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

Optimal. Leaf size=24 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0193849, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 3.47589, size = 24, normalized size = 1. \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00996235, size = 24, normalized size = 1. \[ \frac{2 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.007, size = 46, normalized size = 1.9 \[{1\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(1/2)/(b*x+2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220276, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right )}{\sqrt{b}}, \frac{2 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{\sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[log(sqrt(b*x + 2)*b*sqrt(x) + (b*x + 1)*sqrt(b))/sqrt(b), 2*arctan(sqrt(b*x + 2
)*sqrt(-b)/(b*sqrt(x)))/sqrt(-b)]

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Sympy [A]  time = 3.80638, size = 24, normalized size = 1. \[ \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError