3.1851 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right ) x^4} \, dx\)

Optimal. Leaf size=34 \[ -\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]

[Out]

-(1/(b*x)) - (Sqrt[a]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/b^(3/2)

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Rubi [A]  time = 0.046018, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)*x^4),x]

[Out]

-(1/(b*x)) - (Sqrt[a]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/b^(3/2)

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Rubi in Sympy [A]  time = 7.64168, size = 29, normalized size = 0.85 \[ - \frac{\sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{b^{\frac{3}{2}}} - \frac{1}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0229773, size = 34, normalized size = 1. \[ -\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{3/2}}-\frac{1}{b x} \]

Antiderivative was successfully verified.

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

[Out]

-(1/(b*x)) - (Sqrt[a]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/b^(3/2)

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Maple [A]  time = 0.004, size = 30, normalized size = 0.9 \[ -{\frac{a}{b}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{bx}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)/x^4,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233231, size = 1, normalized size = 0.03 \[ \left [\frac{x \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) - 2}{2 \, b x}, -\frac{x \sqrt{\frac{a}{b}} \arctan \left (\frac{a x}{b \sqrt{\frac{a}{b}}}\right ) + 1}{b x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.32007, size = 65, normalized size = 1.91 \[ \frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (x - \frac{b^{2} \sqrt{- \frac{a}{b^{3}}}}{a} \right )}}{2} - \frac{\sqrt{- \frac{a}{b^{3}}} \log{\left (x + \frac{b^{2} \sqrt{- \frac{a}{b^{3}}}}{a} \right )}}{2} - \frac{1}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.223114, size = 39, normalized size = 1.15 \[ -\frac{a \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} b} - \frac{1}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*b) - 1/(b*x)