Optimal. Leaf size=28 \[ \frac {(2+3 x) \log (2+3 x)}{3 \sqrt {-(2+3 x)^2}} \]
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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 = {253, 15, 29}
\begin {gather*} \frac {(3 x+2) \log (3 x+2)}{3 \sqrt {-(3 x+2)^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 29
Rule 253
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-(2+3 x)^2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt {-x^2}} \, dx,x,2+3 x\right )\\ &=\frac {(2+3 x) \text {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 = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {(2+3 x) \log (2+3 x)}{3 \sqrt {-(2+3 x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 25, normalized size = 0.89
method | result | size |
meijerg | \(-\frac {i \ln \left (1+\frac {3 x}{2}\right )}{3}\) | \(10\) |
default | \(\frac {\left (2+3 x \right ) \ln \left (2+3 x \right )}{3 \sqrt {-\left (2+3 x \right )^{2}}}\) | \(25\) |
risch | \(\frac {\left (2+3 x \right ) \ln \left (2+3 x \right )}{3 \sqrt {-\left (2+3 x \right )^{2}}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.48, size = 6, normalized size = 0.21 \begin {gather*} \frac {1}{3} i \, \log \left (x + \frac {2}{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.36, size = 6, normalized size = 0.21 \begin {gather*} -\frac {1}{3} i \, \log \left (x + \frac {2}{3}\right ) \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}{\sqrt {- \left (3 x + 2\right )^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 1.16, size = 23, normalized size = 0.82 \begin {gather*} \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 )} \end {gather*}
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 \begin {gather*} -\frac {\ln \left (-3\,x-2\right )\,\mathrm {sign}\left (3\,x+2\right )\,1{}\mathrm {i}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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