3.847 \(\int \frac{1}{\sqrt{(-3-x) (5+x)}} \, dx\)

Optimal. Leaf size=4 \[ \sin ^{-1}(x+4) \]

[Out]

ArcSin[4 + x]

________________________________________________________________________________________

Rubi [A]  time = 0.0070078, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1981, 619, 216} \[ \sin ^{-1}(x+4) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(-3 - x)*(5 + x)],x]

[Out]

ArcSin[4 + x]

Rule 1981

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && QuadraticQ[u, x] &&  !QuadraticMatch
Q[u, x]

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{(-3-x) (5+x)}} \, dx &=\int \frac{1}{\sqrt{-15-8 x-x^2}} \, dx\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4}}} \, dx,x,-8-2 x\right )\right )\\ &=\sin ^{-1}(4+x)\\ \end{align*}

Mathematica [B]  time = 0.0027375, size = 18, normalized size = 4.5 \[ -2 \sin ^{-1}\left (\frac{\sqrt{-x-3}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[(-3 - x)*(5 + x)],x]

[Out]

-2*ArcSin[Sqrt[-3 - x]/Sqrt[2]]

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 5, normalized size = 1.3 \begin{align*} \arcsin \left ( 4+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(4+x)

________________________________________________________________________________________

Maxima [B]  time = 2.18335, size = 11, normalized size = 2.75 \begin{align*} -\arcsin \left (-x - 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arcsin(-x - 4)

________________________________________________________________________________________

Fricas [B]  time = 1.48181, size = 77, normalized size = 19.25 \begin{align*} -\arctan \left (\frac{\sqrt{-x^{2} - 8 \, x - 15}{\left (x + 4\right )}}{x^{2} + 8 \, x + 15}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(sqrt(-x^2 - 8*x - 15)*(x + 4)/(x^2 + 8*x + 15))

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\left (- x - 3\right ) \left (x + 5\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt((-x - 3)*(x + 5)), x)

________________________________________________________________________________________

Giac [A]  time = 1.13548, size = 5, normalized size = 1.25 \begin{align*} \arcsin \left (x + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsin(x + 4)