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

Optimal. Leaf size=42 \[ x \sqrt{\frac{a}{x^2}+b}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0618879, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ x \sqrt{\frac{a}{x^2}+b}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(a + b*x^2)/x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 4.93287, size = 34, normalized size = 0.81 \[ - \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^{2}} + b}} \right )} + x \sqrt{\frac{a}{x^{2}} + b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0469191, size = 71, normalized size = 1.69 \[ \frac{x \sqrt{\frac{a}{x^2}+b} \left (\sqrt{a+b x^2}-\sqrt{a} \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\sqrt{a} \log (x)\right )}{\sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b + a/x^2]*x*(Sqrt[a + b*x^2] + Sqrt[a]*Log[x] - Sqrt[a]*Log[a + Sqrt[a]*S
qrt[a + b*x^2]]))/Sqrt[a + b*x^2]

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Maple [A]  time = 0.01, size = 61, normalized size = 1.5 \[{x\sqrt{{\frac{b{x}^{2}+a}{{x}^{2}}}} \left ( \sqrt{b{x}^{2}+a}-\sqrt{a}\ln \left ( 2\,{\frac{\sqrt{a}\sqrt{b{x}^{2}+a}+a}{x}} \right ) \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x^2+a)/x^2)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251397, size = 1, normalized size = 0.02 \[ \left [x \sqrt{\frac{b x^{2} + a}{x^{2}}} + \frac{1}{2} \, \sqrt{a} \log \left (-\frac{b x^{2} - 2 \, \sqrt{a} x \sqrt{\frac{b x^{2} + a}{x^{2}}} + 2 \, a}{x^{2}}\right ), x \sqrt{\frac{b x^{2} + a}{x^{2}}} - \sqrt{-a} \arctan \left (\frac{a}{\sqrt{-a} x \sqrt{\frac{b x^{2} + a}{x^{2}}}}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222623, size = 92, normalized size = 2.19 \[{\left (\frac{a \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \sqrt{b x^{2} + a}\right )}{\rm sign}\left (x\right ) - \frac{{\left (a \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ) + \sqrt{-a} \sqrt{a}\right )}{\rm sign}\left (x\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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