3.1543 \(\int \frac{1}{\sqrt{2-b x} \sqrt{3-b x}} \, dx\)

Optimal. Leaf size=16 \[ -\frac{2 \sinh ^{-1}\left (\sqrt{2-b x}\right )}{b} \]

[Out]

(-2*ArcSinh[Sqrt[2 - b*x]])/b

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Rubi [A]  time = 0.0259772, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{2 \sinh ^{-1}\left (\sqrt{2-b x}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

(-2*ArcSinh[Sqrt[2 - b*x]])/b

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Rubi in Sympy [A]  time = 6.87606, size = 14, normalized size = 0.88 \[ - \frac{2 \operatorname{asinh}{\left (\sqrt{- b x + 2} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*asinh(sqrt(-b*x + 2))/b

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Mathematica [A]  time = 0.0120554, size = 16, normalized size = 1. \[ -\frac{2 \sinh ^{-1}\left (\sqrt{2-b x}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

(-2*ArcSinh[Sqrt[2 - b*x]])/b

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Maple [B]  time = 0.008, size = 70, normalized size = 4.4 \[{1\sqrt{ \left ( -bx+2 \right ) \left ( -bx+3 \right ) }\ln \left ({1 \left ( -{\frac{5\,b}{2}}+{b}^{2}x \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}-5\,bx+6} \right ){\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{-bx+3}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((-b*x+2)*(-b*x+3))^(1/2)/(-b*x+2)^(1/2)/(-b*x+3)^(1/2)*ln((-5/2*b+b^2*x)/(b^2)^
(1/2)+(b^2*x^2-5*b*x+6)^(1/2))/(b^2)^(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 + 3)*sqrt(-b*x + 2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.208236, size = 41, normalized size = 2.56 \[ -\frac{\log \left (-2 \, b x + 2 \, \sqrt{-b x + 3} \sqrt{-b x + 2} + 5\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273011, size = 35, normalized size = 2.19 \[ \frac{2 \,{\rm ln}\left ({\left | -\sqrt{-b x + 3} + \sqrt{-b x + 2} \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*ln(abs(-sqrt(-b*x + 3) + sqrt(-b*x + 2)))/b