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} \[ -\frac {\tanh ^{-1}\left (\frac {5 x+4}{2 \sqrt {2} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {2}} \]
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 \[ -\frac {\tanh ^{-1}\left (\frac {5 x+4}{2 \sqrt {6 x^2+10 x+4}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 43, normalized size = 1.19 \[ \frac {1}{4} \, \sqrt {2} \log \left (-\frac {4 \, \sqrt {2} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x + 4\right )} - 49 \, x^{2} - 80 \, x - 32}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 61, normalized size = 1.69 \[ -\frac {1}{2} \, \sqrt {2} \log \left ({\left | -\sqrt {3} x + \sqrt {2} + \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}\right ) + \frac {1}{2} \, \sqrt {2} \log \left ({\left | -\sqrt {3} x - \sqrt {2} + \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}\right ) \]
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 \[ -\frac {\sqrt {2}\, \arctanh \left (\frac {\left (5 x +4\right ) \sqrt {2}}{4 \sqrt {3 x^{2}+5 x +2}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.99, size = 35, normalized size = 0.97 \[ -\frac {1}{2} \, \sqrt {2} \log \left (\frac {2 \, \sqrt {2} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | x \right |}} + \frac {4}{{\left | x \right |}} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 29, normalized size = 0.81 \[ -\frac {\sqrt {2}\,\ln \left (\frac {5\,x+2\,\sqrt {6\,x^2+10\,x+4}+4}{x}\right )}{2} \]
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 \[ \int \frac {1}{x \sqrt {\left (x + 1\right ) \left (3 x + 2\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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