3.1546 \(\int \frac{1}{\sqrt{-b x} \sqrt{2-b x}} \, dx\)

Optimal. Leaf size=20 \[ -\frac{2 \sinh ^{-1}\left (\frac{\sqrt{-b x}}{\sqrt{2}}\right )}{b} \]

[Out]

(-2*ArcSinh[Sqrt[-(b*x)]/Sqrt[2]])/b

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Rubi [A]  time = 0.0209205, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{2 \sinh ^{-1}\left (\frac{\sqrt{-b x}}{\sqrt{2}}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

(-2*ArcSinh[Sqrt[-(b*x)]/Sqrt[2]])/b

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Rubi in Sympy [A]  time = 5.34128, size = 20, normalized size = 1. \[ - \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{- b x}}{2} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*asinh(sqrt(2)*sqrt(-b*x)/2)/b

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Mathematica [A]  time = 0.0169434, size = 37, normalized size = 1.85 \[ \frac{2 \sqrt{x} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b} \sqrt{-b x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/(Sqrt[b]*Sqrt[-(b*x)])

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Maple [B]  time = 0.009, size = 64, normalized size = 3.2 \[{1\sqrt{-xb \left ( -bx+2 \right ) }\ln \left ({({b}^{2}x-b){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}-2\,bx} \right ){\frac{1}{\sqrt{-bx}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.202253, size = 36, normalized size = 1.8 \[ -\frac{\log \left (-b x + \sqrt{-b x + 2} \sqrt{-b x} + 1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(-b*x + sqrt(-b*x + 2)*sqrt(-b*x) + 1)/b

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Sympy [A]  time = 2.32438, size = 53, normalized size = 2.65 \[ \begin{cases} - \frac{2 \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{2 i \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*acosh(sqrt(2)*sqrt(b)*sqrt(x)/2)/b, Abs(b*x)/2 > 1), (-2*I*asin(sq
rt(2)*sqrt(b)*sqrt(x)/2)/b, True))

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GIAC/XCAS [A]  time = 0.262722, size = 32, normalized size = 1.6 \[ \frac{2 \,{\rm ln}\left ({\left | -\sqrt{-b x + 2} + \sqrt{-b x} \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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