Integrand size = 20, antiderivative size = 20 \[ \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x^4} \, dx=\frac {4 \left (-x^3+x^4\right )^{9/4}}{9 x^9} \]
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Leaf count is larger than twice the leaf count of optimal. \(41\) vs. \(2(20)=40\).
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.05, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2077, 2041, 2039} \[ \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x^4} \, dx=\frac {4 \left (x^4-x^3\right )^{5/4}}{9 x^5}-\frac {4 \left (x^4-x^3\right )^{5/4}}{9 x^6} \]
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Rule 2039
Rule 2041
Rule 2077
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt [4]{-x^3+x^4}}{x^4}+\frac {\sqrt [4]{-x^3+x^4}}{x^3}\right ) \, dx \\ & = -\int \frac {\sqrt [4]{-x^3+x^4}}{x^4} \, dx+\int \frac {\sqrt [4]{-x^3+x^4}}{x^3} \, dx \\ & = -\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}+\frac {4 \left (-x^3+x^4\right )^{5/4}}{5 x^5}-\frac {4}{9} \int \frac {\sqrt [4]{-x^3+x^4}}{x^3} \, dx \\ & = -\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^6}+\frac {4 \left (-x^3+x^4\right )^{5/4}}{9 x^5} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x^4} \, dx=\frac {4 \left ((-1+x) x^3\right )^{9/4}}{9 x^9} \]
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Time = 0.83 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00
method | result | size |
pseudoelliptic | \(\frac {4 \left (x -1\right )^{2} \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}{9 x^{3}}\) | \(20\) |
gosper | \(\frac {4 \left (x -1\right )^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}\) | \(22\) |
trager | \(\frac {4 \left (x^{2}-2 x +1\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}\) | \(25\) |
risch | \(\frac {4 \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}} \left (x^{3}-3 x^{2}+3 x -1\right )}{9 \left (x -1\right ) x^{3}}\) | \(33\) |
meijerg | \(\frac {4 \operatorname {signum}\left (x -1\right )^{\frac {1}{4}} \left (-\frac {4}{5} x^{2}-\frac {1}{5} x +1\right ) \left (1-x \right )^{\frac {1}{4}}}{9 \left (-\operatorname {signum}\left (x -1\right )\right )^{\frac {1}{4}} x^{\frac {9}{4}}}-\frac {4 \operatorname {signum}\left (x -1\right )^{\frac {1}{4}} \left (1-x \right )^{\frac {5}{4}}}{5 \left (-\operatorname {signum}\left (x -1\right )\right )^{\frac {1}{4}} x^{\frac {5}{4}}}\) | \(64\) |
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none
Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x^4} \, dx=\frac {4 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} - 2 \, x + 1\right )}}{9 \, x^{3}} \]
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\[ \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x^4} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x - 1\right )}{x^{4}}\, dx \]
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\[ \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x^4} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x - 1\right )}}{x^{4}} \,d x } \]
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none
Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x^4} \, dx=\frac {4}{9} \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \]
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Time = 5.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.45 \[ \int \frac {(-1+x) \sqrt [4]{-x^3+x^4}}{x^4} \, dx=\frac {4\,x^2\,{\left (x^4-x^3\right )}^{1/4}-8\,x\,{\left (x^4-x^3\right )}^{1/4}+4\,{\left (x^4-x^3\right )}^{1/4}}{9\,x^3} \]
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