3.357 \(\int \frac{\sqrt{a+b x^2}}{x} \, dx\)

Optimal. Leaf size=37 \[ \sqrt{a+b x^2}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0741292, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \sqrt{a+b x^2}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^2]/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.52066, size = 31, normalized size = 0.84 \[ - \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{2}}}{\sqrt{a}} \right )} + \sqrt{a + b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.029536, size = 47, normalized size = 1.27 \[ \sqrt{a+b x^2}-\sqrt{a} \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\sqrt{a} \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^2]/x,x]

[Out]

Sqrt[a + b*x^2] + Sqrt[a]*Log[x] - Sqrt[a]*Log[a + Sqrt[a]*Sqrt[a + b*x^2]]

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Maple [A]  time = 0.006, size = 39, normalized size = 1.1 \[ \sqrt{b{x}^{2}+a}-\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{b{x}^{2}+a} \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/2)/x,x)

[Out]

(b*x^2+a)^(1/2)-a^(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)

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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 + a)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246157, size = 1, normalized size = 0.03 \[ \left [\frac{1}{2} \, \sqrt{a} \log \left (-\frac{b x^{2} - 2 \, \sqrt{b x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + \sqrt{b x^{2} + a}, -\sqrt{-a} \arctan \left (\frac{a}{\sqrt{b x^{2} + a} \sqrt{-a}}\right ) + \sqrt{b x^{2} + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.84016, size = 56, normalized size = 1.51 \[ - \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x} \right )} + \frac{a}{\sqrt{b} x \sqrt{\frac{a}{b x^{2}} + 1}} + \frac{\sqrt{b} x}{\sqrt{\frac{a}{b x^{2}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209539, size = 45, normalized size = 1.22 \[ \frac{a \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \sqrt{b x^{2} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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