Integrand size = 25, antiderivative size = 35 \[ \int \frac {1+x^6}{x^6 \left (1+x^3\right ) \sqrt [4]{x+x^4}} \, dx=\frac {4 \left (x+x^4\right )^{3/4} \left (-3+8 x^3+53 x^6\right )}{63 x^6 \left (1+x^3\right )} \]
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Time = 0.11 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.34, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2081, 1501, 464, 277, 270} \[ \int \frac {1+x^6}{x^6 \left (1+x^3\right ) \sqrt [4]{x+x^4}} \, dx=\frac {212 x}{63 \sqrt [4]{x^4+x}}-\frac {4}{21 \sqrt [4]{x^4+x} x^5}+\frac {32}{63 \sqrt [4]{x^4+x} x^2} \]
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Rule 270
Rule 277
Rule 464
Rule 1501
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^3}\right ) \int \frac {1+x^6}{x^{25/4} \left (1+x^3\right )^{5/4}} \, dx}{\sqrt [4]{x+x^4}} \\ & = -\frac {1}{3 x^2 \sqrt [4]{x+x^4}}-\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^3}\right ) \int \frac {-3+\frac {9 x^3}{4}}{x^{25/4} \left (1+x^3\right )^{5/4}} \, dx}{3 \sqrt [4]{x+x^4}} \\ & = -\frac {4}{21 x^5 \sqrt [4]{x+x^4}}-\frac {1}{3 x^2 \sqrt [4]{x+x^4}}-\frac {\left (53 \sqrt [4]{x} \sqrt [4]{1+x^3}\right ) \int \frac {1}{x^{13/4} \left (1+x^3\right )^{5/4}} \, dx}{28 \sqrt [4]{x+x^4}} \\ & = -\frac {4}{21 x^5 \sqrt [4]{x+x^4}}+\frac {32}{63 x^2 \sqrt [4]{x+x^4}}+\frac {\left (53 \sqrt [4]{x} \sqrt [4]{1+x^3}\right ) \int \frac {1}{\sqrt [4]{x} \left (1+x^3\right )^{5/4}} \, dx}{21 \sqrt [4]{x+x^4}} \\ & = -\frac {4}{21 x^5 \sqrt [4]{x+x^4}}+\frac {32}{63 x^2 \sqrt [4]{x+x^4}}+\frac {212 x}{63 \sqrt [4]{x+x^4}} \\ \end{align*}
Time = 10.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80 \[ \int \frac {1+x^6}{x^6 \left (1+x^3\right ) \sqrt [4]{x+x^4}} \, dx=\frac {4 \left (-3+8 x^3+53 x^6\right )}{63 x^5 \sqrt [4]{x+x^4}} \]
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Time = 1.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.71
method | result | size |
gosper | \(\frac {\frac {212}{63} x^{6}+\frac {32}{63} x^{3}-\frac {4}{21}}{\left (x^{4}+x \right )^{\frac {1}{4}} x^{5}}\) | \(25\) |
pseudoelliptic | \(\frac {\frac {212}{63} x^{6}+\frac {32}{63} x^{3}-\frac {4}{21}}{\left (x^{4}+x \right )^{\frac {1}{4}} x^{5}}\) | \(25\) |
risch | \(\frac {\frac {212}{63} x^{6}+\frac {32}{63} x^{3}-\frac {4}{21}}{x^{5} {\left (x \left (x^{3}+1\right )\right )}^{\frac {1}{4}}}\) | \(27\) |
trager | \(\frac {4 \left (x^{4}+x \right )^{\frac {3}{4}} \left (53 x^{6}+8 x^{3}-3\right )}{63 x^{6} \left (x^{3}+1\right )}\) | \(32\) |
meijerg | \(-\frac {4 \left (-32 x^{6}-8 x^{3}+3\right ) \left (x^{3}+1\right )^{\frac {3}{4}}}{21 x^{\frac {21}{4}} \left (3 x^{3}+3\right )}+\frac {4 x^{\frac {3}{4}}}{3 \left (x^{3}+1\right )^{\frac {1}{4}}}\) | \(47\) |
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Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86 \[ \int \frac {1+x^6}{x^6 \left (1+x^3\right ) \sqrt [4]{x+x^4}} \, dx=\frac {4 \, {\left (53 \, x^{6} + 8 \, x^{3} - 3\right )} {\left (x^{4} + x\right )}^{\frac {3}{4}}}{63 \, {\left (x^{9} + x^{6}\right )}} \]
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\[ \int \frac {1+x^6}{x^6 \left (1+x^3\right ) \sqrt [4]{x+x^4}} \, dx=\int \frac {\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}{x^{6} \sqrt [4]{x \left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
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\[ \int \frac {1+x^6}{x^6 \left (1+x^3\right ) \sqrt [4]{x+x^4}} \, dx=\int { \frac {x^{6} + 1}{{\left (x^{4} + x\right )}^{\frac {1}{4}} {\left (x^{3} + 1\right )} x^{6}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.80 \[ \int \frac {1+x^6}{x^6 \left (1+x^3\right ) \sqrt [4]{x+x^4}} \, dx=-\frac {4}{21} \, {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {7}{4}} + \frac {8}{9} \, {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {3}{4}} + \frac {8}{3 \, {\left (\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}} \]
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Time = 5.70 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {1+x^6}{x^6 \left (1+x^3\right ) \sqrt [4]{x+x^4}} \, dx=\frac {4\,{\left (x^4+x\right )}^{3/4}\,\left (53\,x^6+8\,x^3-3\right )}{63\,x^6\,\left (x^3+1\right )} \]
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