3.1524 \(\int \frac{1}{\sqrt{1+b x} \sqrt{5+b x}} \, dx\)

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

[Out]

(2*ArcSinh[Sqrt[1 + b*x]/2])/b

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Rubi [A]  time = 0.0242426, 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 \[ \frac{2 \sinh ^{-1}\left (\frac{1}{2} \sqrt{b x+1}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[1 + b*x]*Sqrt[5 + b*x]),x]

[Out]

(2*ArcSinh[Sqrt[1 + b*x]/2])/b

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Rubi in Sympy [A]  time = 4.43883, size = 14, normalized size = 0.74 \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{b x + 1}}{2} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+1)**(1/2)/(b*x+5)**(1/2),x)

[Out]

2*asinh(sqrt(b*x + 1)/2)/b

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Mathematica [A]  time = 0.0123833, size = 19, normalized size = 1. \[ \frac{2 \sinh ^{-1}\left (\frac{1}{2} \sqrt{b x+1}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[1 + b*x]*Sqrt[5 + b*x]),x]

[Out]

(2*ArcSinh[Sqrt[1 + b*x]/2])/b

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Maple [B]  time = 0.014, size = 66, normalized size = 3.5 \[{1\sqrt{ \left ( bx+1 \right ) \left ( bx+5 \right ) }\ln \left ({({b}^{2}x+3\,b){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+6\,bx+5} \right ){\frac{1}{\sqrt{bx+1}}}{\frac{1}{\sqrt{bx+5}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+1)^(1/2)/(b*x+5)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219299, size = 36, normalized size = 1.89 \[ -\frac{\log \left (-b x + \sqrt{b x + 5} \sqrt{b x + 1} - 3\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 5)*sqrt(b*x + 1)),x, algorithm="fricas")

[Out]

-log(-b*x + sqrt(b*x + 5)*sqrt(b*x + 1) - 3)/b

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + 1} \sqrt{b x + 5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+1)**(1/2)/(b*x+5)**(1/2),x)

[Out]

Integral(1/(sqrt(b*x + 1)*sqrt(b*x + 5)), x)

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GIAC/XCAS [A]  time = 0.260656, size = 32, normalized size = 1.68 \[ -\frac{2 \,{\rm ln}\left ({\left | -\sqrt{b x + 5} + \sqrt{b x + 1} \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 5)*sqrt(b*x + 1)),x, algorithm="giac")

[Out]

-2*ln(abs(-sqrt(b*x + 5) + sqrt(b*x + 1)))/b