Integrand size = 40, antiderivative size = 47 \[ \int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=\frac {2 \sqrt {-x+x^6}}{-1+x^5}-2 \sqrt {a} \text {arctanh}\left (\frac {x}{\sqrt {a} \sqrt {-x+x^6}}\right ) \]
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\[ \int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=\int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (1+4 x^5\right )}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx \\ & = \frac {\left (\sqrt {x} \sqrt {-1+x^5}\right ) \int \frac {\sqrt {x} \left (1+4 x^5\right )}{\left (-1+x^5\right )^{3/2} \left (-a-x+a x^5\right )} \, dx}{\sqrt {-x+x^6}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \frac {x^2 \left (1+4 x^{10}\right )}{\left (-1+x^{10}\right )^{3/2} \left (-a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \left (\frac {4 x^2}{a \left (-1+x^{10}\right )^{3/2}}+\frac {x^2 \left (5 a+4 x^2\right )}{a \left (-1+x^{10}\right )^{3/2} \left (-a-x^2+a x^{10}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \frac {x^2 \left (5 a+4 x^2\right )}{\left (-1+x^{10}\right )^{3/2} \left (-a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {-x+x^6}}+\frac {\left (8 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^{10}\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{a \sqrt {-x+x^6}} \\ & = -\frac {8 x^2}{5 a \sqrt {-x+x^6}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \left (\frac {5 a x^2}{\left (-1+x^{10}\right )^{3/2} \left (-a-x^2+a x^{10}\right )}+\frac {4 x^4}{\left (-1+x^{10}\right )^{3/2} \left (-a-x^2+a x^{10}\right )}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-x+x^6}}-\frac {\left (16 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x^{10}}} \, dx,x,\sqrt {x}\right )}{5 a \sqrt {-x+x^6}} \\ & = -\frac {8 x^2}{5 a \sqrt {-x+x^6}}-\frac {\left (16 \sqrt {x} \sqrt {1-x^5}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^{10}}} \, dx,x,\sqrt {x}\right )}{5 a \sqrt {-x+x^6}}+\frac {\left (10 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^{10}\right )^{3/2} \left (-a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}}+\frac {\left (8 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \frac {x^4}{\left (-1+x^{10}\right )^{3/2} \left (-a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {-x+x^6}} \\ & = -\frac {8 x^2}{5 a \sqrt {-x+x^6}}-\frac {16 x^2 \sqrt {1-x^5} \operatorname {Hypergeometric2F1}\left (\frac {3}{10},\frac {1}{2},\frac {13}{10},x^5\right )}{15 a \sqrt {-x+x^6}}+\frac {\left (10 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-1+x^{10}\right )^{3/2} \left (-a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^6}}+\frac {\left (8 \sqrt {x} \sqrt {-1+x^5}\right ) \text {Subst}\left (\int \frac {x^4}{\left (-1+x^{10}\right )^{3/2} \left (-a-x^2+a x^{10}\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {-x+x^6}} \\ \end{align*}
\[ \int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=\int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx \]
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Time = 1.63 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96
method | result | size |
pseudoelliptic | \(\frac {-2 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{6}-x}\, \sqrt {a}}{x}\right ) \sqrt {x^{6}-x}+2 x}{\sqrt {x^{6}-x}}\) | \(45\) |
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Time = 0.35 (sec) , antiderivative size = 184, normalized size of antiderivative = 3.91 \[ \int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=\left [\frac {{\left (x^{5} - 1\right )} \sqrt {a} \log \left (-\frac {a^{2} x^{10} - 2 \, a^{2} x^{5} + 6 \, a x^{6} - 4 \, {\left (a x^{5} - a + x\right )} \sqrt {x^{6} - x} \sqrt {a} + a^{2} - 6 \, a x + x^{2}}{a^{2} x^{10} - 2 \, a^{2} x^{5} - 2 \, a x^{6} + a^{2} + 2 \, a x + x^{2}}\right ) + 4 \, \sqrt {x^{6} - x}}{2 \, {\left (x^{5} - 1\right )}}, \frac {{\left (x^{5} - 1\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} - x} \sqrt {-a}}{a x^{5} - a + x}\right ) + 2 \, \sqrt {x^{6} - x}}{x^{5} - 1}\right ] \]
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Timed out. \[ \int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=\text {Timed out} \]
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\[ \int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=\int { \frac {4 \, x^{6} + x}{{\left (a x^{5} - a - x\right )} \sqrt {x^{6} - x} {\left (x^{5} - 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {x+4 x^6}{\left (-1+x^5\right ) \left (-a-x+a x^5\right ) \sqrt {-x+x^6}} \, dx=-\int \frac {4\,x^6+x}{\sqrt {x^6-x}\,\left (x^5-1\right )\,\left (-a\,x^5+x+a\right )} \,d x \]
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