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

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

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Rubi [A]  time = 0.00, antiderivative size = 28, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\sqrt {3 x+2} \log (3 x+2)}{3 \sqrt {-3 x-2}} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} \frac {(3 x+2) \log (3 x+2)}{3 \sqrt {-(3 x+2)^2}} \end {gather*}

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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IntegrateAlgebraic [F]  time = 1.78, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {-2-3 x} \sqrt {2+3 x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[IntegrateAlgebraic][1/(Sqrt[-2 - 3*x]*Sqrt[2 + 3*x]), x]

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fricas [A]  time = 1.27, size = 1, normalized size = 0.04 \begin {gather*} 0 \end {gather*}

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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giac [C]  time = 1.01, size = 11, normalized size = 0.39 \begin {gather*} -\frac {1}{3} i \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \mathrm {sgn}\relax (x) \end {gather*}

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)

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maple [A]  time = 0.00, size = 23, normalized size = 0.82 \begin {gather*} \frac {\sqrt {3 x +2}\, \ln \left (3 x +2\right )}{3 \sqrt {-3 x -2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 2.99, 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.

[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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mupad [B]  time = 0.22, size = 35, normalized size = 1.25 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {-\sqrt {-3\,x-2}+\sqrt {2}\,1{}\mathrm {i}}{\sqrt {2}-\sqrt {3\,x+2}}\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*atan((2^(1/2)*1i - (- 3*x - 2)^(1/2))/(2^(1/2) - (3*x + 2)^(1/2))))/3

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sympy [C]  time = 1.46, size = 53, normalized size = 1.89 \begin {gather*} \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 {gather*}

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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