\(\int \frac {1}{\sqrt {-4-12 x-9 x^2}} \, dx\) [73]

Optimal result

Integrand size = 14, antiderivative size = 29 \[ \int \frac {1}{\sqrt {-4-12 x-9 x^2}} \, dx=\frac {(2+3 x) \log (2+3 x)}{3 \sqrt {-4-12 x-9 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {622, 31} \[ \int \frac {1}{\sqrt {-4-12 x-9 x^2}} \, dx=\frac {(3 x+2) \log (3 x+2)}{3 \sqrt {-9 x^2-12 x-4}} \]

[In]

Int[1/Sqrt[-4 - 12*x - 9*x^2],x]

[Out]

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

Rule 31

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

Rule 622

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(b/2 + c*x)/Sqrt[a + b*x + c*x^2], Int[1/(b/2
+ c*x), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = -\left (-\frac {(-6-9 x) \int \frac {1}{-6-9 x} \, dx}{\sqrt {-4-12 x-9 x^2}}\right ) \\ & = \frac {(2+3 x) \log (2+3 x)}{3 \sqrt {-4-12 x-9 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\sqrt {-4-12 x-9 x^2}} \, dx=\frac {(2+3 x) \log (2+3 x)}{3 \sqrt {-(2+3 x)^2}} \]

[In]

Integrate[1/Sqrt[-4 - 12*x - 9*x^2],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 = 2.22 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.34

method	result	size
meijerg	\(-\frac {i \ln \left (1+\frac {3 x}{2}\right )}{3}\)	\(10\)
default	\(\frac {\left (2+3 x \right ) \ln \left (2+3 x \right )}{3 \sqrt {-\left (2+3 x \right )^{2}}}\)	\(25\)
risch	\(\frac {\left (2+3 x \right ) \ln \left (2+3 x \right )}{3 \sqrt {-\left (2+3 x \right )^{2}}}\)	\(25\)

[In]

int(1/(-(2+3*x)^2)^(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 = 0.26 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.21 \[ \int \frac {1}{\sqrt {-4-12 x-9 x^2}} \, dx=-\frac {1}{3} i \, \log \left (x + \frac {2}{3}\right ) \]

[In]

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

[Out]

-1/3*I*log(x + 2/3)

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {-4-12 x-9 x^2}} \, dx=\frac {\left (x + \frac {2}{3}\right ) \log {\left (x + \frac {2}{3} \right )}}{3 \sqrt {- \left (x + \frac {2}{3}\right )^{2}}} \]

[In]

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

[Out]

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

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 \[ \int \frac {1}{\sqrt {-4-12 x-9 x^2}} \, dx=\frac {1}{3} i \, \log \left (x + \frac {2}{3}\right ) \]

[In]

integrate(1/(-(2+3*x)^2)^(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 = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt {-4-12 x-9 x^2}} \, dx=\frac {i \, \log \left ({\left (-3 i \, x - 2 i\right )} \mathrm {sgn}\left (-3 \, x - 2\right )\right )}{3 \, \mathrm {sgn}\left (-3 \, x - 2\right )} \]

[In]

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

[Out]

1/3*I*log((-3*I*x - 2*I)*sgn(-3*x - 2))/sgn(-3*x - 2)

Mupad [B] (verification not implemented)

Time = 9.12 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\sqrt {-4-12 x-9 x^2}} \, dx=-\frac {\ln \left (-3\,x-2\right )\,\mathrm {sign}\left (3\,x+2\right )\,1{}\mathrm {i}}{3} \]

[In]

int(1/(-(3*x + 2)^2)^(1/2),x)

[Out]

-(log(- 3*x - 2)*sign(3*x + 2)*1i)/3