Integrand size = 27, antiderivative size = 32 \[ \int \frac {1+x^3}{x^3 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=-\frac {4 \left (-1+7 x^3\right ) \left (-x+x^4\right )^{3/4}}{9 x^3 \left (-1+x^3\right )} \]
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Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2081, 464, 270} \[ \int \frac {1+x^3}{x^3 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=\frac {4}{9 x^2 \sqrt [4]{x^4-x}}-\frac {28 x}{9 \sqrt [4]{x^4-x}} \]
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Rule 270
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^{13/4} \left (-1+x^3\right )^{5/4}} \, dx}{\sqrt [4]{-x+x^4}} \\ & = \frac {4}{9 x^2 \sqrt [4]{-x+x^4}}+\frac {\left (7 \sqrt [4]{x} \sqrt [4]{-1+x^3}\right ) \int \frac {1}{\sqrt [4]{x} \left (-1+x^3\right )^{5/4}} \, dx}{3 \sqrt [4]{-x+x^4}} \\ & = \frac {4}{9 x^2 \sqrt [4]{-x+x^4}}-\frac {28 x}{9 \sqrt [4]{-x+x^4}} \\ \end{align*}
Time = 10.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \frac {1+x^3}{x^3 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=-\frac {4 \left (-1+7 x^3\right )}{9 x^2 \sqrt [4]{x \left (-1+x^3\right )}} \]
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Time = 1.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.69
method | result | size |
gosper | \(-\frac {4 \left (7 x^{3}-1\right )}{9 \left (x^{4}-x \right )^{\frac {1}{4}} x^{2}}\) | \(22\) |
risch | \(-\frac {4 \left (7 x^{3}-1\right )}{9 x^{2} {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}}}\) | \(22\) |
pseudoelliptic | \(\frac {-\frac {28 x^{3}}{9}+\frac {4}{9}}{x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}}\) | \(22\) |
trager | \(-\frac {4 \left (7 x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {3}{4}}}{9 x^{3} \left (x^{3}-1\right )}\) | \(29\) |
meijerg | \(\frac {4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (-4 x^{3}+1\right )}{9 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {9}{4}} \left (-x^{3}+1\right )^{\frac {1}{4}}}-\frac {4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} x^{\frac {3}{4}}}{3 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} \left (-x^{3}+1\right )^{\frac {1}{4}}}\) | \(73\) |
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none
Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {1+x^3}{x^3 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=-\frac {4 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} {\left (7 \, x^{3} - 1\right )}}{9 \, {\left (x^{6} - x^{3}\right )}} \]
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\[ \int \frac {1+x^3}{x^3 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=\int \frac {\left (x + 1\right ) \left (x^{2} - x + 1\right )}{x^{3} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
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\[ \int \frac {1+x^3}{x^3 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=\int { \frac {x^{3} + 1}{{\left (x^{4} - x\right )}^{\frac {1}{4}} {\left (x^{3} - 1\right )} x^{3}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {1+x^3}{x^3 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=-\frac {4}{9} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {3}{4}} - \frac {8}{3 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}} \]
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Time = 5.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {1+x^3}{x^3 \left (-1+x^3\right ) \sqrt [4]{-x+x^4}} \, dx=-\frac {4\,{\left (x^4-x\right )}^{3/4}\,\left (7\,x^3-1\right )}{9\,x^3\,\left (x^3-1\right )} \]
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