Integrand size = 13, antiderivative size = 28 \[ \int \frac {\sqrt [4]{x+x^4}}{x^8} \, dx=\frac {4 \sqrt [4]{x+x^4} \left (-5-x^3+4 x^6\right )}{135 x^7} \]
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Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2041, 2039} \[ \int \frac {\sqrt [4]{x+x^4}}{x^8} \, dx=\frac {16 \left (x^4+x\right )^{5/4}}{135 x^5}-\frac {4 \left (x^4+x\right )^{5/4}}{27 x^8} \]
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Rule 2039
Rule 2041
Rubi steps \begin{align*} \text {integral}& = -\frac {4 \left (x+x^4\right )^{5/4}}{27 x^8}-\frac {4}{9} \int \frac {\sqrt [4]{x+x^4}}{x^5} \, dx \\ & = -\frac {4 \left (x+x^4\right )^{5/4}}{27 x^8}+\frac {16 \left (x+x^4\right )^{5/4}}{135 x^5} \\ \end{align*}
Time = 10.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{x+x^4}}{x^8} \, dx=-\frac {4 \left (5-4 x^3\right ) \left (1+x^3\right ) \sqrt [4]{x+x^4}}{135 x^7} \]
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Time = 0.82 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
trager | \(\frac {4 \left (x^{4}+x \right )^{\frac {1}{4}} \left (4 x^{6}-x^{3}-5\right )}{135 x^{7}}\) | \(25\) |
meijerg | \(-\frac {4 \left (-\frac {4}{5} x^{6}+\frac {1}{5} x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {1}{4}}}{27 x^{\frac {27}{4}}}\) | \(25\) |
pseudoelliptic | \(\frac {4 \left (x^{4}+x \right )^{\frac {1}{4}} \left (4 x^{6}-x^{3}-5\right )}{135 x^{7}}\) | \(25\) |
gosper | \(\frac {4 \left (1+x \right ) \left (x^{2}-x +1\right ) \left (4 x^{3}-5\right ) \left (x^{4}+x \right )^{\frac {1}{4}}}{135 x^{7}}\) | \(31\) |
risch | \(\frac {4 {\left (x \left (x^{3}+1\right )\right )}^{\frac {1}{4}} \left (4 x^{9}+3 x^{6}-6 x^{3}-5\right )}{135 x^{7} \left (x^{3}+1\right )}\) | \(39\) |
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none
Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt [4]{x+x^4}}{x^8} \, dx=\frac {4 \, {\left (4 \, x^{6} - x^{3} - 5\right )} {\left (x^{4} + x\right )}^{\frac {1}{4}}}{135 \, x^{7}} \]
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\[ \int \frac {\sqrt [4]{x+x^4}}{x^8} \, dx=\int \frac {\sqrt [4]{x \left (x + 1\right ) \left (x^{2} - x + 1\right )}}{x^{8}}\, dx \]
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none
Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt [4]{x+x^4}}{x^8} \, dx=\frac {4 \, {\left (4 \, x^{7} - x^{4} - 5 \, x\right )} {\left (x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}}}{135 \, x^{\frac {31}{4}}} \]
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none
Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt [4]{x+x^4}}{x^8} \, dx=-\frac {4}{27} \, {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {9}{4}} + \frac {4}{15} \, {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {5}{4}} \]
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Time = 5.48 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt [4]{x+x^4}}{x^8} \, dx=-\frac {4\,{\left (x^4+x\right )}^{1/4}\,\left (-4\,x^6+x^3+5\right )}{135\,x^7} \]
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