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

Optimal. Leaf size=35 \[ 2 \sqrt{a+b x}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

[Out]

2*Sqrt[a + b*x] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

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Rubi [A]  time = 0.0346436, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ 2 \sqrt{a+b x}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x]/x,x]

[Out]

2*Sqrt[a + b*x] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

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Rubi in Sympy [A]  time = 4.70595, size = 31, normalized size = 0.89 \[ - 2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + 2 \sqrt{a + b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a)*atanh(sqrt(a + b*x)/sqrt(a)) + 2*sqrt(a + b*x)

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Mathematica [A]  time = 0.0171722, size = 35, normalized size = 1. \[ 2 \sqrt{a+b x}-2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x]/x,x]

[Out]

2*Sqrt[a + b*x] - 2*Sqrt[a]*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]

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Maple [A]  time = 0.022, size = 28, normalized size = 0.8 \[ -2\,{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \sqrt{a}+2\,\sqrt{bx+a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224733, size = 1, normalized size = 0.03 \[ \left [\sqrt{a} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \, \sqrt{b x + a}, -2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) + 2 \, \sqrt{b x + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.84184, size = 68, normalized size = 1.94 \[ - 2 \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} + \frac{2 a}{\sqrt{b} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{2 \sqrt{b} \sqrt{x}}{\sqrt{\frac{a}{b x} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(1/2)/x,x)

[Out]

-2*sqrt(a)*asinh(sqrt(a)/(sqrt(b)*sqrt(x))) + 2*a/(sqrt(b)*sqrt(x)*sqrt(a/(b*x)
+ 1)) + 2*sqrt(b)*sqrt(x)/sqrt(a/(b*x) + 1)

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GIAC/XCAS [A]  time = 0.205686, size = 43, normalized size = 1.23 \[ \frac{2 \, a \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{b x + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 2*sqrt(b*x + a)