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

Optimal. Leaf size=48 \[ -\frac{1}{9} \sqrt{-9 x^2-12 x-4}-\frac{2 (3 x+2) \log (3 x+2)}{9 \sqrt{-9 x^2-12 x-4}} \]

[Out]

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

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Rubi [A]  time = 0.0412039, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ -\frac{1}{9} \sqrt{-9 x^2-12 x-4}-\frac{2 (3 x+2) \log (3 x+2)}{9 \sqrt{-9 x^2-12 x-4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 5.20679, size = 48, normalized size = 1. \[ - \frac{2 \left (9 x + 6\right ) \log{\left (3 x + 2 \right )}}{27 \sqrt{- 9 x^{2} - 12 x - 4}} - \frac{\sqrt{- 9 x^{2} - 12 x - 4}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(9*x + 6)*log(3*x + 2)/(27*sqrt(-9*x**2 - 12*x - 4)) - sqrt(-9*x**2 - 12*x -
4)/9

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Mathematica [A]  time = 0.0127721, size = 35, normalized size = 0.73 \[ \frac{(3 x+2) (3 x-2 \log (3 x+2)+2)}{9 \sqrt{-(3 x+2)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 31, normalized size = 0.7 \[ -{\frac{ \left ( 2+3\,x \right ) \left ( -3\,x+2\,\ln \left ( 2+3\,x \right ) \right ) }{9}{\frac{1}{\sqrt{- \left ( 2+3\,x \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-(2+3*x)^2)^(1/2),x)

[Out]

-1/9*(2+3*x)*(-3*x+2*ln(2+3*x))/(-(2+3*x)^2)^(1/2)

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Maxima [A]  time = 0.816452, size = 28, normalized size = 0.58 \[ -\frac{1}{9} \, \sqrt{-9 \, x^{2} - 12 \, x - 4} - \frac{2}{9} i \, \log \left (x + \frac{2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*sqrt(-9*x^2 - 12*x - 4) - 2/9*I*log(x + 2/3)

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Fricas [A]  time = 0.225857, size = 14, normalized size = 0.29 \[ -\frac{1}{3} i \, x + \frac{2}{9} i \, \log \left (x + \frac{2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*I*x + 2/9*I*log(x + 2/3)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/sqrt(-(3*x + 2)**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

undef