3.237 \(\int \frac{1}{\sqrt{-2-5 x-3 x^2}} \, dx\)

Optimal. Leaf size=12 \[ \frac{\sin ^{-1}(6 x+5)}{\sqrt{3}} \]

[Out]

ArcSin[5 + 6*x]/Sqrt[3]

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Rubi [A]  time = 0.0044271, antiderivative size = 12, 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 = {619, 216} \[ \frac{\sin ^{-1}(6 x+5)}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-2 - 5*x - 3*x^2],x]

[Out]

ArcSin[5 + 6*x]/Sqrt[3]

Rule 619

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

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-2-5 x-3 x^2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,-5-6 x\right )}{\sqrt{3}}\\ &=\frac{\sin ^{-1}(5+6 x)}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0068594, size = 12, normalized size = 1. \[ \frac{\sin ^{-1}(6 x+5)}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-2 - 5*x - 3*x^2],x]

[Out]

ArcSin[5 + 6*x]/Sqrt[3]

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Maple [A]  time = 0.004, size = 12, normalized size = 1. \begin{align*}{\frac{\arcsin \left ( 6\,x+5 \right ) \sqrt{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-3*x^2-5*x-2)^(1/2),x)

[Out]

1/3*arcsin(6*x+5)*3^(1/2)

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Maxima [A]  time = 1.42477, size = 15, normalized size = 1.25 \begin{align*} \frac{1}{3} \, \sqrt{3} \arcsin \left (6 \, x + 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2-5*x-2)^(1/2),x, algorithm="maxima")

[Out]

1/3*sqrt(3)*arcsin(6*x + 5)

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Fricas [B]  time = 2.03931, size = 115, normalized size = 9.58 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{\sqrt{3} \sqrt{-3 \, x^{2} - 5 \, x - 2}{\left (6 \, x + 5\right )}}{6 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2-5*x-2)^(1/2),x, algorithm="fricas")

[Out]

-1/3*sqrt(3)*arctan(1/6*sqrt(3)*sqrt(-3*x^2 - 5*x - 2)*(6*x + 5)/(3*x^2 + 5*x + 2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- 3 x^{2} - 5 x - 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x**2-5*x-2)**(1/2),x)

[Out]

Integral(1/sqrt(-3*x**2 - 5*x - 2), x)

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Giac [A]  time = 1.06858, size = 15, normalized size = 1.25 \begin{align*} \frac{1}{3} \, \sqrt{3} \arcsin \left (6 \, x + 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-3*x^2-5*x-2)^(1/2),x, algorithm="giac")

[Out]

1/3*sqrt(3)*arcsin(6*x + 5)