\(\int \frac {1}{\sqrt {-2-3 x} \sqrt {2+3 x}} \, dx\) [1027]

Optimal result

Integrand size = 19, antiderivative size = 28 \[ \int \frac {1}{\sqrt {-2-3 x} \sqrt {2+3 x}} \, dx=\frac {\sqrt {2+3 x} \log (2+3 x)}{3 \sqrt {-2-3 x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {23, 31} \[ \int \frac {1}{\sqrt {-2-3 x} \sqrt {2+3 x}} \, dx=\frac {\sqrt {3 x+2} \log (3 x+2)}{3 \sqrt {-3 x-2}} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-2-3 x} \sqrt {2+3 x}} \, dx=\frac {(2+3 x) \log (2+3 x)}{3 \sqrt {-(2+3 x)^2}} \]

[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])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.36

method	result	size
meijerg	\(-\frac {i \ln \left (1+\frac {3 x}{2}\right )}{3}\)	\(10\)
default	\(\frac {\ln \left (2+3 x \right ) \sqrt {2+3 x}}{3 \sqrt {-2-3 x}}\)	\(23\)
risch	\(-\frac {i \sqrt {\frac {-2-3 x}{2+3 x}}\, \sqrt {2+3 x}\, \ln \left (2+3 x \right )}{3 \sqrt {-2-3 x}}\)	\(39\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.23 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.29 \[ \int \frac {1}{\sqrt {-2-3 x} \sqrt {2+3 x}} \, dx=-\frac {1}{3} i \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

-1/3*I*log(3*x + 2)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.65 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36 \[ \int \frac {1}{\sqrt {-2-3 x} \sqrt {2+3 x}} \, dx=\begin {cases} 0 & \text {for}\: \frac {1}{\left |{x + \frac {2}{3}}\right |} < 1 \wedge \left |{x + \frac {2}{3}}\right | < 1 \\- \frac {i \log {\left (x + \frac {2}{3} \right )}}{3} & \text {for}\: \left |{x + \frac {2}{3}}\right | < 1 \\\frac {i \log {\left (\frac {1}{x + \frac {2}{3}} \right )}}{3} & \text {for}\: \frac {1}{\left |{x + \frac {2}{3}}\right |} < 1 \\\frac {i {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x + \frac {2}{3}} \right )}}{3} - \frac {i {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x + \frac {2}{3}} \right )}}{3} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((0, (Abs(x + 2/3) < 1) & (1/Abs(x + 2/3) < 1)), (-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))

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.21 \[ \int \frac {1}{\sqrt {-2-3 x} \sqrt {2+3 x}} \, dx=\frac {1}{3} i \, \log \left (x + \frac {2}{3}\right ) \]

[In]

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

[Out]

1/3*I*log(x + 2/3)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.32 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\sqrt {-2-3 x} \sqrt {2+3 x}} \, dx=-\frac {1}{3} i \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \mathrm {sgn}\left (x\right ) \]

[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)

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\sqrt {-2-3 x} \sqrt {2+3 x}} \, dx=-\frac {4\,\mathrm {atan}\left (\frac {-\sqrt {-3\,x-2}+\sqrt {2}\,1{}\mathrm {i}}{\sqrt {2}-\sqrt {3\,x+2}}\right )}{3} \]

[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