3.1523 \(\int \frac{1}{\sqrt{2+b x} \sqrt{6+b x}} \, dx\)

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

[Out]

(2*ArcSinh[Sqrt[2 + b*x]/2])/b

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Rubi [A]  time = 0.0237856, antiderivative size = 19, 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 \[ \frac{2 \sinh ^{-1}\left (\frac{1}{2} \sqrt{b x+2}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[2 + b*x]*Sqrt[6 + b*x]),x]

[Out]

(2*ArcSinh[Sqrt[2 + b*x]/2])/b

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Rubi in Sympy [A]  time = 4.56024, size = 14, normalized size = 0.74 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b x + 2}}{2} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+2)**(1/2)/(b*x+6)**(1/2),x)

[Out]

2*asinh(sqrt(b*x + 2)/2)/b

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Mathematica [A]  time = 0.0121331, size = 19, normalized size = 1. \[ \frac{2 \sinh ^{-1}\left (\frac{1}{2} \sqrt{b x+2}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[2 + b*x]*Sqrt[6 + b*x]),x]

[Out]

(2*ArcSinh[Sqrt[2 + b*x]/2])/b

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Maple [B]  time = 0.013, size = 66, normalized size = 3.5 \[{1\sqrt{ \left ( bx+2 \right ) \left ( bx+6 \right ) }\ln \left ({({b}^{2}x+4\,b){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+8\,bx+12} \right ){\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{bx+6}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+2)^(1/2)/(b*x+6)^(1/2),x)

[Out]

((b*x+2)*(b*x+6))^(1/2)/(b*x+2)^(1/2)/(b*x+6)^(1/2)*ln((b^2*x+4*b)/(b^2)^(1/2)+(
b^2*x^2+8*b*x+12)^(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 + 6)*sqrt(b*x + 2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23159, size = 36, normalized size = 1.89 \[ -\frac{\log \left (-b x + \sqrt{b x + 6} \sqrt{b x + 2} - 4\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 6)*sqrt(b*x + 2)),x, algorithm="fricas")

[Out]

-log(-b*x + sqrt(b*x + 6)*sqrt(b*x + 2) - 4)/b

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + 2} \sqrt{b x + 6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+2)**(1/2)/(b*x+6)**(1/2),x)

[Out]

Integral(1/(sqrt(b*x + 2)*sqrt(b*x + 6)), x)

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GIAC/XCAS [A]  time = 0.267314, size = 32, normalized size = 1.68 \[ -\frac{2 \,{\rm ln}\left ({\left | -\sqrt{b x + 6} + \sqrt{b x + 2} \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 6)*sqrt(b*x + 2)),x, algorithm="giac")

[Out]

-2*ln(abs(-sqrt(b*x + 6) + sqrt(b*x + 2)))/b