3.1853 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right ) x^6} \, dx\)

Optimal. Leaf size=43 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{5/2}}+\frac{a}{b^2 x}-\frac{1}{3 b x^3} \]

[Out]

-1/(3*b*x^3) + a/(b^2*x) + (a^(3/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/b^(5/2)

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Rubi [A]  time = 0.0637973, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{5/2}}+\frac{a}{b^2 x}-\frac{1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)*x^6),x]

[Out]

-1/(3*b*x^3) + a/(b^2*x) + (a^(3/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/b^(5/2)

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Rubi in Sympy [A]  time = 11.1869, size = 37, normalized size = 0.86 \[ \frac{a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{b^{\frac{5}{2}}} + \frac{a}{b^{2} x} - \frac{1}{3 b x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)/x**6,x)

[Out]

a**(3/2)*atan(sqrt(a)*x/sqrt(b))/b**(5/2) + a/(b**2*x) - 1/(3*b*x**3)

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Mathematica [A]  time = 0.0382578, size = 43, normalized size = 1. \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{5/2}}+\frac{a}{b^2 x}-\frac{1}{3 b x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)*x^6),x]

[Out]

-1/(3*b*x^3) + a/(b^2*x) + (a^(3/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/b^(5/2)

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Maple [A]  time = 0.005, size = 39, normalized size = 0.9 \[ -{\frac{1}{3\,b{x}^{3}}}+{\frac{a}{{b}^{2}x}}+{\frac{{a}^{2}}{{b}^{2}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)/x^6,x)

[Out]

-1/3/b/x^3+a/b^2/x+a^2/b^2/(a*b)^(1/2)*arctan(a*x/(a*b)^(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(1/((a + b/x^2)*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238518, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a x^{3} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) + 6 \, a x^{2} - 2 \, b}{6 \, b^{2} x^{3}}, \frac{3 \, a x^{3} \sqrt{\frac{a}{b}} \arctan \left (\frac{a x}{b \sqrt{\frac{a}{b}}}\right ) + 3 \, a x^{2} - b}{3 \, b^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)*x^6),x, algorithm="fricas")

[Out]

[1/6*(3*a*x^3*sqrt(-a/b)*log((a*x^2 + 2*b*x*sqrt(-a/b) - b)/(a*x^2 + b)) + 6*a*x
^2 - 2*b)/(b^2*x^3), 1/3*(3*a*x^3*sqrt(a/b)*arctan(a*x/(b*sqrt(a/b))) + 3*a*x^2
- b)/(b^2*x^3)]

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Sympy [A]  time = 1.48079, size = 87, normalized size = 2.02 \[ - \frac{\sqrt{- \frac{a^{3}}{b^{5}}} \log{\left (x - \frac{b^{3} \sqrt{- \frac{a^{3}}{b^{5}}}}{a^{2}} \right )}}{2} + \frac{\sqrt{- \frac{a^{3}}{b^{5}}} \log{\left (x + \frac{b^{3} \sqrt{- \frac{a^{3}}{b^{5}}}}{a^{2}} \right )}}{2} + \frac{3 a x^{2} - b}{3 b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)/x**6,x)

[Out]

-sqrt(-a**3/b**5)*log(x - b**3*sqrt(-a**3/b**5)/a**2)/2 + sqrt(-a**3/b**5)*log(x
+ b**3*sqrt(-a**3/b**5)/a**2)/2 + (3*a*x**2 - b)/(3*b**2*x**3)

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GIAC/XCAS [A]  time = 0.227875, size = 54, normalized size = 1.26 \[ \frac{a^{2} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{3 \, a x^{2} - b}{3 \, b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)*x^6),x, algorithm="giac")

[Out]

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