3.1924 \(\int \frac{1}{\sqrt{a+\frac{b}{x^2}} x^4} \, dx\)

Optimal. Leaf size=53 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 b^{3/2}}-\frac{\sqrt{a+\frac{b}{x^2}}}{2 b x} \]

[Out]

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

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Rubi [A]  time = 0.0777898, antiderivative size = 53, 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 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 b^{3/2}}-\frac{\sqrt{a+\frac{b}{x^2}}}{2 b x} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x^2]*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.40605, size = 41, normalized size = 0.77 \[ \frac{a \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{2 b^{\frac{3}{2}}} - \frac{\sqrt{a + \frac{b}{x^{2}}}}{2 b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**(1/2)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0749784, size = 93, normalized size = 1.75 \[ \frac{-\sqrt{b} \left (a x^2+b\right )-a x^2 \log (x) \sqrt{a x^2+b}+a x^2 \sqrt{a x^2+b} \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )}{2 b^{3/2} x^3 \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b/x^2]*x^4),x]

[Out]

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

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Maple [A]  time = 0.01, size = 73, normalized size = 1.4 \[ -{\frac{1}{2\,{x}^{3}}\sqrt{a{x}^{2}+b} \left ( -a\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{2}b+\sqrt{a{x}^{2}+b}{b}^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}{b}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(1/2)/x^4,x)

[Out]

-1/2*(a*x^2+b)^(1/2)*(-a*ln(2*(b^(1/2)*(a*x^2+b)^(1/2)+b)/x)*x^2*b+(a*x^2+b)^(1/
2)*b^(3/2))/((a*x^2+b)/x^2)^(1/2)/x^3/b^(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(1/(sqrt(a + b/x^2)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.251512, size = 1, normalized size = 0.02 \[ \left [\frac{a \sqrt{b} x \log \left (-\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} +{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) - 2 \, b \sqrt{\frac{a x^{2} + b}{x^{2}}}}{4 \, b^{2} x}, -\frac{a \sqrt{-b} x \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) + b \sqrt{\frac{a x^{2} + b}{x^{2}}}}{2 \, b^{2} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^2)*x^4),x, algorithm="fricas")

[Out]

[1/4*(a*sqrt(b)*x*log(-(2*b*x*sqrt((a*x^2 + b)/x^2) + (a*x^2 + 2*b)*sqrt(b))/x^2
) - 2*b*sqrt((a*x^2 + b)/x^2))/(b^2*x), -1/2*(a*sqrt(-b)*x*arctan(sqrt(-b)/(x*sq
rt((a*x^2 + b)/x^2))) + b*sqrt((a*x^2 + b)/x^2))/(b^2*x)]

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Sympy [A]  time = 9.00321, size = 42, normalized size = 0.79 \[ - \frac{\sqrt{a} \sqrt{1 + \frac{b}{a x^{2}}}}{2 b x} + \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{2 b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**(1/2)/x**4,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{2}}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^2)*x^4),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a + b/x^2)*x^4), x)