Integrand size = 55, antiderivative size = 62 \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {4 \left (1+x^4\right ) \left (x+x^5\right )^{3/4}}{7 x^6}+2 \arctan \left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right )+2 \text {arctanh}\left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right ) \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^{25/4} \sqrt [4]{1+x^4} \left (1-x^3+x^4\right )} \, dx}{\sqrt [4]{x+x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {\left (-3+x^{16}\right ) \left (1-x^{12}+2 x^{16}-x^{24}-x^{28}+x^{32}\right )}{x^{22} \sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \left (-\frac {3}{x^{22} \sqrt [4]{1+x^{16}}}-\frac {2}{x^6 \sqrt [4]{1+x^{16}}}-\frac {x^2}{\sqrt [4]{1+x^{16}}}+\frac {x^{10}}{\sqrt [4]{1+x^{16}}}-\frac {x^2 \left (-4+x^{12}\right )}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}} \\ & = -\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [4]{1+x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {x^2 \left (-4+x^{12}\right )}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}-\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^6 \sqrt [4]{1+x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}-\frac {\left (12 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^{22} \sqrt [4]{1+x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}} \\ & = \frac {4 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {21}{16},\frac {1}{4},-\frac {5}{16},-x^4\right )}{7 x^5 \sqrt [4]{x+x^5}}+\frac {8 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{16},\frac {1}{4},\frac {11}{16},-x^4\right )}{5 x \sqrt [4]{x+x^5}}-\frac {4 x \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {3}{16},\frac {1}{4},\frac {19}{16},-x^4\right )}{3 \sqrt [4]{x+x^5}}+\frac {4 x^3 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{16},\frac {27}{16},-x^4\right )}{11 \sqrt [4]{x+x^5}}-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \left (-\frac {4 x^2}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )}+\frac {x^{14}}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}} \\ & = \frac {4 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {21}{16},\frac {1}{4},-\frac {5}{16},-x^4\right )}{7 x^5 \sqrt [4]{x+x^5}}+\frac {8 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {5}{16},\frac {1}{4},\frac {11}{16},-x^4\right )}{5 x \sqrt [4]{x+x^5}}-\frac {4 x \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {3}{16},\frac {1}{4},\frac {19}{16},-x^4\right )}{3 \sqrt [4]{x+x^5}}+\frac {4 x^3 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {11}{16},\frac {27}{16},-x^4\right )}{11 \sqrt [4]{x+x^5}}-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}+\frac {\left (16 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}} \\ \end{align*}
Time = 33.80 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {4 \left (1+x^4\right ) \left (x+x^5\right )^{3/4}}{7 x^6}+2 \arctan \left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right )+2 \text {arctanh}\left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right ) \]
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Time = 5.46 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.53
method | result | size |
pseudoelliptic | \(\frac {-7 \ln \left (\frac {{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}-x}{x}\right ) x^{6}+7 \ln \left (\frac {{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}+x}{x}\right ) x^{6}-14 \arctan \left (\frac {{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}{x}\right ) x^{6}+4 x^{4} {\left (x \left (x^{4}+1\right )\right )}^{\frac {3}{4}}+4 {\left (x \left (x^{4}+1\right )\right )}^{\frac {3}{4}}}{7 x^{6}}\) | \(95\) |
trager | \(\frac {4 \left (x^{4}+1\right ) \left (x^{5}+x \right )^{\frac {3}{4}}}{7 x^{6}}-\ln \left (\frac {-x^{4}+2 \left (x^{5}+x \right )^{\frac {3}{4}}-2 x \sqrt {x^{5}+x}+2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}-x^{3}-1}{x^{4}-x^{3}+1}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{5}+x}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{5}+x \right )^{\frac {3}{4}}-2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}+1}\right )\) | \(168\) |
risch | \(\frac {\frac {4}{7} x^{8}+\frac {8}{7} x^{4}+\frac {4}{7}}{x^{5} {\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}-\ln \left (-\frac {-x^{4}+2 \left (x^{5}+x \right )^{\frac {3}{4}}-2 x \sqrt {x^{5}+x}+2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}-x^{3}-1}{x^{4}-x^{3}+1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{5}+x}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{5}+x \right )^{\frac {3}{4}}-2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}+1}\right )\) | \(172\) |
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Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (54) = 108\).
Time = 24.66 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.90 \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {7 \, x^{6} \arctan \left (\frac {{\left (x^{5} + x\right )}^{\frac {3}{4}} x - {\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}{2 \, {\left (x^{5} + x\right )}}\right ) + 7 \, x^{6} \log \left (-\frac {x^{4} + x^{3} + 2 \, {\left (x^{5} + x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{5} + x} x + 2 \, {\left (x^{5} + x\right )}^{\frac {3}{4}} + 1}{x^{4} - x^{3} + 1}\right ) + 4 \, {\left (x^{5} + x\right )}^{\frac {3}{4}} {\left (x^{4} + 1\right )}}{7 \, x^{6}} \]
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Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int { \frac {{\left (x^{8} - x^{7} - x^{6} + 2 \, x^{4} - x^{3} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} - x^{3} + 1\right )} x^{6}} \,d x } \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int { \frac {{\left (x^{8} - x^{7} - x^{6} + 2 \, x^{4} - x^{3} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} - x^{3} + 1\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int -\frac {\left (x^4-3\right )\,\left (-x^8+x^7+x^6-2\,x^4+x^3-1\right )}{x^6\,{\left (x^5+x\right )}^{1/4}\,\left (x^4-x^3+1\right )} \,d x \]
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