3.1891 \(\int \sqrt{a+\frac{b}{x^2}} x \, dx\)

Optimal. Leaf size=47 \[ \frac{1}{2} x^2 \sqrt{a+\frac{b}{x^2}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.0746072, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{1}{2} x^2 \sqrt{a+\frac{b}{x^2}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^2]*x,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.80272, size = 39, normalized size = 0.83 \[ \frac{x^{2} \sqrt{a + \frac{b}{x^{2}}}}{2} + \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{2 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0508789, size = 58, normalized size = 1.23 \[ \frac{1}{2} x \sqrt{a+\frac{b}{x^2}} \left (\frac{b \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{\sqrt{a} \sqrt{a x^2+b}}+x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b/x^2]*x,x]

[Out]

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

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Maple [A]  time = 0.007, size = 62, normalized size = 1.3 \[{\frac{x}{2}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ( x\sqrt{a{x}^{2}+b}\sqrt{a}+b\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ) \right ){\frac{1}{\sqrt{a{x}^{2}+b}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^(1/2)*x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253134, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} + \sqrt{a} b \log \left (-2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (2 \, a x^{2} + b\right )} \sqrt{a}\right )}{4 \, a}, \frac{a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - \sqrt{-a} b \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{2} + b}{x^{2}}}}\right )}{2 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(2*a*x^2*sqrt((a*x^2 + b)/x^2) + sqrt(a)*b*log(-2*a*x^2*sqrt((a*x^2 + b)/x^
2) - (2*a*x^2 + b)*sqrt(a)))/a, 1/2*(a*x^2*sqrt((a*x^2 + b)/x^2) - sqrt(-a)*b*ar
ctan(sqrt(-a)/sqrt((a*x^2 + b)/x^2)))/a]

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Sympy [A]  time = 6.99278, size = 41, normalized size = 0.87 \[ \frac{\sqrt{b} x \sqrt{\frac{a x^{2}}{b} + 1}}{2} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.23426, size = 72, normalized size = 1.53 \[ \frac{b{\rm ln}\left (\sqrt{b}\right ){\rm sign}\left (x\right )}{2 \, \sqrt{a}} + \frac{1}{2} \,{\left (\sqrt{a x^{2} + b} x - \frac{b{\rm ln}\left ({\left | -\sqrt{a} x + \sqrt{a x^{2} + b} \right |}\right )}{\sqrt{a}}\right )}{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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