Integrand size = 18, antiderivative size = 88 \[ \int \frac {\sqrt {-2-3 x+5 x^2}}{x} \, dx=\sqrt {-2-3 x+5 x^2}+\sqrt {2} \arctan \left (\frac {4+3 x}{2 \sqrt {2} \sqrt {-2-3 x+5 x^2}}\right )+\frac {3 \text {arctanh}\left (\frac {3-10 x}{2 \sqrt {5} \sqrt {-2-3 x+5 x^2}}\right )}{2 \sqrt {5}} \]
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Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {748, 857, 635, 212, 738, 210} \[ \int \frac {\sqrt {-2-3 x+5 x^2}}{x} \, dx=\sqrt {2} \arctan \left (\frac {3 x+4}{2 \sqrt {2} \sqrt {5 x^2-3 x-2}}\right )+\frac {3 \text {arctanh}\left (\frac {3-10 x}{2 \sqrt {5} \sqrt {5 x^2-3 x-2}}\right )}{2 \sqrt {5}}+\sqrt {5 x^2-3 x-2} \]
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Rule 210
Rule 212
Rule 635
Rule 738
Rule 748
Rule 857
Rubi steps \begin{align*} \text {integral}& = \sqrt {-2-3 x+5 x^2}-\frac {1}{2} \int \frac {4+3 x}{x \sqrt {-2-3 x+5 x^2}} \, dx \\ & = \sqrt {-2-3 x+5 x^2}-\frac {3}{2} \int \frac {1}{\sqrt {-2-3 x+5 x^2}} \, dx-2 \int \frac {1}{x \sqrt {-2-3 x+5 x^2}} \, dx \\ & = \sqrt {-2-3 x+5 x^2}-3 \text {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-3+10 x}{\sqrt {-2-3 x+5 x^2}}\right )+4 \text {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,\frac {-4-3 x}{\sqrt {-2-3 x+5 x^2}}\right ) \\ & = \sqrt {-2-3 x+5 x^2}+\sqrt {2} \tan ^{-1}\left (\frac {4+3 x}{2 \sqrt {2} \sqrt {-2-3 x+5 x^2}}\right )+\frac {3 \tanh ^{-1}\left (\frac {3-10 x}{2 \sqrt {5} \sqrt {-2-3 x+5 x^2}}\right )}{2 \sqrt {5}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {-2-3 x+5 x^2}}{x} \, dx=\sqrt {-2-3 x+5 x^2}+2 \sqrt {2} \arctan \left (\frac {\sqrt {-2-3 x+5 x^2}}{\sqrt {2} (-1+x)}\right )-\frac {3 \text {arctanh}\left (\frac {\sqrt {5} \sqrt {-2-3 x+5 x^2}}{2+5 x}\right )}{\sqrt {5}} \]
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Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.81
method | result | size |
default | \(\sqrt {5 x^{2}-3 x -2}-\frac {3 \ln \left (\frac {\left (-\frac {3}{2}+5 x \right ) \sqrt {5}}{5}+\sqrt {5 x^{2}-3 x -2}\right ) \sqrt {5}}{10}-\sqrt {2}\, \arctan \left (\frac {\left (-3 x -4\right ) \sqrt {2}}{4 \sqrt {5 x^{2}-3 x -2}}\right )\) | \(71\) |
trager | \(\sqrt {5 x^{2}-3 x -2}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (10 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {5 x^{2}-3 x -2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )\right )}{10}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x +4 \sqrt {5 x^{2}-3 x -2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{x}\right )\) | \(99\) |
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Time = 0.37 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {-2-3 x+5 x^2}}{x} \, dx=\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (3 \, x + 4\right )}}{4 \, \sqrt {5 \, x^{2} - 3 \, x - 2}}\right ) + \frac {3}{20} \, \sqrt {5} \log \left (-4 \, \sqrt {5} \sqrt {5 \, x^{2} - 3 \, x - 2} {\left (10 \, x - 3\right )} + 200 \, x^{2} - 120 \, x - 31\right ) + \sqrt {5 \, x^{2} - 3 \, x - 2} \]
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\[ \int \frac {\sqrt {-2-3 x+5 x^2}}{x} \, dx=\int \frac {\sqrt {\left (x - 1\right ) \left (5 x + 2\right )}}{x}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {-2-3 x+5 x^2}}{x} \, dx=\sqrt {2} \arcsin \left (\frac {3 \, x}{7 \, {\left | x \right |}} + \frac {4}{7 \, {\left | x \right |}}\right ) - \frac {3}{10} \, \sqrt {5} \log \left (2 \, \sqrt {5} \sqrt {5 \, x^{2} - 3 \, x - 2} + 10 \, x - 3\right ) + \sqrt {5 \, x^{2} - 3 \, x - 2} \]
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Time = 0.35 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {-2-3 x+5 x^2}}{x} \, dx=-2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {5} x - \sqrt {5 \, x^{2} - 3 \, x - 2}\right )}\right ) + \frac {3}{10} \, \sqrt {5} \log \left ({\left | -10 \, \sqrt {5} x + 3 \, \sqrt {5} + 10 \, \sqrt {5 \, x^{2} - 3 \, x - 2} \right |}\right ) + \sqrt {5 \, x^{2} - 3 \, x - 2} \]
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Time = 0.22 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {-2-3 x+5 x^2}}{x} \, dx=\sqrt {5\,x^2-3\,x-2}-\frac {3\,\sqrt {5}\,\ln \left (\sqrt {5\,x^2-3\,x-2}+\frac {\sqrt {5}\,\left (5\,x-\frac {3}{2}\right )}{5}\right )}{10}-\sqrt {2}\,\ln \left (-\frac {2}{x}-\frac {3}{2}+\frac {\sqrt {2}\,\sqrt {5\,x^2-3\,x-2}\,1{}\mathrm {i}}{x}\right )\,1{}\mathrm {i} \]
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