3.1868 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^2 x^6} \, dx\)

Optimal. Leaf size=57 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{5/2}}+\frac{1}{2 b x \left (a x^2+b\right )}-\frac{3}{2 b^2 x} \]

[Out]

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

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Rubi [A]  time = 0.0660439, antiderivative size = 57, 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 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{5/2}}+\frac{1}{2 b x \left (a x^2+b\right )}-\frac{3}{2 b^2 x} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^2*x^6),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.9878, size = 48, normalized size = 0.84 \[ - \frac{3 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2 b^{\frac{5}{2}}} + \frac{1}{2 b x \left (a x^{2} + b\right )} - \frac{3}{2 b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0651415, size = 54, normalized size = 0.95 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{5/2}}-\frac{a x}{2 b^2 \left (a x^2+b\right )}-\frac{1}{b^2 x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 46, normalized size = 0.8 \[ -{\frac{1}{{b}^{2}x}}-{\frac{ax}{2\,{b}^{2} \left ( a{x}^{2}+b \right ) }}-{\frac{3\,a}{2\,{b}^{2}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^2/x^6,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233679, size = 1, normalized size = 0.02 \[ \left [-\frac{6 \, a x^{2} - 3 \,{\left (a x^{3} + b x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) + 4 \, b}{4 \,{\left (a b^{2} x^{3} + b^{3} x\right )}}, -\frac{3 \, a x^{2} + 3 \,{\left (a x^{3} + b x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{a x}{b \sqrt{\frac{a}{b}}}\right ) + 2 \, b}{2 \,{\left (a b^{2} x^{3} + b^{3} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.75089, size = 90, normalized size = 1.58 \[ \frac{3 \sqrt{- \frac{a}{b^{5}}} \log{\left (x - \frac{b^{3} \sqrt{- \frac{a}{b^{5}}}}{a} \right )}}{4} - \frac{3 \sqrt{- \frac{a}{b^{5}}} \log{\left (x + \frac{b^{3} \sqrt{- \frac{a}{b^{5}}}}{a} \right )}}{4} - \frac{3 a x^{2} + 2 b}{2 a b^{2} x^{3} + 2 b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*sqrt(-a/b**5)*log(x - b**3*sqrt(-a/b**5)/a)/4 - 3*sqrt(-a/b**5)*log(x + b**3*s
qrt(-a/b**5)/a)/4 - (3*a*x**2 + 2*b)/(2*a*b**2*x**3 + 2*b**3*x)

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GIAC/XCAS [A]  time = 0.224528, size = 63, normalized size = 1.11 \[ -\frac{3 \, a \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{2}} - \frac{3 \, a x^{2} + 2 \, b}{2 \,{\left (a x^{3} + b x\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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