\(\int \frac {1}{\sqrt {-2-5 x-3 x^2}} \, dx\) [237]

Optimal result

Integrand size = 14, antiderivative size = 12 \[ \int \frac {1}{\sqrt {-2-5 x-3 x^2}} \, dx=\frac {\arcsin (5+6 x)}{\sqrt {3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, 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 = {633, 222} \[ \int \frac {1}{\sqrt {-2-5 x-3 x^2}} \, dx=\frac {\arcsin (6 x+5)}{\sqrt {3}} \]

[In]

Int[1/Sqrt[-2 - 5*x - 3*x^2],x]

[Out]

ArcSin[5 + 6*x]/Sqrt[3]

Rule 222

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 633

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(33\) vs. \(2(12)=24\).

Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.75 \[ \int \frac {1}{\sqrt {-2-5 x-3 x^2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {-2-5 x-3 x^2}}{\sqrt {3} (1+x)}\right )}{\sqrt {3}} \]

[In]

Integrate[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]

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (11) = 22\).

Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 3.33 \[ \int \frac {1}{\sqrt {-2-5 x-3 x^2}} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-3 \, x^{2} - 5 \, x - 2} {\left (6 \, x + 5\right )}}{6 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}}\right ) \]

[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))

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {-2-5 x-3 x^2}} \, dx=\frac {\sqrt {3} \operatorname {asin}{\left (6 x + 5 \right )}}{3} \]

[In]

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

[Out]

sqrt(3)*asin(6*x + 5)/3

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {-2-5 x-3 x^2}} \, dx=\frac {1}{3} \, \sqrt {3} \arcsin \left (6 \, x + 5\right ) \]

[In]

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

[Out]

1/3*sqrt(3)*arcsin(6*x + 5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (11) = 22\).

Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.58 \[ \int \frac {1}{\sqrt {-2-5 x-3 x^2}} \, dx=\frac {1}{12} \, \sqrt {-3 \, x^{2} - 5 \, x - 2} {\left (6 \, x + 5\right )} + \frac {1}{72} \, \sqrt {3} \arcsin \left (6 \, x + 5\right ) \]

[In]

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

[Out]

1/12*sqrt(-3*x^2 - 5*x - 2)*(6*x + 5) + 1/72*sqrt(3)*arcsin(6*x + 5)

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {-2-5 x-3 x^2}} \, dx=\frac {\sqrt {3}\,\mathrm {asin}\left (6\,x+5\right )}{3} \]

[In]

int(1/(- 5*x - 3*x^2 - 2)^(1/2),x)

[Out]

(3^(1/2)*asin(6*x + 5))/3