Integrand size = 27, antiderivative size = 37 \[ \int \frac {1+x^3}{x^6 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=-\frac {4 \left (-x+x^4\right )^{3/4} \left (-1-5 x^3+20 x^6\right )}{21 x^6 \left (-1+x^3\right )} \]
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Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.43, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2081, 464, 277, 270} \[ \int \frac {1+x^3}{x^6 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=-\frac {80 x}{21 \sqrt [4]{x^4-x}}+\frac {4}{21 \sqrt [4]{x^4-x} x^5}+\frac {20}{21 \sqrt [4]{x^4-x} x^2} \]
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Rule 270
Rule 277
Rule 464
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1+x^3}{x^{25/4} \left (-1+x^3\right )^{5/4}} \, dx}{\sqrt [4]{-x+x^4}} \\ & = \frac {4}{21 x^5 \sqrt [4]{-x+x^4}}+\frac {\left (15 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1}{x^{13/4} \left (-1+x^3\right )^{5/4}} \, dx}{7 \sqrt [4]{-x+x^4}} \\ & = \frac {4}{21 x^5 \sqrt [4]{-x+x^4}}+\frac {20}{21 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (20 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1}{\sqrt [4]{x} \left (-1+x^3\right )^{5/4}} \, dx}{7 \sqrt [4]{-x+x^4}} \\ & = \frac {4}{21 x^5 \sqrt [4]{-x+x^4}}+\frac {20}{21 x^2 \sqrt [4]{-x+x^4}}-\frac {80 x}{21 \sqrt [4]{-x+x^4}} \\ \end{align*}
Time = 10.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81 \[ \int \frac {1+x^3}{x^6 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=\frac {4+20 x^3-80 x^6}{21 x^5 \sqrt [4]{x \left (-1+x^3\right )}} \]
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Time = 1.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.73
method | result | size |
gosper | \(-\frac {4 \left (20 x^{6}-5 x^{3}-1\right )}{21 \left (x^{4}-x \right )^{\frac {1}{4}} x^{5}}\) | \(27\) |
risch | \(-\frac {4 \left (20 x^{6}-5 x^{3}-1\right )}{21 x^{5} {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}}}\) | \(27\) |
pseudoelliptic | \(\frac {-\frac {80}{21} x^{6}+\frac {20}{21} x^{3}+\frac {4}{21}}{\left (x^{4}-x \right )^{\frac {1}{4}} x^{5}}\) | \(27\) |
trager | \(-\frac {4 \left (x^{4}-x \right )^{\frac {3}{4}} \left (20 x^{6}-5 x^{3}-1\right )}{21 x^{6} \left (x^{3}-1\right )}\) | \(34\) |
meijerg | \(\frac {4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (-32 x^{6}+8 x^{3}+3\right ) \left (-x^{3}+1\right )^{\frac {3}{4}}}{21 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {21}{4}} \left (-3 x^{3}+3\right )}+\frac {4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (-4 x^{3}+1\right )}{9 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {9}{4}} \left (-x^{3}+1\right )^{\frac {1}{4}}}\) | \(94\) |
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Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int \frac {1+x^3}{x^6 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=-\frac {4 \, {\left (20 \, x^{6} - 5 \, x^{3} - 1\right )} {\left (x^{4} - x\right )}^{\frac {3}{4}}}{21 \, {\left (x^{9} - x^{6}\right )}} \]
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\[ \int \frac {1+x^3}{x^6 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=\int \frac {\left (x + 1\right ) \left (x^{2} - x + 1\right )}{x^{6} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
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\[ \int \frac {1+x^3}{x^6 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=\int { \frac {x^{3} + 1}{{\left (x^{4} - x\right )}^{\frac {1}{4}} {\left (x^{3} - 1\right )} x^{6}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92 \[ \int \frac {1+x^3}{x^6 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=\frac {4}{21} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {7}{4}} - \frac {4}{3} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {3}{4}} - \frac {8}{3 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}} \]
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Time = 5.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int \frac {1+x^3}{x^6 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=\frac {4\,{\left (x^4-x\right )}^{3/4}\,\left (-20\,x^6+5\,x^3+1\right )}{21\,x^6\,\left (x^3-1\right )} \]
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