x^6}+\left (-1-2 x^2+x^3+x^6\right )^{2/3}\right )\right )}{2 \left (-x \left (-1-2 x^2+x^3+x^6\right )\right )^{2/3}} \]
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Time = 15.78 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.93
method | result | size |
pseudoelliptic | \(\frac {\ln \left (\frac {{\left (-x \left (x^{6}+x^{3}-2 x^{2}-1\right )\right )}^{\frac {2}{3}}+{\left (-x \left (x^{6}+x^{3}-2 x^{2}-1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (2 {\left (-x \left (x^{6}+x^{3}-2 x^{2}-1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )-\ln \left (\frac {{\left (-x \left (x^{6}+x^{3}-2 x^{2}-1\right )\right )}^{\frac {1}{3}}-x}{x}\right )\) | \(115\) |
trager | \(\text {Expression too large to display}\) | \(536\) |
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Time = 1.65 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.40 \[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=\sqrt {3} \arctan \left (-\frac {70 \, \sqrt {3} {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} x - \sqrt {3} {\left (32 \, x^{6} + 32 \, x^{3} - 39 \, x^{2} - 32\right )} - 56 \, \sqrt {3} {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {2}{3}}}{64 \, x^{6} + 64 \, x^{3} - 253 \, x^{2} - 64}\right ) - \frac {1}{2} \, \log \left (\frac {x^{6} + x^{3} - x^{2} - 3 \, {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} x + 3 \, {\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {2}{3}} - 1}{x^{6} + x^{3} - x^{2} - 1}\right ) \]
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\[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=\int \frac {\sqrt [3]{- x \left (x^{6} + x^{3} - 2 x^{2} - 1\right )} \left (4 x^{6} + x^{3} + 2\right )}{\left (x - 1\right ) \left (x^{6} + x^{3} - 2 x^{2} - 1\right ) \left (x^{5} + x^{4} + x^{3} + 2 x^{2} + x + 1\right )}\, dx \]
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\[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=\int { \frac {{\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} {\left (4 \, x^{6} + x^{3} + 2\right )}}{{\left (x^{6} + x^{3} - x^{2} - 1\right )} {\left (x^{6} + x^{3} - 2 \, x^{2} - 1\right )}} \,d x } \]
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\[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=\int { \frac {{\left (-x^{7} - x^{4} + 2 \, x^{3} + x\right )}^{\frac {1}{3}} {\left (4 \, x^{6} + x^{3} + 2\right )}}{{\left (x^{6} + x^{3} - x^{2} - 1\right )} {\left (x^{6} + x^{3} - 2 \, x^{2} - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {\left (2+x^3+4 x^6\right ) \sqrt [3]{x+2 x^3-x^4-x^7}}{\left (-1-2 x^2+x^3+x^6\right ) \left (-1-x^2+x^3+x^6\right )} \, dx=\int \frac {\left (4\,x^6+x^3+2\right )\,{\left (-x^7-x^4+2\,x^3+x\right )}^{1/3}}{\left (-x^6-x^3+2\,x^2+1\right )\,\left (-x^6-x^3+x^2+1\right )} \,d x \]
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