Optimal. Leaf size=36 \[ -\frac {\tanh ^{-1}\left (\frac {5 x+4}{2 \sqrt {2} \sqrt {-3 x^2+5 x+2}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {724, 206} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {5 x+4}{2 \sqrt {2} \sqrt {-3 x^2+5 x+2}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {2+5 x-3 x^2}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {4+5 x}{\sqrt {2+5 x-3 x^2}}\right )\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {4+5 x}{2 \sqrt {2} \sqrt {2+5 x-3 x^2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.86 \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {5 x+4}{2 \sqrt {-6 x^2+10 x+4}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 32, normalized size = 0.89 \begin {gather*} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {-3 x^2+5 x+2}}{x-2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 43, normalized size = 1.19 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (-\frac {4 \, \sqrt {2} \sqrt {-3 \, x^{2} + 5 \, x + 2} {\left (5 \, x + 4\right )} - x^{2} - 80 \, x - 32}{x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 84, normalized size = 2.33 \begin {gather*} -\frac {1}{6} \, \sqrt {6} \sqrt {3} \log \left (\frac {{\left | -4 \, \sqrt {6} + \frac {10 \, {\left (2 \, \sqrt {3} \sqrt {-3 \, x^{2} + 5 \, x + 2} - 7\right )}}{6 \, x - 5} - 14 \right |}}{{\left | 4 \, \sqrt {6} + \frac {10 \, {\left (2 \, \sqrt {3} \sqrt {-3 \, x^{2} + 5 \, x + 2} - 7\right )}}{6 \, x - 5} - 14 \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 29, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {2}\, \arctanh \left (\frac {\left (5 x +4\right ) \sqrt {2}}{4 \sqrt {-3 x^{2}+5 x +2}}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.87, size = 35, normalized size = 0.97 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \log \left (\frac {2 \, \sqrt {2} \sqrt {-3 \, x^{2} + 5 \, x + 2}}{{\left | x \right |}} + \frac {4}{{\left | x \right |}} + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 29, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {2}\,\ln \left (\frac {5\,x+2\,\sqrt {-6\,x^2+10\,x+4}+4}{x}\right )}{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 {1}{x \sqrt {- \left (x - 2\right ) \left (3 x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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