Optimal. Leaf size=88 \[ \sqrt {-2-3 x+5 x^2}+\sqrt {2} \tan ^{-1}\left (\frac {4+3 x}{2 \sqrt {2} \sqrt {-2-3 x+5 x^2}}\right )+\frac {3 \tanh ^{-1}\left (\frac {3-10 x}{2 \sqrt {5} \sqrt {-2-3 x+5 x^2}}\right )}{2 \sqrt {5}} \]
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Rubi [A]
time = 0.03, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {748, 857, 635,
212, 738, 210} \begin {gather*} \sqrt {2} \text {ArcTan}\left (\frac {3 x+4}{2 \sqrt {2} \sqrt {5 x^2-3 x-2}}\right )+\sqrt {5 x^2-3 x-2}+\frac {3 \tanh ^{-1}\left (\frac {3-10 x}{2 \sqrt {5} \sqrt {5 x^2-3 x-2}}\right )}{2 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 635
Rule 738
Rule 748
Rule 857
Rubi steps
\begin {align*} \int \frac {\sqrt {-2-3 x+5 x^2}}{x} \, dx &=\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*}
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Mathematica [A]
time = 0.13, size = 83, normalized size = 0.94 \begin {gather*} \sqrt {-2-3 x+5 x^2}+2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {-2-3 x+5 x^2}}{\sqrt {2} (-1+x)}\right )-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {5} \sqrt {-2-3 x+5 x^2}}{2+5 x}\right )}{\sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.69, size = 71, normalized size = 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 | \(\sqrt {5 x^{2}-3 x -2}+\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {3 \RootOf \left (\textit {\_Z}^{2}+2\right ) x +4 \RootOf \left (\textit {\_Z}^{2}+2\right )-4 \sqrt {5 x^{2}-3 x -2}}{x}\right )+\frac {3 \RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (-10 \RootOf \left (\textit {\_Z}^{2}-5\right ) x +3 \RootOf \left (\textit {\_Z}^{2}-5\right )+10 \sqrt {5 x^{2}-3 x -2}\right )}{10}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 60, normalized size = 0.68 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.03, size = 78, normalized size = 0.89 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\left (x - 1\right ) \left (5 x + 2\right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.22, size = 77, normalized size = 0.88 \begin {gather*} -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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 77, normalized size = 0.88 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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