Integrand size = 13, antiderivative size = 28 \[ \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx=-\frac {3 \left (x+x^3\right )^{2/3} \left (5-6 x^2+9 x^4\right )}{80 x^6} \]
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Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2041, 2039} \[ \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx=-\frac {3 \left (x^3+x\right )^{2/3}}{16 x^6}+\frac {9 \left (x^3+x\right )^{2/3}}{40 x^4}-\frac {27 \left (x^3+x\right )^{2/3}}{80 x^2} \]
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Rule 2039
Rule 2041
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \left (x+x^3\right )^{2/3}}{16 x^6}-\frac {3}{4} \int \frac {1}{x^4 \sqrt [3]{x+x^3}} \, dx \\ & = -\frac {3 \left (x+x^3\right )^{2/3}}{16 x^6}+\frac {9 \left (x+x^3\right )^{2/3}}{40 x^4}+\frac {9}{20} \int \frac {1}{x^2 \sqrt [3]{x+x^3}} \, dx \\ & = -\frac {3 \left (x+x^3\right )^{2/3}}{16 x^6}+\frac {9 \left (x+x^3\right )^{2/3}}{40 x^4}-\frac {27 \left (x+x^3\right )^{2/3}}{80 x^2} \\ \end{align*}
Time = 4.88 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx=-\frac {3 \left (x+x^3\right )^{2/3} \left (5-6 x^2+9 x^4\right )}{80 x^6} \]
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Time = 0.83 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
trager | \(-\frac {3 \left (x^{3}+x \right )^{\frac {2}{3}} \left (9 x^{4}-6 x^{2}+5\right )}{80 x^{6}}\) | \(25\) |
meijerg | \(-\frac {3 \left (\frac {9}{5} x^{4}-\frac {6}{5} x^{2}+1\right ) \left (x^{2}+1\right )^{\frac {2}{3}}}{16 x^{\frac {16}{3}}}\) | \(25\) |
pseudoelliptic | \(-\frac {3 {\left (\left (x^{2}+1\right ) x \right )}^{\frac {2}{3}} \left (9 x^{4}-6 x^{2}+5\right )}{80 x^{6}}\) | \(27\) |
gosper | \(-\frac {3 \left (x^{2}+1\right ) \left (9 x^{4}-6 x^{2}+5\right )}{80 x^{5} \left (x^{3}+x \right )^{\frac {1}{3}}}\) | \(30\) |
risch | \(-\frac {3 \left (9 x^{6}+3 x^{4}-x^{2}+5\right )}{80 x^{5} {\left (\left (x^{2}+1\right ) x \right )}^{\frac {1}{3}}}\) | \(32\) |
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Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx=-\frac {3 \, {\left (9 \, x^{4} - 6 \, x^{2} + 5\right )} {\left (x^{3} + x\right )}^{\frac {2}{3}}}{80 \, x^{6}} \]
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\[ \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx=\int \frac {1}{x^{6} \sqrt [3]{x \left (x^{2} + 1\right )}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx=-\frac {3 \, {\left (9 \, x^{7} + 3 \, x^{5} - x^{3} + 5 \, x\right )}}{80 \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} x^{\frac {19}{3}}} \]
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Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx=-\frac {3}{16} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {8}{3}} + \frac {3}{5} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {5}{3}} - \frac {3}{4} \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} \]
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Time = 5.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^6 \sqrt [3]{x+x^3}} \, dx=-\frac {3\,{\left (x^3+x\right )}^{2/3}\,\left (9\,x^4-6\,x^2+5\right )}{80\,x^6} \]
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