Integrand size = 33, antiderivative size = 24 \[ \int \frac {2+3 x^5}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {x}{\sqrt {a} \sqrt {-1+x^5}}\right )}{\sqrt {a}} \]
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\[ \int \frac {2+3 x^5}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx=\int \frac {2+3 x^5}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{a \sqrt {-1+x^5}}+\frac {5 a+3 x^2}{a \sqrt {-1+x^5} \left (-a-x^2+a x^5\right )}\right ) \, dx \\ & = \frac {\int \frac {5 a+3 x^2}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx}{a}+\frac {3 \int \frac {1}{\sqrt {-1+x^5}} \, dx}{a} \\ & = \frac {\int \left (\frac {5 a}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )}+\frac {3 x^2}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )}\right ) \, dx}{a}+\frac {\left (3 \sqrt {1-x^5}\right ) \int \frac {1}{\sqrt {1-x^5}} \, dx}{a \sqrt {-1+x^5}} \\ & = \frac {3 x \sqrt {1-x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{5},\frac {1}{2},\frac {6}{5},x^5\right )}{a \sqrt {-1+x^5}}+5 \int \frac {1}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx+\frac {3 \int \frac {x^2}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx}{a} \\ \end{align*}
Time = 1.80 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {2+3 x^5}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {x}{\sqrt {a} \sqrt {-1+x^5}}\right )}{\sqrt {a}} \]
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Time = 0.90 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{5}-1}\, \sqrt {a}}{x}\right )}{\sqrt {a}}\) | \(21\) |
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (18) = 36\).
Time = 0.30 (sec) , antiderivative size = 151, normalized size of antiderivative = 6.29 \[ \int \frac {2+3 x^5}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx=\left [\frac {\log \left (\frac {a^{2} x^{10} + 6 \, a x^{7} - 2 \, a^{2} x^{5} + x^{4} - 6 \, a x^{2} - 4 \, {\left (a x^{6} + x^{3} - a x\right )} \sqrt {x^{5} - 1} \sqrt {a} + a^{2}}{a^{2} x^{10} - 2 \, a x^{7} - 2 \, a^{2} x^{5} + x^{4} + 2 \, a x^{2} + a^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (a x^{5} + x^{2} - a\right )} \sqrt {x^{5} - 1} \sqrt {-a}}{2 \, {\left (a x^{6} - a x\right )}}\right )}{a}\right ] \]
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\[ \int \frac {2+3 x^5}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx=\int \frac {3 x^{5} + 2}{\sqrt {\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (a x^{5} - a - x^{2}\right )}\, dx \]
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\[ \int \frac {2+3 x^5}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx=\int { \frac {3 \, x^{5} + 2}{{\left (a x^{5} - x^{2} - a\right )} \sqrt {x^{5} - 1}} \,d x } \]
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\[ \int \frac {2+3 x^5}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx=\int { \frac {3 \, x^{5} + 2}{{\left (a x^{5} - x^{2} - a\right )} \sqrt {x^{5} - 1}} \,d x } \]
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Time = 6.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.17 \[ \int \frac {2+3 x^5}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx=\frac {\ln \left (\frac {a^4\,\left (x^5-1\right )+a^3\,x^2-2\,a^{7/2}\,x\,\sqrt {x^5-1}}{4\,x^2-4\,a\,\left (x^5-1\right )}\right )}{\sqrt {a}} \]
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