3.2234 \(\int \frac{\sqrt{a+b \sqrt{x}}}{x} \, dx\)

Optimal. Leaf size=43 \[ 4 \sqrt{a+b \sqrt{x}}-4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0665888, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ 4 \sqrt{a+b \sqrt{x}}-4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[x]]/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.03237, size = 37, normalized size = 0.86 \[ - 4 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b \sqrt{x}}}{\sqrt{a}} \right )} + 4 \sqrt{a + b \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0251753, size = 43, normalized size = 1. \[ 4 \sqrt{a+b \sqrt{x}}-4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 32, normalized size = 0.7 \[ -4\,{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{x}}}{\sqrt{a}}} \right ) \sqrt{a}+4\,\sqrt{a+b\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/2))^(1/2)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257928, size = 1, normalized size = 0.02 \[ \left [2 \, \sqrt{a} \log \left (\frac{b \sqrt{x} - 2 \, \sqrt{b \sqrt{x} + a} \sqrt{a} + 2 \, a}{\sqrt{x}}\right ) + 4 \, \sqrt{b \sqrt{x} + a}, -4 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right ) + 4 \, \sqrt{b \sqrt{x} + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[2*sqrt(a)*log((b*sqrt(x) - 2*sqrt(b*sqrt(x) + a)*sqrt(a) + 2*a)/sqrt(x)) + 4*sq
rt(b*sqrt(x) + a), -4*sqrt(-a)*arctan(sqrt(b*sqrt(x) + a)/sqrt(-a)) + 4*sqrt(b*s
qrt(x) + a)]

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Sympy [A]  time = 5.2011, size = 75, normalized size = 1.74 \[ - 4 \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt [4]{x}} \right )} + \frac{4 a}{\sqrt{b} \sqrt [4]{x} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{4 \sqrt{b} \sqrt [4]{x}}{\sqrt{\frac{a}{b \sqrt{x}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.252703, size = 49, normalized size = 1.14 \[ \frac{4 \, a \arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 4 \, \sqrt{b \sqrt{x} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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