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

Optimal. Leaf size=29 \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{-9 x^2-12 x-4}} \]

[Out]

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

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Rubi [A]  time = 0.0056675, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {608, 31} \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{-9 x^2-12 x-4}} \]

Antiderivative was successfully verified.

[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 608

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]

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{-4-12 x-9 x^2}} \, dx &=-\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]  time = 0.00384, size = 28, normalized size = 0.97 \[ \frac{(3 x+2) \log (3 x+2)}{3 \sqrt{-(3 x+2)^2}} \]

Antiderivative was successfully verified.

[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])

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Maple [A]  time = 0.122, size = 25, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2+3\,x \right ) \ln \left ( 2+3\,x \right ) }{3}{\frac{1}{\sqrt{- \left ( 2+3\,x \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 1.69936, size = 8, normalized size = 0.28 \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)^2)^(1/2),x, algorithm="maxima")

[Out]

1/3*I*log(x + 2/3)

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Fricas [C]  time = 2.02109, size = 28, normalized size = 0.97 \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)^2)^(1/2),x, algorithm="fricas")

[Out]

-1/3*I*log(x + 2/3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- \left (3 x + 2\right )^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-(3*x + 2)**2), x)

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Giac [C]  time = 1.3579, size = 31, normalized size = 1.07 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)