3.73 \(\int \frac{1}{\sqrt{-4-12 x-9 x^2}} \, dx\)

Optimal. Leaf size=29 \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{-9 x^2-12 x-4}} \]

[Out]

((2 + 3*x)*Log[2 + 3*x])/(3*Sqrt[-4 - 12*x - 9*x^2])

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Rubi [A]  time = 0.0174394, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{-9 x^2-12 x-4}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-4 - 12*x - 9*x^2],x]

[Out]

((2 + 3*x)*Log[2 + 3*x])/(3*Sqrt[-4 - 12*x - 9*x^2])

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Rubi in Sympy [A]  time = 1.88976, size = 27, normalized size = 0.93 \[ \frac{\left (9 x + 6\right ) \log{\left (3 x + 2 \right )}}{9 \sqrt{- 9 x^{2} - 12 x - 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-(2+3*x)**2)**(1/2),x)

[Out]

(9*x + 6)*log(3*x + 2)/(9*sqrt(-9*x**2 - 12*x - 4))

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Mathematica [A]  time = 0.00728601, size = 28, normalized size = 0.97 \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{-(3 x+2)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[-4 - 12*x - 9*x^2],x]

[Out]

((2 + 3*x)*Log[2 + 3*x])/(3*Sqrt[-(2 + 3*x)^2])

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Maple [A]  time = 0.003, size = 25, normalized size = 0.9 \[{\frac{ \left ( 2+3\,x \right ) \ln \left ( 2+3\,x \right ) }{3}{\frac{1}{\sqrt{- \left ( 2+3\,x \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-(2+3*x)^2)^(1/2),x)

[Out]

1/3*(2+3*x)*ln(2+3*x)/(-(2+3*x)^2)^(1/2)

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Maxima [A]  time = 0.795971, size = 8, normalized size = 0.28 \[ \frac{1}{3} i \, \log \left (x + \frac{2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-(3*x + 2)^2),x, algorithm="maxima")

[Out]

1/3*I*log(x + 2/3)

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Fricas [A]  time = 0.21563, size = 8, normalized size = 0.28 \[ -\frac{1}{3} i \, \log \left (x + \frac{2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-(3*x + 2)^2),x, algorithm="fricas")

[Out]

-1/3*I*log(x + 2/3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- \left (3 x + 2\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-(2+3*x)**2)**(1/2),x)

[Out]

Integral(1/sqrt(-(3*x + 2)**2), x)

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GIAC/XCAS [A]  time = 0.21228, size = 35, normalized size = 1.21 \[ \frac{i{\rm ln}\left (-i{\left (3 \, x + 2\right )}{\rm sign}\left (-3 \, x - 2\right )\right )}{3 \,{\rm sign}\left (-3 \, x - 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-(3*x + 2)^2),x, algorithm="giac")

[Out]

1/3*i*ln(-i*(3*x + 2)*sign(-3*x - 2))/sign(-3*x - 2)