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

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

[Out]

ArcSin[4 + x]

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Rubi [A]  time = 0.0146539, 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 \[ \sin ^{-1}(x+4) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[(-3 - x)*(5 + x)],x]

[Out]

ArcSin[4 + x]

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Rubi in Sympy [A]  time = 0.774144, size = 24, normalized size = 6. \[ \operatorname{atan}{\left (- \frac{- 2 x - 8}{2 \sqrt{- x^{2} - 8 x - 15}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((-3-x)*(5+x))**(1/2),x)

[Out]

atan(-(-2*x - 8)/(2*sqrt(-x**2 - 8*x - 15)))

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Mathematica [B]  time = 0.00652221, size = 42, normalized size = 10.5 \[ \frac{2 \sqrt{x+3} \sqrt{x+5} \sinh ^{-1}\left (\frac{\sqrt{x+3}}{\sqrt{2}}\right )}{\sqrt{-(x+3) (x+5)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[(-3 - x)*(5 + x)],x]

[Out]

(2*Sqrt[3 + x]*Sqrt[5 + x]*ArcSinh[Sqrt[3 + x]/Sqrt[2]])/Sqrt[-((3 + x)*(5 + x))
]

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Maple [A]  time = 0.007, size = 5, normalized size = 1.3 \[ \arcsin \left ( 4+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((-3-x)*(5+x))^(1/2),x)

[Out]

arcsin(4+x)

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Maxima [A]  time = 0.749723, size = 11, normalized size = 2.75 \[ -\arcsin \left (-x - 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-(x + 5)*(x + 3)),x, algorithm="maxima")

[Out]

-arcsin(-x - 4)

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Fricas [A]  time = 0.272978, size = 23, normalized size = 5.75 \[ \arctan \left (\frac{x + 4}{\sqrt{-x^{2} - 8 \, x - 15}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-(x + 5)*(x + 3)),x, algorithm="fricas")

[Out]

arctan((x + 4)/sqrt(-x^2 - 8*x - 15))

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

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)

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GIAC/XCAS [A]  time = 0.267951, size = 5, normalized size = 1.25 \[ \arcsin \left (x + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-(x + 5)*(x + 3)),x, algorithm="giac")

[Out]

arcsin(x + 4)