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 = 0.0030387, 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, Rules used = {23, 31} \[ \frac{\sqrt{3 x+2} \log (3 x+2)}{3 \sqrt{-3 x-2}} \]

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])

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-2-3 x} \sqrt{2+3 x}} \, dx &=\frac{\sqrt{2+3 x} \int \frac{1}{2+3 x} \, dx}{\sqrt{-2-3 x}}\\ &=\frac{\sqrt{2+3 x} \log (2+3 x)}{3 \sqrt{-2-3 x}}\\ \end{align*}

Mathematica [A]  time = 0.0059376, 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 = 0.002, size = 23, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( 2+3\,x \right ) }{3}\sqrt{2+3\,x}{\frac{1}{\sqrt{-2-3\,x}}}} \end{align*}

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 [C]  time = 1.55498, size = 8, normalized size = 0.29 \begin{align*} \frac{1}{3} i \, \log \left (x + \frac{2}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*I*log(x + 2/3)

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Fricas [A]  time = 1.48654, size = 4, normalized size = 0.14 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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Sympy [C]  time = 1.40529, size = 53, normalized size = 1.89 \begin{align*} \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}\: \frac{1}{\left |{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} \end{align*}

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, 1/Abs(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 [C]  time = 1.06409, size = 15, normalized size = 0.54 \begin{align*} -\frac{1}{3} i \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*I*log(abs(3*x + 2))*sgn(x)