3.2240 \(\int \frac{1}{\sqrt{a+b \sqrt{x}} x} \, dx\)

Optimal. Leaf size=27 \[ -\frac{4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

(-4*ArcTanh[Sqrt[a + b*Sqrt[x]]/Sqrt[a]])/Sqrt[a]

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Rubi [A]  time = 0.0515509, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ -\frac{4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*ArcTanh[Sqrt[a + b*Sqrt[x]]/Sqrt[a]])/Sqrt[a]

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Rubi in Sympy [A]  time = 5.35677, size = 26, normalized size = 0.96 \[ - \frac{4 \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{x}}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*atanh(sqrt(a + b*sqrt(x))/sqrt(a))/sqrt(a)

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Mathematica [A]  time = 0.0179539, size = 27, normalized size = 1. \[ -\frac{4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*ArcTanh[Sqrt[a + b*Sqrt[x]]/Sqrt[a]])/Sqrt[a]

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Maple [A]  time = 0.005, size = 20, normalized size = 0.7 \[ -4\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{x}}}{\sqrt{a}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*x^(1/2))^(1/2),x)

[Out]

-4*arctanh((a+b*x^(1/2))^(1/2)/a^(1/2))/a^(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*sqrt(x) + a)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.258527, size = 1, normalized size = 0.04 \[ \left [\frac{2 \, \log \left (\frac{\sqrt{a} b \sqrt{x} - 2 \, \sqrt{b \sqrt{x} + a} a + 2 \, a^{\frac{3}{2}}}{\sqrt{x}}\right )}{\sqrt{a}}, \frac{4 \, \arctan \left (\frac{a}{\sqrt{b \sqrt{x} + a} \sqrt{-a}}\right )}{\sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[2*log((sqrt(a)*b*sqrt(x) - 2*sqrt(b*sqrt(x) + a)*a + 2*a^(3/2))/sqrt(x))/sqrt(a
), 4*arctan(a/(sqrt(b*sqrt(x) + a)*sqrt(-a)))/sqrt(-a)]

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Sympy [A]  time = 4.54514, size = 24, normalized size = 0.89 \[ - \frac{4 \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt [4]{x}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*asinh(sqrt(a)/(sqrt(b)*x**(1/4)))/sqrt(a)

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GIAC/XCAS [A]  time = 0.2676, size = 31, normalized size = 1.15 \[ \frac{4 \, \arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*arctan(sqrt(b*sqrt(x) + a)/sqrt(-a))/sqrt(-a)