Integrand size = 24, antiderivative size = 19 \[ \int \frac {x}{\sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\frac {5 (-1+x) (5+x)}{6 \sqrt [5]{(-1+x)^4}} \]
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Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1694, 15, 45} \[ \int \frac {x}{\sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\frac {5 (x-1)^2}{6 \sqrt [5]{(x-1)^4}}+\frac {5 (x-1)}{\sqrt [5]{(x-1)^4}} \]
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Rule 15
Rule 45
Rule 1694
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1+x}{\sqrt [5]{x^4}} \, dx,x,-1+x\right ) \\ & = \frac {(-1+x)^{4/5} \text {Subst}\left (\int \frac {1+x}{x^{4/5}} \, dx,x,-1+x\right )}{\sqrt [5]{(-1+x)^4}} \\ & = \frac {(-1+x)^{4/5} \text {Subst}\left (\int \left (\frac {1}{x^{4/5}}+\sqrt [5]{x}\right ) \, dx,x,-1+x\right )}{\sqrt [5]{(-1+x)^4}} \\ & = -\frac {5 (1-x)}{\sqrt [5]{(-1+x)^4}}+\frac {5 (1-x)^2}{6 \sqrt [5]{(-1+x)^4}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\frac {5 (-1+x) (5+x)}{6 \sqrt [5]{(-1+x)^4}} \]
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Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84
method | result | size |
risch | \(\frac {5 \left (x -1\right ) \left (x +5\right )}{6 \left (\left (x -1\right )^{4}\right )^{\frac {1}{5}}}\) | \(16\) |
gosper | \(\frac {5 \left (x -1\right ) \left (x +5\right )}{6 \left (x^{4}-4 x^{3}+6 x^{2}-4 x +1\right )^{\frac {1}{5}}}\) | \(29\) |
trager | \(\frac {5 \left (x +5\right ) \left (x^{4}-4 x^{3}+6 x^{2}-4 x +1\right )^{\frac {4}{5}}}{6 \left (x -1\right )^{3}}\) | \(31\) |
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Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (15) = 30\).
Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11 \[ \int \frac {x}{\sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\frac {5 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}^{\frac {4}{5}} {\left (x + 5\right )}}{6 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} \]
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\[ \int \frac {x}{\sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\int \frac {x}{\sqrt [5]{\left (x - 1\right )^{4}}}\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {x}{\sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\frac {5 \, {\left (x^{2} + 4 \, x - 5\right )}}{6 \, {\left (x - 1\right )}^{\frac {4}{5}}} \]
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\[ \int \frac {x}{\sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\int { \frac {x}{{\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}^{\frac {1}{5}}} \,d x } \]
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Time = 4.99 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.58 \[ \int \frac {x}{\sqrt [5]{1-4 x+6 x^2-4 x^3+x^4}} \, dx=\frac {5\,\left (x+5\right )\,{\left (x^4-4\,x^3+6\,x^2-4\,x+1\right )}^{4/5}}{6\,{\left (x-1\right )}^3} \]
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