Integrand size = 20, antiderivative size = 48 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^3+x^4}}{x^8} \, dx=\frac {4 \sqrt [4]{x^3+x^4} \left (13923+663 x-780 x^2+960 x^3-39955 x^4-5687 x^5+22748 x^6\right )}{348075 x^7} \]
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Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(48)=96\).
Time = 0.17 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.27, number of steps used = 10, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2077, 2041, 2039} \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^3+x^4}}{x^8} \, dx=\frac {4 \left (x^4+x^3\right )^{5/4}}{25 x^{10}}-\frac {16 \left (x^4+x^3\right )^{5/4}}{105 x^9}+\frac {256 \left (x^4+x^3\right )^{5/4}}{1785 x^8}-\frac {1024 \left (x^4+x^3\right )^{5/4}}{7735 x^7}-\frac {22748 \left (x^4+x^3\right )^{5/4}}{69615 x^6}+\frac {90992 \left (x^4+x^3\right )^{5/4}}{348075 x^5} \]
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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^8}+\frac {\sqrt [4]{x^3+x^4}}{x^4}\right ) \, dx \\ & = -\int \frac {\sqrt [4]{x^3+x^4}}{x^8} \, dx+\int \frac {\sqrt [4]{x^3+x^4}}{x^4} \, dx \\ & = \frac {4 \left (x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {4 \left (x^3+x^4\right )^{5/4}}{9 x^6}-\frac {4}{9} \int \frac {\sqrt [4]{x^3+x^4}}{x^3} \, dx+\frac {4}{5} \int \frac {\sqrt [4]{x^3+x^4}}{x^7} \, dx \\ & = \frac {4 \left (x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {16 \left (x^3+x^4\right )^{5/4}}{105 x^9}-\frac {4 \left (x^3+x^4\right )^{5/4}}{9 x^6}+\frac {16 \left (x^3+x^4\right )^{5/4}}{45 x^5}-\frac {64}{105} \int \frac {\sqrt [4]{x^3+x^4}}{x^6} \, dx \\ & = \frac {4 \left (x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {16 \left (x^3+x^4\right )^{5/4}}{105 x^9}+\frac {256 \left (x^3+x^4\right )^{5/4}}{1785 x^8}-\frac {4 \left (x^3+x^4\right )^{5/4}}{9 x^6}+\frac {16 \left (x^3+x^4\right )^{5/4}}{45 x^5}+\frac {256}{595} \int \frac {\sqrt [4]{x^3+x^4}}{x^5} \, dx \\ & = \frac {4 \left (x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {16 \left (x^3+x^4\right )^{5/4}}{105 x^9}+\frac {256 \left (x^3+x^4\right )^{5/4}}{1785 x^8}-\frac {1024 \left (x^3+x^4\right )^{5/4}}{7735 x^7}-\frac {4 \left (x^3+x^4\right )^{5/4}}{9 x^6}+\frac {16 \left (x^3+x^4\right )^{5/4}}{45 x^5}-\frac {2048 \int \frac {\sqrt [4]{x^3+x^4}}{x^4} \, dx}{7735} \\ & = \frac {4 \left (x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {16 \left (x^3+x^4\right )^{5/4}}{105 x^9}+\frac {256 \left (x^3+x^4\right )^{5/4}}{1785 x^8}-\frac {1024 \left (x^3+x^4\right )^{5/4}}{7735 x^7}-\frac {22748 \left (x^3+x^4\right )^{5/4}}{69615 x^6}+\frac {16 \left (x^3+x^4\right )^{5/4}}{45 x^5}+\frac {8192 \int \frac {\sqrt [4]{x^3+x^4}}{x^3} \, dx}{69615} \\ & = \frac {4 \left (x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {16 \left (x^3+x^4\right )^{5/4}}{105 x^9}+\frac {256 \left (x^3+x^4\right )^{5/4}}{1785 x^8}-\frac {1024 \left (x^3+x^4\right )^{5/4}}{7735 x^7}-\frac {22748 \left (x^3+x^4\right )^{5/4}}{69615 x^6}+\frac {90992 \left (x^3+x^4\right )^{5/4}}{348075 x^5} \\ \end{align*}
Time = 0.94 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.79 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^3+x^4}}{x^8} \, dx=\frac {4 \left (x^3 (1+x)\right )^{9/4} \left (13923-27183 x+39663 x^2-51183 x^3+22748 x^4\right )}{348075 x^{13}} \]
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Result contains higher order function than in optimal. Order 5 vs. order 2.
Time = 0.84 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.75
method | result | size |
meijerg | \(\frac {4 \operatorname {hypergeom}\left (\left [-\frac {25}{4}, -\frac {1}{4}\right ], \left [-\frac {21}{4}\right ], -x \right )}{25 x^{\frac {25}{4}}}-\frac {4 \left (-\frac {4}{5} x^{2}+\frac {1}{5} x +1\right ) \left (1+x \right )^{\frac {1}{4}}}{9 x^{\frac {9}{4}}}\) | \(36\) |
gosper | \(\frac {4 \left (1+x \right )^{2} \left (22748 x^{4}-51183 x^{3}+39663 x^{2}-27183 x +13923\right ) \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}{348075 x^{7}}\) | \(40\) |
pseudoelliptic | \(\frac {4 \left (1+x \right )^{2} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} \left (22748 x^{4}-51183 x^{3}+39663 x^{2}-27183 x +13923\right )}{348075 x^{7}}\) | \(40\) |
trager | \(\frac {4 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} \left (22748 x^{6}-5687 x^{5}-39955 x^{4}+960 x^{3}-780 x^{2}+663 x +13923\right )}{348075 x^{7}}\) | \(45\) |
risch | \(\frac {4 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} \left (22748 x^{7}+17061 x^{6}-45642 x^{5}-38995 x^{4}+180 x^{3}-117 x^{2}+14586 x +13923\right )}{348075 x^{7} \left (1+x \right )}\) | \(55\) |
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Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^3+x^4}}{x^8} \, dx=\frac {4 \, {\left (22748 \, x^{6} - 5687 \, x^{5} - 39955 \, x^{4} + 960 \, x^{3} - 780 \, x^{2} + 663 \, x + 13923\right )} {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{348075 \, x^{7}} \]
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\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^3+x^4}}{x^8} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{8}}\, dx \]
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\[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^3+x^4}}{x^8} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{8}} \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^3+x^4}}{x^8} \, dx=\frac {4}{25} \, {\left (\frac {1}{x} + 1\right )}^{\frac {25}{4}} - \frac {20}{21} \, {\left (\frac {1}{x} + 1\right )}^{\frac {21}{4}} + \frac {40}{17} \, {\left (\frac {1}{x} + 1\right )}^{\frac {17}{4}} - \frac {40}{13} \, {\left (\frac {1}{x} + 1\right )}^{\frac {13}{4}} + \frac {16}{9} \, {\left (\frac {1}{x} + 1\right )}^{\frac {9}{4}} \]
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Time = 5.88 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.06 \[ \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^3+x^4}}{x^8} \, dx=\frac {90992\,{\left (x^4+x^3\right )}^{1/4}}{348075\,x}-\frac {22748\,{\left (x^4+x^3\right )}^{1/4}}{348075\,x^2}-\frac {31964\,{\left (x^4+x^3\right )}^{1/4}}{69615\,x^3}+\frac {256\,{\left (x^4+x^3\right )}^{1/4}}{23205\,x^4}-\frac {16\,{\left (x^4+x^3\right )}^{1/4}}{1785\,x^5}+\frac {4\,{\left (x^4+x^3\right )}^{1/4}}{525\,x^6}+\frac {4\,{\left (x^4+x^3\right )}^{1/4}}{25\,x^7} \]
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