3.1027 \(\int \frac{1}{\sqrt{-2-3 x} \sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=28 \[ \frac{\sqrt{3 x+2} \log (3 x+2)}{3 \sqrt{-3 x-2}} \]

[Out]

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

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Rubi [A]  time = 0.0104922, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\sqrt{3 x+2} \log (3 x+2)}{3 \sqrt{-3 x-2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.81897, size = 26, normalized size = 0.93 \[ \frac{\sqrt{3 x + 2} \log{\left (3 x + 2 \right )}}{3 \sqrt{- 3 x - 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3*x + 2)*log(3*x + 2)/(3*sqrt(-3*x - 2))

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Mathematica [A]  time = 0.00959629, size = 28, normalized size = 1. \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{-(3 x+2)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[-2 - 3*x]*Sqrt[2 + 3*x]),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 = 23, normalized size = 0.8 \[{\frac{\ln \left ( 2+3\,x \right ) }{3}\sqrt{2+3\,x}{\frac{1}{\sqrt{-2-3\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.52115, size = 8, normalized size = 0.29 \[ \frac{1}{3} i \, \log \left (x + \frac{2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*I*log(x + 2/3)

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Fricas [A]  time = 0.211185, size = 1, normalized size = 0.04 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

0

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Sympy [A]  time = 3.92112, size = 53, normalized size = 1.89 \[ \begin{cases} - \frac{i \log{\left (x + \frac{2}{3} \right )}}{3} & \text{for}\: \left |{x + \frac{2}{3}}\right | < 1 \\\frac{i \log{\left (\frac{1}{x + \frac{2}{3}} \right )}}{3} & \text{for}\: \left |{\frac{1}{x + \frac{2}{3}}}\right | < 1 \\\frac{i{G_{2, 2}^{2, 0}\left (\begin{matrix} & 1, 1 \\0, 0 & \end{matrix} \middle |{x + \frac{2}{3}} \right )}}{3} - \frac{i{G_{2, 2}^{0, 2}\left (\begin{matrix} 1, 1 & \\ & 0, 0 \end{matrix} \middle |{x + \frac{2}{3}} \right )}}{3} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-I*log(x + 2/3)/3, Abs(x + 2/3) < 1), (I*log(1/(x + 2/3))/3, Abs(1/(x
+ 2/3)) < 1), (I*meijerg(((), (1, 1)), ((0, 0), ()), x + 2/3)/3 - I*meijerg(((1
, 1), ()), ((), (0, 0)), x + 2/3)/3, True))

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GIAC/XCAS [A]  time = 0.206485, size = 12, normalized size = 0.43 \[ -\frac{1}{3} i \,{\rm ln}\left ({\left | 3 \, x + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*I*ln(abs(3*x + 2))