Integrand size = 15, antiderivative size = 68 \[ \int x^7 \sqrt [4]{-x+x^4} \, dx=\frac {1}{288} \sqrt [4]{-x+x^4} \left (-7 x^2-4 x^5+32 x^8\right )+\frac {7}{192} \arctan \left (\frac {x}{\sqrt [4]{-x+x^4}}\right )-\frac {7}{192} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x+x^4}}\right ) \]
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Leaf count is larger than twice the leaf count of optimal. \(145\) vs. \(2(68)=136\).
Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 2.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2046, 2049, 2057, 335, 281, 338, 304, 209, 212} \[ \int x^7 \sqrt [4]{-x+x^4} \, dx=\frac {7 \left (x^3-1\right )^{3/4} x^{3/4} \arctan \left (\frac {x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{192 \left (x^4-x\right )^{3/4}}-\frac {7 \left (x^3-1\right )^{3/4} x^{3/4} \text {arctanh}\left (\frac {x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{192 \left (x^4-x\right )^{3/4}}+\frac {1}{9} \sqrt [4]{x^4-x} x^8-\frac {1}{72} \sqrt [4]{x^4-x} x^5-\frac {7}{288} \sqrt [4]{x^4-x} x^2 \]
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Rule 209
Rule 212
Rule 281
Rule 304
Rule 335
Rule 338
Rule 2046
Rule 2049
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {1}{12} \int \frac {x^8}{\left (-x+x^4\right )^{3/4}} \, dx \\ & = -\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {7}{96} \int \frac {x^5}{\left (-x+x^4\right )^{3/4}} \, dx \\ & = -\frac {7}{288} x^2 \sqrt [4]{-x+x^4}-\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {7}{128} \int \frac {x^2}{\left (-x+x^4\right )^{3/4}} \, dx \\ & = -\frac {7}{288} x^2 \sqrt [4]{-x+x^4}-\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {\left (7 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \int \frac {x^{5/4}}{\left (-1+x^3\right )^{3/4}} \, dx}{128 \left (-x+x^4\right )^{3/4}} \\ & = -\frac {7}{288} x^2 \sqrt [4]{-x+x^4}-\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {\left (7 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^8}{\left (-1+x^{12}\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{32 \left (-x+x^4\right )^{3/4}} \\ & = -\frac {7}{288} x^2 \sqrt [4]{-x+x^4}-\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {\left (7 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,x^{3/4}\right )}{96 \left (-x+x^4\right )^{3/4}} \\ & = -\frac {7}{288} x^2 \sqrt [4]{-x+x^4}-\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {\left (7 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{96 \left (-x+x^4\right )^{3/4}} \\ & = -\frac {7}{288} x^2 \sqrt [4]{-x+x^4}-\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}-\frac {\left (7 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{192 \left (-x+x^4\right )^{3/4}}+\frac {\left (7 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{192 \left (-x+x^4\right )^{3/4}} \\ & = -\frac {7}{288} x^2 \sqrt [4]{-x+x^4}-\frac {1}{72} x^5 \sqrt [4]{-x+x^4}+\frac {1}{9} x^8 \sqrt [4]{-x+x^4}+\frac {7 x^{3/4} \left (-1+x^3\right )^{3/4} \arctan \left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{192 \left (-x+x^4\right )^{3/4}}-\frac {7 x^{3/4} \left (-1+x^3\right )^{3/4} \text {arctanh}\left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{192 \left (-x+x^4\right )^{3/4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.01 \[ \int x^7 \sqrt [4]{-x+x^4} \, dx=\frac {x^2 \sqrt [4]{x \left (-1+x^3\right )} \left (\sqrt [4]{1-x^3} \left (-7-x^3+8 x^6\right )+7 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{4},\frac {7}{4},x^3\right )\right )}{72 \sqrt [4]{1-x^3}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.48 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.49
method | result | size |
meijerg | \(\frac {4 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {33}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {11}{4}\right ], \left [\frac {15}{4}\right ], x^{3}\right )}{33 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}}}\) | \(33\) |
pseudoelliptic | \(-\frac {x^{3} \left (\left (-128 x^{8}+16 x^{5}+28 x^{2}\right ) \left (x^{4}-x \right )^{\frac {1}{4}}+42 \arctan \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}}{x}\right )-21 \ln \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}-x}{x}\right )+21 \ln \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}+x}{x}\right )\right )}{1152 {\left (-\left (x^{4}-x \right )^{\frac {1}{4}}+x \right )}^{3} \left (x^{2}+\sqrt {x^{4}-x}\right )^{3} {\left (\left (x^{4}-x \right )^{\frac {1}{4}}+x \right )}^{3}}\) | \(130\) |
trager | \(\frac {x^{2} \left (32 x^{6}-4 x^{3}-7\right ) \left (x^{4}-x \right )^{\frac {1}{4}}}{288}+\frac {7 \ln \left (-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x \sqrt {x^{4}-x}-2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+2 x^{3}-1\right )}{384}+\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \sqrt {x^{4}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{384}\) | \(145\) |
risch | \(\frac {x^{2} \left (32 x^{6}-4 x^{3}-7\right ) {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}}}{288}+\frac {\left (-\frac {7 \ln \left (\frac {2 x^{9}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}-5 x^{6}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}-4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}-1}{\left (x^{2}+x +1\right )^{2} \left (x -1\right )^{2}}\right )}{384}+\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 x^{9}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}-5 x^{6}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}-1}{\left (x^{2}+x +1\right )^{2} \left (x -1\right )^{2}}\right )}{384}\right ) {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (x^{3} \left (x^{3}-1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{3}-1\right )}\) | \(455\) |
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Time = 1.32 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.53 \[ \int x^7 \sqrt [4]{-x+x^4} \, dx=\frac {1}{288} \, {\left (32 \, x^{8} - 4 \, x^{5} - 7 \, x^{2}\right )} {\left (x^{4} - x\right )}^{\frac {1}{4}} - \frac {7}{384} \, \arctan \left (2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}}\right ) + \frac {7}{384} \, \log \left (2 \, x^{3} - 2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} - x} x - 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} - 1\right ) \]
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\[ \int x^7 \sqrt [4]{-x+x^4} \, dx=\int x^{7} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
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\[ \int x^7 \sqrt [4]{-x+x^4} \, dx=\int { {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{7} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.29 \[ \int x^7 \sqrt [4]{-x+x^4} \, dx=-\frac {1}{288} \, {\left (7 \, {\left (\frac {1}{x^{3}} - 1\right )}^{2} {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} - 18 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {5}{4}} - 21 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right )} x^{9} - \frac {7}{192} \, \arctan \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {7}{384} \, \log \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {7}{384} \, \log \left ({\left | {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
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Timed out. \[ \int x^7 \sqrt [4]{-x+x^4} \, dx=\int x^7\,{\left (x^4-x\right )}^{1/4} \,d x \]
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