Integrand size = 50, antiderivative size = 26 \[ \int \frac {\left (-1+x^3-x^5-2 x^7\right )^{2/3} \left (1-x^3+x^5+2 x^7\right ) \left (-3+2 x^5+8 x^7\right )}{x^9} \, dx=\frac {3 \left (-1+x^3-x^5-2 x^7\right )^{8/3}}{8 x^8} \]
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Time = 0.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {6820, 1604} \[ \int \frac {\left (-1+x^3-x^5-2 x^7\right )^{2/3} \left (1-x^3+x^5+2 x^7\right ) \left (-3+2 x^5+8 x^7\right )}{x^9} \, dx=\frac {3 \left (-2 x^7-x^5+x^3-1\right )^{8/3}}{8 x^8} \]
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Rule 1604
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (3-2 x^5-8 x^7\right ) \left (-1+x^3-x^5-2 x^7\right )^{5/3}}{x^9} \, dx \\ & = \frac {3 \left (-1+x^3-x^5-2 x^7\right )^{8/3}}{8 x^8} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3-x^5-2 x^7\right )^{2/3} \left (1-x^3+x^5+2 x^7\right ) \left (-3+2 x^5+8 x^7\right )}{x^9} \, dx=\frac {3 \left (-1+x^3-x^5-2 x^7\right )^{8/3}}{8 x^8} \]
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Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
gosper | \(\frac {3 \left (-2 x^{7}-x^{5}+x^{3}-1\right )^{\frac {8}{3}}}{8 x^{8}}\) | \(23\) |
pseudoelliptic | \(\frac {3 \left (-2 x^{7}-x^{5}+x^{3}-1\right )^{\frac {8}{3}}}{8 x^{8}}\) | \(23\) |
trager | \(\frac {3 \left (4 x^{14}+4 x^{12}-3 x^{10}-2 x^{8}+4 x^{7}+x^{6}+2 x^{5}-2 x^{3}+1\right ) \left (-2 x^{7}-x^{5}+x^{3}-1\right )^{\frac {2}{3}}}{8 x^{8}}\) | \(63\) |
risch | \(-\frac {3 \left (8 x^{21}+12 x^{19}-6 x^{17}-11 x^{15}+12 x^{14}+3 x^{13}+12 x^{12}+3 x^{11}-9 x^{10}-x^{9}-6 x^{8}+6 x^{7}+3 x^{6}+3 x^{5}-3 x^{3}+1\right )}{8 x^{8} \left (-2 x^{7}-x^{5}+x^{3}-1\right )^{\frac {1}{3}}}\) | \(100\) |
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (22) = 44\).
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.38 \[ \int \frac {\left (-1+x^3-x^5-2 x^7\right )^{2/3} \left (1-x^3+x^5+2 x^7\right ) \left (-3+2 x^5+8 x^7\right )}{x^9} \, dx=\frac {3 \, {\left (4 \, x^{14} + 4 \, x^{12} - 3 \, x^{10} - 2 \, x^{8} + 4 \, x^{7} + x^{6} + 2 \, x^{5} - 2 \, x^{3} + 1\right )} {\left (-2 \, x^{7} - x^{5} + x^{3} - 1\right )}^{\frac {2}{3}}}{8 \, x^{8}} \]
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\[ \int \frac {\left (-1+x^3-x^5-2 x^7\right )^{2/3} \left (1-x^3+x^5+2 x^7\right ) \left (-3+2 x^5+8 x^7\right )}{x^9} \, dx=\int \frac {\left (8 x^{7} + 2 x^{5} - 3\right ) \left (- 2 x^{7} - x^{5} + x^{3} - 1\right )^{\frac {2}{3}} \cdot \left (2 x^{7} + x^{5} - x^{3} + 1\right )}{x^{9}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (22) = 44\).
Time = 0.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.38 \[ \int \frac {\left (-1+x^3-x^5-2 x^7\right )^{2/3} \left (1-x^3+x^5+2 x^7\right ) \left (-3+2 x^5+8 x^7\right )}{x^9} \, dx=\frac {3 \, {\left (4 \, x^{14} + 4 \, x^{12} - 3 \, x^{10} - 2 \, x^{8} + 4 \, x^{7} + x^{6} + 2 \, x^{5} - 2 \, x^{3} + 1\right )} {\left (-2 \, x^{7} - x^{5} + x^{3} - 1\right )}^{\frac {2}{3}}}{8 \, x^{8}} \]
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\[ \int \frac {\left (-1+x^3-x^5-2 x^7\right )^{2/3} \left (1-x^3+x^5+2 x^7\right ) \left (-3+2 x^5+8 x^7\right )}{x^9} \, dx=\int { \frac {{\left (8 \, x^{7} + 2 \, x^{5} - 3\right )} {\left (2 \, x^{7} + x^{5} - x^{3} + 1\right )} {\left (-2 \, x^{7} - x^{5} + x^{3} - 1\right )}^{\frac {2}{3}}}{x^{9}} \,d x } \]
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Time = 6.39 (sec) , antiderivative size = 147, normalized size of antiderivative = 5.65 \[ \int \frac {\left (-1+x^3-x^5-2 x^7\right )^{2/3} \left (1-x^3+x^5+2 x^7\right ) \left (-3+2 x^5+8 x^7\right )}{x^9} \, dx=\frac {3\,{\left (-2\,x^7-x^5+x^3-1\right )}^{2/3}}{2\,x}+\frac {3\,{\left (-2\,x^7-x^5+x^3-1\right )}^{2/3}}{8\,x^2}+\frac {3\,{\left (-2\,x^7-x^5+x^3-1\right )}^{2/3}}{4\,x^3}-\frac {3\,{\left (-2\,x^7-x^5+x^3-1\right )}^{2/3}}{4\,x^5}+\frac {3\,{\left (-2\,x^7-x^5+x^3-1\right )}^{2/3}}{8\,x^8}-\left (-\frac {3\,x^6}{2}-\frac {3\,x^4}{2}+\frac {9\,x^2}{8}+\frac {3}{4}\right )\,{\left (-2\,x^7-x^5+x^3-1\right )}^{2/3} \]
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