3.1855 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right ) x^8} \, dx\)

Optimal. Leaf size=58 \[ -\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{7/2}}-\frac{a^2}{b^3 x}+\frac{a}{3 b^2 x^3}-\frac{1}{5 b x^5} \]

[Out]

-1/(5*b*x^5) + a/(3*b^2*x^3) - a^2/(b^3*x) - (a^(5/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]
])/b^(7/2)

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Rubi [A]  time = 0.0835069, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{7/2}}-\frac{a^2}{b^3 x}+\frac{a}{3 b^2 x^3}-\frac{1}{5 b x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)*x^8),x]

[Out]

-1/(5*b*x^5) + a/(3*b^2*x^3) - a^2/(b^3*x) - (a^(5/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]
])/b^(7/2)

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Rubi in Sympy [A]  time = 15.4592, size = 49, normalized size = 0.84 \[ - \frac{a^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{b^{\frac{7}{2}}} - \frac{a^{2}}{b^{3} x} + \frac{a}{3 b^{2} x^{3}} - \frac{1}{5 b x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)/x**8,x)

[Out]

-a**(5/2)*atan(sqrt(a)*x/sqrt(b))/b**(7/2) - a**2/(b**3*x) + a/(3*b**2*x**3) - 1
/(5*b*x**5)

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Mathematica [A]  time = 0.0500393, size = 58, normalized size = 1. \[ -\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{b^{7/2}}-\frac{a^2}{b^3 x}+\frac{a}{3 b^2 x^3}-\frac{1}{5 b x^5} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)*x^8),x]

[Out]

-1/(5*b*x^5) + a/(3*b^2*x^3) - a^2/(b^3*x) - (a^(5/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]
])/b^(7/2)

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Maple [A]  time = 0.006, size = 52, normalized size = 0.9 \[ -{\frac{1}{5\,b{x}^{5}}}-{\frac{{a}^{2}}{{b}^{3}x}}+{\frac{a}{3\,{b}^{2}{x}^{3}}}-{\frac{{a}^{3}}{{b}^{3}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)/x^8,x)

[Out]

-1/5/b/x^5-a^2/b^3/x+1/3*a/b^2/x^3-a^3/b^3/(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^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229053, size = 1, normalized size = 0.02 \[ \left [\frac{15 \, a^{2} x^{5} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) - 30 \, a^{2} x^{4} + 10 \, a b x^{2} - 6 \, b^{2}}{30 \, b^{3} x^{5}}, -\frac{15 \, a^{2} x^{5} \sqrt{\frac{a}{b}} \arctan \left (\frac{a x}{b \sqrt{\frac{a}{b}}}\right ) + 15 \, a^{2} x^{4} - 5 \, a b x^{2} + 3 \, b^{2}}{15 \, b^{3} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)*x^8),x, algorithm="fricas")

[Out]

[1/30*(15*a^2*x^5*sqrt(-a/b)*log((a*x^2 - 2*b*x*sqrt(-a/b) - b)/(a*x^2 + b)) - 3
0*a^2*x^4 + 10*a*b*x^2 - 6*b^2)/(b^3*x^5), -1/15*(15*a^2*x^5*sqrt(a/b)*arctan(a*
x/(b*sqrt(a/b))) + 15*a^2*x^4 - 5*a*b*x^2 + 3*b^2)/(b^3*x^5)]

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Sympy [A]  time = 1.79658, size = 100, normalized size = 1.72 \[ \frac{\sqrt{- \frac{a^{5}}{b^{7}}} \log{\left (x - \frac{b^{4} \sqrt{- \frac{a^{5}}{b^{7}}}}{a^{3}} \right )}}{2} - \frac{\sqrt{- \frac{a^{5}}{b^{7}}} \log{\left (x + \frac{b^{4} \sqrt{- \frac{a^{5}}{b^{7}}}}{a^{3}} \right )}}{2} - \frac{15 a^{2} x^{4} - 5 a b x^{2} + 3 b^{2}}{15 b^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)/x**8,x)

[Out]

sqrt(-a**5/b**7)*log(x - b**4*sqrt(-a**5/b**7)/a**3)/2 - sqrt(-a**5/b**7)*log(x
+ b**4*sqrt(-a**5/b**7)/a**3)/2 - (15*a**2*x**4 - 5*a*b*x**2 + 3*b**2)/(15*b**3*
x**5)

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GIAC/XCAS [A]  time = 0.225045, size = 70, normalized size = 1.21 \[ -\frac{a^{3} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{3}} - \frac{15 \, a^{2} x^{4} - 5 \, a b x^{2} + 3 \, b^{2}}{15 \, b^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)*x^8),x, algorithm="giac")

[Out]

-a^3*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*b^3) - 1/15*(15*a^2*x^4 - 5*a*b*x^2 + 3*b^
2)/(b^3*x^5)