Integrand size = 13, antiderivative size = 61 \[ \int \frac {\left (-10+x^2\right )^{5/2}}{x} \, dx=100 \sqrt {-10+x^2}-\frac {10}{3} \left (-10+x^2\right )^{3/2}+\frac {1}{5} \left (-10+x^2\right )^{5/2}-100 \sqrt {10} \arctan \left (\frac {\sqrt {-10+x^2}}{\sqrt {10}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 52, 65, 209} \[ \int \frac {\left (-10+x^2\right )^{5/2}}{x} \, dx=-100 \sqrt {10} \arctan \left (\frac {\sqrt {x^2-10}}{\sqrt {10}}\right )+\frac {1}{5} \left (x^2-10\right )^{5/2}-\frac {10}{3} \left (x^2-10\right )^{3/2}+100 \sqrt {x^2-10} \]
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Rule 52
Rule 65
Rule 209
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(-10+x)^{5/2}}{x} \, dx,x,x^2\right ) \\ & = \frac {1}{5} \left (-10+x^2\right )^{5/2}-5 \text {Subst}\left (\int \frac {(-10+x)^{3/2}}{x} \, dx,x,x^2\right ) \\ & = -\frac {10}{3} \left (-10+x^2\right )^{3/2}+\frac {1}{5} \left (-10+x^2\right )^{5/2}+50 \text {Subst}\left (\int \frac {\sqrt {-10+x}}{x} \, dx,x,x^2\right ) \\ & = 100 \sqrt {-10+x^2}-\frac {10}{3} \left (-10+x^2\right )^{3/2}+\frac {1}{5} \left (-10+x^2\right )^{5/2}-500 \text {Subst}\left (\int \frac {1}{\sqrt {-10+x} x} \, dx,x,x^2\right ) \\ & = 100 \sqrt {-10+x^2}-\frac {10}{3} \left (-10+x^2\right )^{3/2}+\frac {1}{5} \left (-10+x^2\right )^{5/2}-1000 \text {Subst}\left (\int \frac {1}{10+x^2} \, dx,x,\sqrt {-10+x^2}\right ) \\ & = 100 \sqrt {-10+x^2}-\frac {10}{3} \left (-10+x^2\right )^{3/2}+\frac {1}{5} \left (-10+x^2\right )^{5/2}-100 \sqrt {10} \arctan \left (\frac {\sqrt {-10+x^2}}{\sqrt {10}}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.80 \[ \int \frac {\left (-10+x^2\right )^{5/2}}{x} \, dx=\frac {1}{15} \sqrt {-10+x^2} \left (2300-110 x^2+3 x^4\right )-100 \sqrt {10} \arctan \left (\frac {\sqrt {-10+x^2}}{\sqrt {10}}\right ) \]
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Time = 0.43 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.67
method | result | size |
pseudoelliptic | \(-100 \arctan \left (\frac {\sqrt {x^{2}-10}\, \sqrt {10}}{10}\right ) \sqrt {10}+\frac {\sqrt {x^{2}-10}\, \left (3 x^{4}-110 x^{2}+2300\right )}{15}\) | \(41\) |
default | \(\frac {\left (x^{2}-10\right )^{\frac {5}{2}}}{5}-\frac {10 \left (x^{2}-10\right )^{\frac {3}{2}}}{3}+100 \sqrt {x^{2}-10}+100 \sqrt {10}\, \arctan \left (\frac {\sqrt {10}}{\sqrt {x^{2}-10}}\right )\) | \(46\) |
trager | \(\left (\frac {1}{5} x^{4}-\frac {22}{3} x^{2}+\frac {460}{3}\right ) \sqrt {x^{2}-10}-100 \operatorname {RootOf}\left (\textit {\_Z}^{2}+10\right ) \ln \left (\frac {\sqrt {x^{2}-10}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+10\right )}{x}\right )\) | \(49\) |
meijerg | \(-\frac {375 \sqrt {2}\, \sqrt {5}\, \operatorname {signum}\left (-1+\frac {x^{2}}{10}\right )^{\frac {5}{2}} \left (-\frac {8 \left (\frac {46}{15}-3 \ln \left (2\right )+2 \ln \left (x \right )-\ln \left (5\right )+i \pi \right ) \sqrt {\pi }}{15}+\frac {368 \sqrt {\pi }}{225}-\frac {4 \sqrt {\pi }\, \left (\frac {3}{25} x^{4}-\frac {22}{5} x^{2}+92\right ) \sqrt {1-\frac {x^{2}}{10}}}{225}+\frac {16 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1-\frac {x^{2}}{10}}}{2}\right )}{15}\right )}{4 \sqrt {\pi }\, {\left (-\operatorname {signum}\left (-1+\frac {x^{2}}{10}\right )\right )}^{\frac {5}{2}}}\) | \(108\) |
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Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77 \[ \int \frac {\left (-10+x^2\right )^{5/2}}{x} \, dx=\frac {1}{15} \, {\left (3 \, x^{4} - 110 \, x^{2} + 2300\right )} \sqrt {x^{2} - 10} - 200 \, \sqrt {10} \arctan \left (-\frac {1}{10} \, \sqrt {10} x + \frac {1}{10} \, \sqrt {10} \sqrt {x^{2} - 10}\right ) \]
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Result contains complex when optimal does not.
Time = 3.43 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.70 \[ \int \frac {\left (-10+x^2\right )^{5/2}}{x} \, dx=\begin {cases} \frac {x^{4} \sqrt {x^{2} - 10}}{5} - \frac {22 x^{2} \sqrt {x^{2} - 10}}{3} + \frac {460 \sqrt {x^{2} - 10}}{3} - 100 \sqrt {10} i \log {\left (x \right )} + 50 \sqrt {10} i \log {\left (x^{2} \right )} + 100 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {10}}{x} \right )} & \text {for}\: \left |{x^{2}}\right | > 10 \\\frac {i x^{4} \sqrt {10 - x^{2}}}{5} - \frac {22 i x^{2} \sqrt {10 - x^{2}}}{3} + \frac {460 i \sqrt {10 - x^{2}}}{3} + 50 \sqrt {10} i \log {\left (x^{2} \right )} - 100 \sqrt {10} i \log {\left (\sqrt {1 - \frac {x^{2}}{10}} + 1 \right )} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.69 \[ \int \frac {\left (-10+x^2\right )^{5/2}}{x} \, dx=\frac {1}{5} \, {\left (x^{2} - 10\right )}^{\frac {5}{2}} - \frac {10}{3} \, {\left (x^{2} - 10\right )}^{\frac {3}{2}} + 100 \, \sqrt {10} \arcsin \left (\frac {\sqrt {10}}{{\left | x \right |}}\right ) + 100 \, \sqrt {x^{2} - 10} \]
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Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-10+x^2\right )^{5/2}}{x} \, dx=\frac {1}{5} \, {\left (x^{2} - 10\right )}^{\frac {5}{2}} - \frac {10}{3} \, {\left (x^{2} - 10\right )}^{\frac {3}{2}} - 100 \, \sqrt {10} \arctan \left (\frac {1}{10} \, \sqrt {10} \sqrt {x^{2} - 10}\right ) + 100 \, \sqrt {x^{2} - 10} \]
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Time = 0.49 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-10+x^2\right )^{5/2}}{x} \, dx=100\,\sqrt {x^2-10}-100\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\sqrt {x^2-10}}{10}\right )-\frac {10\,{\left (x^2-10\right )}^{3/2}}{3}+\frac {{\left (x^2-10\right )}^{5/2}}{5} \]
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