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

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

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \[ \frac {\sin ^{-1}(6 x+5)}{\sqrt {3}} \]

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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IntegrateAlgebraic [B]  time = 0.10, size = 33, normalized size = 2.75 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {-3 x^2-5 x-2}}{\sqrt {3} (x+1)}\right )}{\sqrt {3}} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[-2 - 5*x - 3*x^2],x]

[Out]

(-2*ArcTan[Sqrt[-2 - 5*x - 3*x^2]/(Sqrt[3]*(1 + x))])/Sqrt[3]

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fricas [B]  time = 0.63, size = 40, normalized size = 3.33 \[ -\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-3 \, x^{2} - 5 \, x - 2} {\left (6 \, x + 5\right )}}{6 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}}\right ) \]

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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giac [A]  time = 0.63, size = 11, normalized size = 0.92 \[ \frac {1}{3} \, \sqrt {3} \arcsin \left (6 \, x + 5\right ) \]

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)

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maple [A]  time = 0.29, size = 12, normalized size = 1.00

method	result	size
default	\(\frac {\arcsin \left (5+6 x \right ) \sqrt {3}}{3}\)	\(12\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-6 \RootOf \left (\textit {\_Z}^{2}+3\right ) x -5 \RootOf \left (\textit {\_Z}^{2}+3\right )+6 \sqrt {-3 x^{2}-5 x -2}\right )}{3}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.10, size = 11, normalized size = 0.92 \[ \frac {1}{3} \, \sqrt {3} \arcsin \left (6 \, x + 5\right ) \]

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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mupad [B]  time = 0.22, size = 11, normalized size = 0.92 \[ \frac {\sqrt {3}\,\mathrm {asin}\left (6\,x+5\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(- 5*x - 3*x^2 - 2)^(1/2),x)

[Out]

(3^(1/2)*asin(6*x + 5))/3

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- 3 x^{2} - 5 x - 2}}\, dx \]

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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