Integrand size = 27, antiderivative size = 54 \[ \int \frac {\sqrt {-5+x} \sqrt {3+x}}{(-1+x) \left (-25+x^2\right )} \, dx=\frac {1}{6} \arctan \left (\frac {1}{4} \sqrt {-5+x} \sqrt {3+x}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {5} \sqrt {3+x}}{\sqrt {-5+x}}\right )}{3 \sqrt {5}} \]
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Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1600, 184, 94, 209, 95, 212} \[ \int \frac {\sqrt {-5+x} \sqrt {3+x}}{(-1+x) \left (-25+x^2\right )} \, dx=\frac {1}{6} \arctan \left (\frac {1}{4} \sqrt {x-5} \sqrt {x+3}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {5} \sqrt {x+3}}{\sqrt {x-5}}\right )}{3 \sqrt {5}} \]
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Rule 94
Rule 95
Rule 184
Rule 209
Rule 212
Rule 1600
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {3+x}}{\sqrt {-5+x} (-1+x) (5+x)} \, dx \\ & = \frac {1}{3} \int \frac {1}{\sqrt {-5+x} \sqrt {3+x} (5+x)} \, dx+\frac {2}{3} \int \frac {1}{\sqrt {-5+x} (-1+x) \sqrt {3+x}} \, dx \\ & = \frac {2}{3} \text {Subst}\left (\int \frac {1}{2-10 x^2} \, dx,x,\frac {\sqrt {3+x}}{\sqrt {-5+x}}\right )+\frac {2}{3} \text {Subst}\left (\int \frac {1}{16+x^2} \, dx,x,\sqrt {-5+x} \sqrt {3+x}\right ) \\ & = \frac {1}{6} \arctan \left (\frac {1}{4} \sqrt {-5+x} \sqrt {3+x}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {5} \sqrt {3+x}}{\sqrt {-5+x}}\right )}{3 \sqrt {5}} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {-5+x} \sqrt {3+x}}{(-1+x) \left (-25+x^2\right )} \, dx=\frac {1}{15} \left (-5 \arctan \left (\frac {1}{\sqrt {\frac {-5+x}{3+x}}}\right )+\sqrt {5} \text {arctanh}\left (\frac {\sqrt {5}}{\sqrt {\frac {-5+x}{3+x}}}\right )\right ) \]
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Time = 0.28 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.19
method | result | size |
default | \(\frac {\sqrt {x -5}\, \sqrt {3+x}\, \left (\sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (5+3 x \right ) \sqrt {5}}{5 \sqrt {x^{2}-2 x -15}}\right )-5 \arctan \left (\frac {4}{\sqrt {x^{2}-2 x -15}}\right )\right )}{30 \sqrt {x^{2}-2 x -15}}\) | \(64\) |
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none
Time = 0.24 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {-5+x} \sqrt {3+x}}{(-1+x) \left (-25+x^2\right )} \, dx=\frac {1}{30} \, \sqrt {5} \log \left (\frac {\sqrt {x + 3} \sqrt {x - 5} {\left (3 \, \sqrt {5} + 5\right )} + \sqrt {5} {\left (3 \, x + 5\right )} + 9 \, x + 15}{x + 5}\right ) + \frac {1}{3} \, \arctan \left (\frac {1}{4} \, \sqrt {x + 3} \sqrt {x - 5} - \frac {1}{4} \, x + \frac {1}{4}\right ) \]
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\[ \int \frac {\sqrt {-5+x} \sqrt {3+x}}{(-1+x) \left (-25+x^2\right )} \, dx=\int \frac {\sqrt {x + 3}}{\sqrt {x - 5} \left (x - 1\right ) \left (x + 5\right )}\, dx \]
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\[ \int \frac {\sqrt {-5+x} \sqrt {3+x}}{(-1+x) \left (-25+x^2\right )} \, dx=\int { \frac {\sqrt {x + 3} \sqrt {x - 5}}{{\left (x^{2} - 25\right )} {\left (x - 1\right )}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (36) = 72\).
Time = 0.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {-5+x} \sqrt {3+x}}{(-1+x) \left (-25+x^2\right )} \, dx=-\frac {1}{30} \, \sqrt {5} \log \left (\frac {{\left (\sqrt {x + 3} - \sqrt {x - 5}\right )}^{2} - 4 \, \sqrt {5} + 12}{{\left (\sqrt {x + 3} - \sqrt {x - 5}\right )}^{2} + 4 \, \sqrt {5} + 12}\right ) - \frac {1}{3} \, \arctan \left (\frac {1}{8} \, {\left (\sqrt {x + 3} - \sqrt {x - 5}\right )}^{2}\right ) \]
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Time = 0.65 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.76 \[ \int \frac {\sqrt {-5+x} \sqrt {3+x}}{(-1+x) \left (-25+x^2\right )} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {x+3}\,\sqrt {x-5}-2\,\sqrt {2}\,\sqrt {x-5}}{x-2\,\sqrt {2}\,\sqrt {x+3}+3}\right )}{3}-\frac {\sqrt {5}\,\mathrm {atanh}\left (-\frac {\sqrt {5}\,\sqrt {x+3}\,\sqrt {x-5}-2\,\sqrt {2}\,\sqrt {5}\,\sqrt {x-5}}{5\,x-10\,\sqrt {2}\,\sqrt {x+3}+15}\right )}{15} \]
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