3.1542 \(\int \frac{1}{\sqrt{-3-b x} \sqrt{2+b x}} \, dx\)

Optimal. Leaf size=26 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{-b x-3}}{\sqrt{b x+2}}\right )}{b} \]

[Out]

(-2*ArcTan[Sqrt[-3 - b*x]/Sqrt[2 + b*x]])/b

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Rubi [A]  time = 0.0332318, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{-b x-3}}{\sqrt{b x+2}}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[-3 - b*x]*Sqrt[2 + b*x]),x]

[Out]

(-2*ArcTan[Sqrt[-3 - b*x]/Sqrt[2 + b*x]])/b

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Rubi in Sympy [A]  time = 4.589, size = 22, normalized size = 0.85 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{b x + 2}}{\sqrt{- b x - 3}} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atan(sqrt(b*x + 2)/sqrt(-b*x - 3))/b

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Mathematica [A]  time = 0.0132892, size = 33, normalized size = 1.27 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{-b x-3} \sqrt{b x+2}}{b x+3}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[-3 - b*x]*Sqrt[2 + b*x]),x]

[Out]

(-2*ArcTan[(Sqrt[-3 - b*x]*Sqrt[2 + b*x])/(3 + b*x)])/b

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Maple [B]  time = 0.013, size = 66, normalized size = 2.5 \[{1\sqrt{ \left ( -bx-3 \right ) \left ( bx+2 \right ) }\arctan \left ({1\sqrt{{b}^{2}} \left ( x+{\frac{5}{2\,b}} \right ){\frac{1}{\sqrt{-{b}^{2}{x}^{2}-5\,bx-6}}}} \right ){\frac{1}{\sqrt{-bx-3}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x-3)^(1/2)/(b*x+2)^(1/2),x)

[Out]

((-b*x-3)*(b*x+2))^(1/2)/(-b*x-3)^(1/2)/(b*x+2)^(1/2)/(b^2)^(1/2)*arctan((b^2)^(
1/2)*(x+5/2/b)/(-b^2*x^2-5*b*x-6)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 2)*sqrt(-b*x - 3)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.215201, size = 38, normalized size = 1.46 \[ \frac{\arctan \left (\frac{2 \, b x + 5}{2 \, \sqrt{b x + 2} \sqrt{-b x - 3}}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 2)*sqrt(-b*x - 3)),x, algorithm="fricas")

[Out]

arctan(1/2*(2*b*x + 5)/(sqrt(b*x + 2)*sqrt(-b*x - 3)))/b

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- b x - 3} \sqrt{b x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-b*x-3)**(1/2)/(b*x+2)**(1/2),x)

[Out]

Integral(1/(sqrt(-b*x - 3)*sqrt(b*x + 2)), x)

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GIAC/XCAS [A]  time = 0.216893, size = 20, normalized size = 0.77 \[ -\frac{2 i \, \arcsin \left (i \, \sqrt{b x + 2}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 2)*sqrt(-b*x - 3)),x, algorithm="giac")

[Out]

-2*I*arcsin(I*sqrt(b*x + 2))/b