Integrand size = 20, antiderivative size = 23 \[ \int \frac {1+2 x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=-\frac {4 \left (3+10 x^3\right ) \left (x+x^4\right )^{3/4}}{63 x^6} \]
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Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2063, 2039} \[ \int \frac {1+2 x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=-\frac {4 \left (x^4+x\right )^{3/4}}{21 x^6}-\frac {40 \left (x^4+x\right )^{3/4}}{63 x^3} \]
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Rule 2039
Rule 2063
Rubi steps \begin{align*} \text {integral}& = -\frac {4 \left (x+x^4\right )^{3/4}}{21 x^6}+\frac {10}{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 {40 \left (x+x^4\right )^{3/4}}{63 x^3} \\ \end{align*}
Time = 10.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1+2 x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=-\frac {4 \left (3+10 x^3\right ) \left (x+x^4\right )^{3/4}}{63 x^6} \]
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Time = 0.84 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87
method | result | size |
trager | \(-\frac {4 \left (10 x^{3}+3\right ) \left (x^{4}+x \right )^{\frac {3}{4}}}{63 x^{6}}\) | \(20\) |
pseudoelliptic | \(-\frac {4 \left (10 x^{3}+3\right ) \left (x^{4}+x \right )^{\frac {3}{4}}}{63 x^{6}}\) | \(20\) |
risch | \(-\frac {4 \left (10 x^{6}+13 x^{3}+3\right )}{63 x^{5} {\left (x \left (x^{3}+1\right )\right )}^{\frac {1}{4}}}\) | \(27\) |
gosper | \(-\frac {4 \left (1+x \right ) \left (x^{2}-x +1\right ) \left (10 x^{3}+3\right )}{63 x^{5} \left (x^{4}+x \right )^{\frac {1}{4}}}\) | \(31\) |
meijerg | \(-\frac {4 \left (1-\frac {4 x^{3}}{3}\right ) \left (x^{3}+1\right )^{\frac {3}{4}}}{21 x^{\frac {21}{4}}}-\frac {8 \left (x^{3}+1\right )^{\frac {3}{4}}}{9 x^{\frac {9}{4}}}\) | \(33\) |
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Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1+2 x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=-\frac {4 \, {\left (x^{4} + x\right )}^{\frac {3}{4}} {\left (10 \, x^{3} + 3\right )}}{63 \, x^{6}} \]
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\[ \int \frac {1+2 x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=\int \frac {2 x^{3} + 1}{x^{6} \sqrt [4]{x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.52 \[ \int \frac {1+2 x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=-\frac {8 \, {\left (x^{4} + x\right )}}{9 \, {\left (x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}} x^{\frac {13}{4}}} + \frac {4 \, {\left (4 \, x^{7} + x^{4} - 3 \, x\right )}}{63 \, {\left (x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}} x^{\frac {25}{4}}} \]
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Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int \frac {1+2 x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=-\frac {4}{21} \, {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {7}{4}} - \frac {4}{9} \, {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {3}{4}} \]
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Time = 5.35 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {1+2 x^3}{x^6 \sqrt [4]{x+x^4}} \, dx=-\frac {12\,{\left (x^4+x\right )}^{3/4}+40\,x^3\,{\left (x^4+x\right )}^{3/4}}{63\,x^6} \]
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