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

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

[Out]

ArcSin[4 + x]

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Rubi [A]  time = 0.0059976, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {53, 619, 216} \[ \sin ^{-1}(x+4) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-3 - x]*Sqrt[5 + x]),x]

[Out]

ArcSin[4 + x]

Rule 53

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Int[1/Sqrt[a*c - b*(a - c)*x - b^2*x^2]
, x] /; FreeQ[{a, b, c, d}, x] && EqQ[b + d, 0] && GtQ[a + c, 0]

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} \sqrt{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 = 0.0103522, 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]*Sqrt[5 + x]),x]

[Out]

-2*ArcSin[Sqrt[-3 - x]/Sqrt[2]]

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Maple [B]  time = 0.006, size = 29, normalized size = 7.3 \begin{align*}{\arcsin \left ( 4+x \right ) \sqrt{ \left ( -3-x \right ) \left ( 5+x \right ) }{\frac{1}{\sqrt{-3-x}}}{\frac{1}{\sqrt{5+x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((-3-x)*(5+x))^(1/2)/(-3-x)^(1/2)/(5+x)^(1/2)*arcsin(4+x)

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Maxima [B]  time = 2.03335, 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)^(1/2)/(5+x)^(1/2),x, algorithm="maxima")

[Out]

-arcsin(-x - 4)

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Fricas [B]  time = 1.51496, size = 81, normalized size = 20.25 \begin{align*} -\arctan \left (\frac{\sqrt{x + 5}{\left (x + 4\right )} \sqrt{-x - 3}}{x^{2} + 8 \, x + 15}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(sqrt(x + 5)*(x + 4)*sqrt(-x - 3)/(x^2 + 8*x + 15))

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Sympy [B]  time = 1.01342, size = 41, normalized size = 10.25 \begin{align*} \begin{cases} - 2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 5}}{2} \right )} & \text{for}\: \frac{\left |{x + 5}\right |}{2} > 1 \\2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 5}}{2} \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*I*acosh(sqrt(2)*sqrt(x + 5)/2), Abs(x + 5)/2 > 1), (2*asin(sqrt(2)*sqrt(x + 5)/2), True))

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Giac [B]  time = 1.1544, size = 18, normalized size = 4.5 \begin{align*} 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arcsin(1/2*sqrt(2)*sqrt(x + 5))