3.1914 \(\int \frac{x}{\sqrt{a+\frac{b}{x^2}}} \, dx\)

Optimal. Leaf size=50 \[ \frac{x^2 \sqrt{a+\frac{b}{x^2}}}{2 a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 a^{3/2}} \]

[Out]

(Sqrt[a + b/x^2]*x^2)/(2*a) - (b*ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]])/(2*a^(3/2))

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Rubi [A]  time = 0.0764839, antiderivative size = 50, 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 \[ \frac{x^2 \sqrt{a+\frac{b}{x^2}}}{2 a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a + b/x^2],x]

[Out]

(Sqrt[a + b/x^2]*x^2)/(2*a) - (b*ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]])/(2*a^(3/2))

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Rubi in Sympy [A]  time = 6.87324, size = 41, normalized size = 0.82 \[ \frac{x^{2} \sqrt{a + \frac{b}{x^{2}}}}{2 a} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*sqrt(a + b/x**2)/(2*a) - b*atanh(sqrt(a + b/x**2)/sqrt(a))/(2*a**(3/2))

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Mathematica [A]  time = 0.0445525, size = 74, normalized size = 1.48 \[ \frac{\sqrt{a} x \left (a x^2+b\right )-b \sqrt{a x^2+b} \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{2 a^{3/2} x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[a + b/x^2],x]

[Out]

(Sqrt[a]*x*(b + a*x^2) - b*Sqrt[b + a*x^2]*Log[a*x + Sqrt[a]*Sqrt[b + a*x^2]])/(
2*a^(3/2)*Sqrt[a + b/x^2]*x)

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Maple [A]  time = 0.013, size = 66, normalized size = 1.3 \[{\frac{1}{2\,x}\sqrt{a{x}^{2}+b} \left ( x\sqrt{a{x}^{2}+b}{a}^{{\frac{3}{2}}}-b\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ) a \right ){\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}{a}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x^2)^(1/2),x)

[Out]

1/2*(a*x^2+b)^(1/2)*(x*(a*x^2+b)^(1/2)*a^(3/2)-b*ln(a^(1/2)*x+(a*x^2+b)^(1/2))*a
)/((a*x^2+b)/x^2)^(1/2)/x/a^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.257976, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} + \sqrt{a} b \log \left (2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (2 \, a x^{2} + b\right )} \sqrt{a}\right )}{4 \, a^{2}}, \frac{a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} + \sqrt{-a} b \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{2} + b}{x^{2}}}}\right )}{2 \, a^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(a + b/x^2),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^2, 1/2*(a*x^2*sqrt((a*x^2 + b)/x^2) + sqrt(-a)*b*a
rctan(sqrt(-a)/sqrt((a*x^2 + b)/x^2)))/a^2]

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Sympy [A]  time = 7.91433, size = 42, normalized size = 0.84 \[ \frac{\sqrt{b} x \sqrt{\frac{a x^{2}}{b} + 1}}{2 a} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b)*x*sqrt(a*x**2/b + 1)/(2*a) - b*asinh(sqrt(a)*x/sqrt(b))/(2*a**(3/2))

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GIAC/XCAS [A]  time = 0.240789, size = 90, normalized size = 1.8 \[ \frac{1}{2} \, b{\left (\frac{\arctan \left (\frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{{\left (a - \frac{a x^{2} + b}{x^{2}}\right )} a}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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