Integrand size = 20, antiderivative size = 25 \[ \int \frac {1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \left (3+11 x^3\right ) \left (-x+x^4\right )^{3/4}}{63 x^6} \]
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Time = 0.05 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2063, 2039} \[ \int \frac {1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \left (x^4-x\right )^{3/4}}{21 x^6}+\frac {44 \left (x^4-x\right )^{3/4}}{63 x^3} \]
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Rule 2039
Rule 2063
Rubi steps \begin{align*} \text {integral}& = \frac {4 \left (-x+x^4\right )^{3/4}}{21 x^6}+\frac {11}{7} \int \frac {1}{x^3 \sqrt [4]{-x+x^4}} \, dx \\ & = \frac {4 \left (-x+x^4\right )^{3/4}}{21 x^6}+\frac {44 \left (-x+x^4\right )^{3/4}}{63 x^3} \\ \end{align*}
Time = 10.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \left (x \left (-1+x^3\right )\right )^{3/4} \left (3+11 x^3\right )}{63 x^6} \]
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Time = 0.79 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
trager | \(\frac {4 \left (11 x^{3}+3\right ) \left (x^{4}-x \right )^{\frac {3}{4}}}{63 x^{6}}\) | \(22\) |
pseudoelliptic | \(\frac {4 \left (11 x^{3}+3\right ) \left (x^{4}-x \right )^{\frac {3}{4}}}{63 x^{6}}\) | \(22\) |
risch | \(\frac {\frac {44}{63} x^{6}-\frac {32}{63} x^{3}-\frac {4}{21}}{x^{5} {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}}}\) | \(27\) |
gosper | \(\frac {4 \left (11 x^{3}+3\right ) \left (x -1\right ) \left (x^{2}+x +1\right )}{63 \left (x^{4}-x \right )^{\frac {1}{4}} x^{5}}\) | \(31\) |
meijerg | \(-\frac {4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (1+\frac {4 x^{3}}{3}\right ) \left (-x^{3}+1\right )^{\frac {3}{4}}}{21 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {21}{4}}}-\frac {4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (-x^{3}+1\right )^{\frac {3}{4}}}{9 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {9}{4}}}\) | \(73\) |
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Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} {\left (11 \, x^{3} + 3\right )}}{63 \, x^{6}} \]
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\[ \int \frac {1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\int \frac {\left (x + 1\right ) \left (x^{2} - x + 1\right )}{x^{6} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (21) = 42\).
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \frac {1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {4 \, {\left (x^{4} - x\right )}}{9 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{\frac {13}{4}}} + \frac {4 \, {\left (4 \, x^{7} - x^{4} - 3 \, x\right )}}{63 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{\frac {25}{4}}} \]
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Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=-\frac {4}{21} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {7}{4}} + \frac {8}{9} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {3}{4}} \]
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Time = 5.18 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx=\frac {12\,{\left (x^4-x\right )}^{3/4}+44\,x^3\,{\left (x^4-x\right )}^{3/4}}{63\,x^6} \]
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