Optimal. Leaf size=28 \[ \frac {(3 x+2) \log (3 x+2)}{3 \sqrt {-(3 x+2)^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {247, 15, 29} \[ \frac {(3 x+2) \log (3 x+2)}{3 \sqrt {-(3 x+2)^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 29
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-(2+3 x)^2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-x^2}} \, dx,x,2+3 x\right )\\ &=\frac {(2+3 x) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,2+3 x\right )}{3 \sqrt {-(2+3 x)^2}}\\ &=\frac {(2+3 x) \log (2+3 x)}{3 \sqrt {-(2+3 x)^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \[ \frac {(3 x+2) \log (3 x+2)}{3 \sqrt {-(3 x+2)^2}} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.81, size = 6, normalized size = 0.21 \[ -\frac {1}{3} i \, \log \left (x + \frac {2}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.24, size = 23, normalized size = 0.82 \[ \frac {i \, \log \left ({\left (-3 i \, x - 2 i\right )} \mathrm {sgn}\left (-3 \, x - 2\right )\right )}{3 \, \mathrm {sgn}\left (-3 \, x - 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.89 \[ \frac {\left (3 x +2\right ) \ln \left (3 x +2\right )}{3 \sqrt {-\left (3 x +2\right )^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.29, size = 6, normalized size = 0.21 \[ \frac {1}{3} i \, \log \left (x + \frac {2}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 15, normalized size = 0.54 \[ -\frac {\ln \left (-3\,x-2\right )\,\mathrm {sign}\left (3\,x+2\right )\,1{}\mathrm {i}}{3} \]
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 \[ \int \frac {1}{\sqrt {- \left (3 x + 2\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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