Integrand size = 18, antiderivative size = 16 \[ \int \frac {1+x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=-\frac {4 \left (x+x^4\right )^{7/4}}{21 x^7} \]
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Leaf count is larger than twice the leaf count of optimal. \(33\) vs. \(2(16)=32\).
Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.06, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2077, 2041, 2039} \[ \int \frac {1+x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=-\frac {4 \left (x^4+x\right )^{3/4}}{21 x^6}-\frac {4 \left (x^4+x\right )^{3/4}}{21 x^3} \]
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Rule 2039
Rule 2041
Rule 2077
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{x^6 \sqrt [4]{x+x^4}}+\frac {1}{x^3 \sqrt [4]{x+x^4}}\right ) \, dx \\ & = \int \frac {1}{x^6 \sqrt [4]{x+x^4}} \, dx+\int \frac {1}{x^3 \sqrt [4]{x+x^4}} \, dx \\ & = -\frac {4 \left (x+x^4\right )^{3/4}}{21 x^6}-\frac {4 \left (x+x^4\right )^{3/4}}{9 x^3}-\frac {4}{7} \int \frac {1}{x^3 \sqrt [4]{x+x^4}} \, dx \\ & = -\frac {4 \left (x+x^4\right )^{3/4}}{21 x^6}-\frac {4 \left (x+x^4\right )^{3/4}}{21 x^3} \\ \end{align*}
Time = 9.86 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.31 \[ \int \frac {1+x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=-\frac {4 \left (1+x^3\right ) \left (x+x^4\right )^{3/4}}{21 x^6} \]
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Time = 0.88 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12
method | result | size |
trager | \(-\frac {4 \left (x^{3}+1\right ) \left (x^{4}+x \right )^{\frac {3}{4}}}{21 x^{6}}\) | \(18\) |
pseudoelliptic | \(-\frac {4 \left (x^{3}+1\right ) \left (x^{4}+x \right )^{\frac {3}{4}}}{21 x^{6}}\) | \(18\) |
risch | \(-\frac {4 \left (x^{6}+2 x^{3}+1\right )}{21 x^{5} {\left (x \left (x^{3}+1\right )\right )}^{\frac {1}{4}}}\) | \(25\) |
gosper | \(-\frac {4 \left (1+x \right ) \left (x^{2}-x +1\right ) \left (x^{3}+1\right )}{21 x^{5} \left (x^{4}+x \right )^{\frac {1}{4}}}\) | \(29\) |
meijerg | \(-\frac {4 \left (1-\frac {4 x^{3}}{3}\right ) \left (x^{3}+1\right )^{\frac {3}{4}}}{21 x^{\frac {21}{4}}}-\frac {4 \left (x^{3}+1\right )^{\frac {3}{4}}}{9 x^{\frac {9}{4}}}\) | \(33\) |
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Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \frac {1+x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=-\frac {4 \, {\left (x^{4} + x\right )}^{\frac {3}{4}} {\left (x^{3} + 1\right )}}{21 \, x^{6}} \]
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\[ \int \frac {1+x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=\int \frac {\left (x + 1\right ) \left (x^{2} - x + 1\right )}{x^{6} \sqrt [4]{x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (12) = 24\).
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.62 \[ \int \frac {1+x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=-\frac {4 \, {\left (x^{4} + x\right )}}{9 \, {\left (x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}} x^{\frac {13}{4}}} + \frac {4 \, {\left (4 \, x^{7} + x^{4} - 3 \, x\right )}}{63 \, {\left (x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}} x^{\frac {25}{4}}} \]
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Time = 0.30 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.56 \[ \int \frac {1+x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=-\frac {4}{21} \, {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {7}{4}} \]
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Time = 5.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.69 \[ \int \frac {1+x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=-\frac {4\,{\left (x^4+x\right )}^{3/4}+4\,x^3\,{\left (x^4+x\right )}^{3/4}}{21\,x^6} \]
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