3.509 \(\int \frac{\sqrt{2+b x}}{x^{3/2}} \, dx\)

Optimal. Leaf size=41 \[ 2 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{2 \sqrt{b x+2}}{\sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0322514, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ 2 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{2 \sqrt{b x+2}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 + b*x]/x^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.85653, size = 39, normalized size = 0.95 \[ 2 \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} - \frac{2 \sqrt{b x + 2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0228564, size = 41, normalized size = 1. \[ 2 \sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )-\frac{2 \sqrt{b x+2}}{\sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.033, size = 59, normalized size = 1.4 \[ -2\,{\frac{\sqrt{bx+2}}{\sqrt{x}}}+{1\sqrt{b}\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ) \sqrt{x \left ( bx+2 \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+2)^(1/2)/x^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21834, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b} x \log \left (b x + \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 1\right ) - 2 \, \sqrt{b x + 2} \sqrt{x}}{x}, \frac{2 \,{\left (\sqrt{-b} x \arctan \left (\frac{\sqrt{b x + 2}}{\sqrt{-b} \sqrt{x}}\right ) - \sqrt{b x + 2} \sqrt{x}\right )}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.46726, size = 48, normalized size = 1.17 \[ - 2 \sqrt{b} \sqrt{1 + \frac{2}{b x}} - \sqrt{b} \log{\left (\frac{1}{b x} \right )} + 2 \sqrt{b} \log{\left (\sqrt{1 + \frac{2}{b x}} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(b)*sqrt(1 + 2/(b*x)) - sqrt(b)*log(1/(b*x)) + 2*sqrt(b)*log(sqrt(1 + 2/(
b*x)) + 1)

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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(sqrt(b*x + 2)/x^(3/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError