3.1525 \(\int \frac{1}{\sqrt{b x} \sqrt{4+b x}} \, dx\)

Optimal. Leaf size=17 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b x}}{2}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0173335, antiderivative size = 17, 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 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b x}}{2}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[b*x]*Sqrt[4 + b*x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.58326, size = 12, normalized size = 0.71 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b x}}{2} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x)**(1/2)/(b*x+4)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0202466, size = 34, normalized size = 2. \[ \frac{2 \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{2}\right )}{\sqrt{b} \sqrt{b x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[b*x]*Sqrt[4 + b*x]),x]

[Out]

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

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Maple [B]  time = 0.01, size = 60, normalized size = 3.5 \[{1\sqrt{xb \left ( bx+4 \right ) }\ln \left ({({b}^{2}x+2\,b){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+4\,bx} \right ){\frac{1}{\sqrt{bx}}}{\frac{1}{\sqrt{bx+4}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x)^(1/2)/(b*x+4)^(1/2),x)

[Out]

(x*b*(b*x+4))^(1/2)/(b*x)^(1/2)/(b*x+4)^(1/2)*ln((b^2*x+2*b)/(b^2)^(1/2)+(b^2*x^
2+4*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 + 4)*sqrt(b*x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.215262, size = 34, normalized size = 2. \[ -\frac{\log \left (-b x + \sqrt{b x + 4} \sqrt{b x} - 2\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 4)*sqrt(b*x)),x, algorithm="fricas")

[Out]

-log(-b*x + sqrt(b*x + 4)*sqrt(b*x) - 2)/b

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Sympy [A]  time = 2.17145, size = 15, normalized size = 0.88 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{2} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x)**(1/2)/(b*x+4)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.254723, size = 30, normalized size = 1.76 \[ -\frac{2 \,{\rm ln}\left ({\left | -\sqrt{b x + 4} + \sqrt{b x} \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 4)*sqrt(b*x)),x, algorithm="giac")

[Out]

-2*ln(abs(-sqrt(b*x + 4) + sqrt(b*x)))/b