Integrand size = 23, antiderivative size = 16 \[ \int \frac {\sqrt [3]{x+x^4} \left (-2-x^3+x^6\right )}{x^6} \, dx=\frac {3 \left (x+x^4\right )^{7/3}}{7 x^7} \]
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Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(16)=32\).
Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.94, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2077, 2029, 2057, 371, 2045, 2050} \[ \int \frac {\sqrt [3]{x+x^4} \left (-2-x^3+x^6\right )}{x^6} \, dx=\frac {3}{7} \sqrt [3]{x^4+x} x+\frac {3 \sqrt [3]{x^4+x}}{7 x^5}+\frac {6 \sqrt [3]{x^4+x}}{7 x^2} \]
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Rule 371
Rule 2029
Rule 2045
Rule 2050
Rule 2057
Rule 2077
Rubi steps \begin{align*} \text {integral}& = \int \left (\sqrt [3]{x+x^4}-\frac {2 \sqrt [3]{x+x^4}}{x^6}-\frac {\sqrt [3]{x+x^4}}{x^3}\right ) \, dx \\ & = -\left (2 \int \frac {\sqrt [3]{x+x^4}}{x^6} \, dx\right )+\int \sqrt [3]{x+x^4} \, dx-\int \frac {\sqrt [3]{x+x^4}}{x^3} \, dx \\ & = \frac {3 \sqrt [3]{x+x^4}}{7 x^5}+\frac {3 \sqrt [3]{x+x^4}}{5 x^2}+\frac {3}{7} x \sqrt [3]{x+x^4}-\frac {3}{7} \int \frac {1}{x^2 \left (x+x^4\right )^{2/3}} \, dx+\frac {3}{7} \int \frac {x}{\left (x+x^4\right )^{2/3}} \, dx-\frac {3}{5} \int \frac {x}{\left (x+x^4\right )^{2/3}} \, dx \\ & = \frac {3 \sqrt [3]{x+x^4}}{7 x^5}+\frac {6 \sqrt [3]{x+x^4}}{7 x^2}+\frac {3}{7} x \sqrt [3]{x+x^4}+\frac {6}{35} \int \frac {x}{\left (x+x^4\right )^{2/3}} \, dx+\frac {\left (3 x^{2/3} \left (1+x^3\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^3\right )^{2/3}} \, dx}{7 \left (x+x^4\right )^{2/3}}-\frac {\left (3 x^{2/3} \left (1+x^3\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^3\right )^{2/3}} \, dx}{5 \left (x+x^4\right )^{2/3}} \\ & = \frac {3 \sqrt [3]{x+x^4}}{7 x^5}+\frac {6 \sqrt [3]{x+x^4}}{7 x^2}+\frac {3}{7} x \sqrt [3]{x+x^4}-\frac {9 x^2 \left (1+x^3\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {4}{9},\frac {2}{3},\frac {13}{9},-x^3\right )}{70 \left (x+x^4\right )^{2/3}}+\frac {\left (6 x^{2/3} \left (1+x^3\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^3\right )^{2/3}} \, dx}{35 \left (x+x^4\right )^{2/3}} \\ & = \frac {3 \sqrt [3]{x+x^4}}{7 x^5}+\frac {6 \sqrt [3]{x+x^4}}{7 x^2}+\frac {3}{7} x \sqrt [3]{x+x^4} \\ \end{align*}
Time = 5.91 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt [3]{x+x^4} \left (-2-x^3+x^6\right )}{x^6} \, dx=\frac {3 \left (1+x^3\right )^2 \sqrt [3]{x+x^4}}{7 x^5} \]
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Time = 0.88 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.25
method | result | size |
pseudoelliptic | \(\frac {3 \left (x^{3}+1\right )^{2} \left (x^{4}+x \right )^{\frac {1}{3}}}{7 x^{5}}\) | \(20\) |
trager | \(\frac {3 \left (x^{6}+2 x^{3}+1\right ) \left (x^{4}+x \right )^{\frac {1}{3}}}{7 x^{5}}\) | \(23\) |
gosper | \(\frac {3 \left (x^{3}+1\right ) \left (x^{4}+x \right )^{\frac {1}{3}} \left (1+x \right ) \left (x^{2}-x +1\right )}{7 x^{5}}\) | \(29\) |
risch | \(\frac {3 {\left (x \left (x^{3}+1\right )\right )}^{\frac {1}{3}} \left (x^{9}+3 x^{6}+3 x^{3}+1\right )}{7 x^{5} \left (x^{3}+1\right )}\) | \(37\) |
meijerg | \(\frac {3 x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {4}{9}\right ], \left [\frac {13}{9}\right ], -x^{3}\right )}{4}+\frac {3 \operatorname {hypergeom}\left (\left [-\frac {5}{9}, -\frac {1}{3}\right ], \left [\frac {4}{9}\right ], -x^{3}\right )}{5 x^{\frac {5}{3}}}+\frac {3 \operatorname {hypergeom}\left (\left [-\frac {14}{9}, -\frac {1}{3}\right ], \left [-\frac {5}{9}\right ], -x^{3}\right )}{7 x^{\frac {14}{3}}}\) | \(50\) |
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Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt [3]{x+x^4} \left (-2-x^3+x^6\right )}{x^6} \, dx=\frac {3 \, {\left (x^{6} + 2 \, x^{3} + 1\right )} {\left (x^{4} + x\right )}^{\frac {1}{3}}}{7 \, x^{5}} \]
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\[ \int \frac {\sqrt [3]{x+x^4} \left (-2-x^3+x^6\right )}{x^6} \, dx=\int \frac {\sqrt [3]{x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 1\right ) \left (x^{3} - 2\right ) \left (x^{2} - x + 1\right )}{x^{6}}\, dx \]
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\[ \int \frac {\sqrt [3]{x+x^4} \left (-2-x^3+x^6\right )}{x^6} \, dx=\int { \frac {{\left (x^{6} - x^{3} - 2\right )} {\left (x^{4} + x\right )}^{\frac {1}{3}}}{x^{6}} \,d x } \]
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\[ \int \frac {\sqrt [3]{x+x^4} \left (-2-x^3+x^6\right )}{x^6} \, dx=\int { \frac {{\left (x^{6} - x^{3} - 2\right )} {\left (x^{4} + x\right )}^{\frac {1}{3}}}{x^{6}} \,d x } \]
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Time = 5.45 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt [3]{x+x^4} \left (-2-x^3+x^6\right )}{x^6} \, dx=\frac {3\,{\left (x^3+1\right )}^2\,{\left (x^4+x\right )}^{1/3}}{7\,x^5} \]
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