3.1529 \(\int \frac{1}{\sqrt{2+b x} \sqrt{3+b x}} \, dx\)

Optimal. Leaf size=15 \[ \frac{2 \sinh ^{-1}\left (\sqrt{b x+2}\right )}{b} \]

[Out]

(2*ArcSinh[Sqrt[2 + b*x]])/b

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Rubi [A]  time = 0.0219876, antiderivative size = 15, 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 \[ \frac{2 \sinh ^{-1}\left (\sqrt{b x+2}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[2 + b*x]*Sqrt[3 + b*x]),x]

[Out]

(2*ArcSinh[Sqrt[2 + b*x]])/b

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Rubi in Sympy [A]  time = 4.45438, size = 12, normalized size = 0.8 \[ \frac{2 \operatorname{asinh}{\left (\sqrt{b x + 2} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+2)**(1/2)/(b*x+3)**(1/2),x)

[Out]

2*asinh(sqrt(b*x + 2))/b

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Mathematica [A]  time = 0.00979564, size = 15, normalized size = 1. \[ \frac{2 \sinh ^{-1}\left (\sqrt{b x+2}\right )}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[2 + b*x]*Sqrt[3 + b*x]),x]

[Out]

(2*ArcSinh[Sqrt[2 + b*x]])/b

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Maple [B]  time = 0.01, size = 66, normalized size = 4.4 \[{1\sqrt{ \left ( bx+2 \right ) \left ( bx+3 \right ) }\ln \left ({1 \left ({\frac{5\,b}{2}}+{b}^{2}x \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+5\,bx+6} \right ){\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{bx+3}}}{\frac{1}{\sqrt{{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+2)^(1/2)/(b*x+3)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224498, size = 38, normalized size = 2.53 \[ -\frac{\log \left (-2 \, b x + 2 \, \sqrt{b x + 3} \sqrt{b x + 2} - 5\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 3)*sqrt(b*x + 2)),x, algorithm="fricas")

[Out]

-log(-2*b*x + 2*sqrt(b*x + 3)*sqrt(b*x + 2) - 5)/b

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b x + 2} \sqrt{b x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+2)**(1/2)/(b*x+3)**(1/2),x)

[Out]

Integral(1/(sqrt(b*x + 2)*sqrt(b*x + 3)), x)

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GIAC/XCAS [A]  time = 0.265542, size = 32, normalized size = 2.13 \[ -\frac{2 \,{\rm ln}\left ({\left | -\sqrt{b x + 3} + \sqrt{b x + 2} \right |}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b*x + 3)*sqrt(b*x + 2)),x, algorithm="giac")

[Out]

-2*ln(abs(-sqrt(b*x + 3) + sqrt(b*x + 2)))/b