3.1532 \(\int \frac{1}{\sqrt{b x} \sqrt{2+b x}} \, dx\)

Optimal. Leaf size=19 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b x}}{\sqrt{2}}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0195618, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b x}}{\sqrt{2}}\right )}{b} \]

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 = 3.60663, size = 17, normalized size = 0.89 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b x}}{2} \right )}}{b} \]

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 = 0.0188604, size = 36, normalized size = 1.89 \[ \frac{2 \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b} \sqrt{b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.007, size = 58, normalized size = 3.1 \[{1\sqrt{xb \left ( bx+2 \right ) }\ln \left ({({b}^{2}x+b){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+2\,bx} \right ){\frac{1}{\sqrt{bx}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]

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 = 0.204132, size = 34, normalized size = 1.79 \[ -\frac{\log \left (-b x + \sqrt{b x + 2} \sqrt{b x} - 1\right )}{b} \]

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 = 2.21744, size = 20, normalized size = 1.05 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.25721, size = 30, normalized size = 1.58 \[ -\frac{2 \,{\rm ln}\left ({\left | -\sqrt{b x + 2} + \sqrt{b x} \right |}\right )}{b} \]

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