Integrand size = 43, antiderivative size = 69 \[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {4 \left (3+7 x^3+3 x^4\right ) \left (x+x^5\right )^{3/4}}{21 x^6}-4 \arctan \left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right )-4 \text {arctanh}\left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right ) \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx \]
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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+2 x^4+x^6+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+2 x^{16}+x^{24}+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 {3}{x^{10} \sqrt [4]{1+x^{16}}}-\frac {2}{x^6 \sqrt [4]{1+x^{16}}}+\frac {2 x^2}{\sqrt [4]{1+x^{16}}}+\frac {x^6}{\sqrt [4]{1+x^{16}}}+\frac {x^{10}}{\sqrt [4]{1+x^{16}}}+\frac {2 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^6}{\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 (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 (8 \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 (8 \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 (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 {\left (12 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \text {Subst}\left (\int \frac {1}{x^{10} \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 {4 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {9}{16},\frac {1}{4},\frac {7}{16},-x^4\right )}{3 x^2 \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 {8 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^2 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {7}{16},\frac {23}{16},-x^4\right )}{7 \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 (8 \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 {4 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (-\frac {9}{16},\frac {1}{4},\frac {7}{16},-x^4\right )}{3 x^2 \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 {8 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^2 \sqrt [4]{1+x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {7}{16},\frac {23}{16},-x^4\right )}{7 \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 (8 \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 (32 \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 = 11.51 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.04 \[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\frac {4 \left (3+7 x^3+3 x^4\right ) \left (x+x^5\right )^{3/4}}{21 x^6}+2 \left (-2 \arctan \left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right )-2 \text {arctanh}\left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right )\right ) \]
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Time = 4.83 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.32
method | result | size |
pseudoelliptic | \(\frac {2 \ln \left (\frac {{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}-x}{x}\right ) x^{6}-2 \ln \left (\frac {{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}+x}{x}\right ) x^{6}+4 \arctan \left (\frac {{\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}{x}\right ) x^{6}+\frac {4 \left (x^{4}+\frac {7}{3} x^{3}+1\right ) {\left (x \left (x^{4}+1\right )\right )}^{\frac {3}{4}}}{7}}{x^{6}}\) | \(91\) |
trager | \(\frac {4 \left (3 x^{4}+7 x^{3}+3\right ) \left (x^{5}+x \right )^{\frac {3}{4}}}{21 x^{6}}-2 \ln \left (-\frac {x^{4}+2 \left (x^{5}+x \right )^{\frac {3}{4}}+2 x \sqrt {x^{5}+x}+2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}+x^{3}+1}{x^{4}-x^{3}+1}\right )-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{5}+x}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{5}+x \right )^{\frac {3}{4}}-2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}+1}\right )\) | \(168\) |
risch | \(\frac {\frac {4}{7} x^{8}+\frac {8}{7} x^{4}+\frac {4}{7}+\frac {4}{3} x^{7}+\frac {4}{3} x^{3}}{x^{5} {\left (x \left (x^{4}+1\right )\right )}^{\frac {1}{4}}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{5}+x}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{5}+x \right )^{\frac {3}{4}}-2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}+1}\right )+2 \ln \left (-\frac {-x^{4}+2 \left (x^{5}+x \right )^{\frac {3}{4}}-2 x \sqrt {x^{5}+x}+2 \left (x^{5}+x \right )^{\frac {1}{4}} x^{2}-x^{3}-1}{x^{4}-x^{3}+1}\right )\) | \(189\) |
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Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (61) = 122\).
Time = 25.35 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.81 \[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=-\frac {2 \, {\left (21 \, x^{6} \arctan \left (\frac {{\left (x^{5} + x\right )}^{\frac {3}{4}} x - {\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}{2 \, {\left (x^{5} + x\right )}}\right ) - 21 \, x^{6} \log \left (-\frac {x^{4} + x^{3} - 2 \, {\left (x^{5} + x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{5} + x} x - 2 \, {\left (x^{5} + x\right )}^{\frac {3}{4}} + 1}{x^{4} - x^{3} + 1}\right ) - 2 \, {\left (x^{5} + x\right )}^{\frac {3}{4}} {\left (3 \, x^{4} + 7 \, x^{3} + 3\right )}\right )}}{21 \, x^{6}} \]
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Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\text {Timed out} \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int { \frac {{\left (x^{8} + x^{6} + 2 \, x^{4} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} - x^{3} + 1\right )} x^{6}} \,d x } \]
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\[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int { \frac {{\left (x^{8} + x^{6} + 2 \, x^{4} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} - x^{3} + 1\right )} x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1+2 x^4+x^6+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx=\int \frac {\left (x^4-3\right )\,\left (x^8+x^6+2\,x^4+1\right )}{x^6\,{\left (x^5+x\right )}^{1/4}\,\left (x^4-x^3+1\right )} \,d x \]
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