3.1883 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x^6} \, dx\)

Optimal. Leaf size=62 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 \sqrt{a} b^{5/2}}+\frac{3 x}{8 b^2 \left (a x^2+b\right )}+\frac{x}{4 b \left (a x^2+b\right )^2} \]

[Out]

x/(4*b*(b + a*x^2)^2) + (3*x)/(8*b^2*(b + a*x^2)) + (3*ArcTan[(Sqrt[a]*x)/Sqrt[b
]])/(8*Sqrt[a]*b^(5/2))

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Rubi [A]  time = 0.0573739, antiderivative size = 62, 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 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 \sqrt{a} b^{5/2}}+\frac{3 x}{8 b^2 \left (a x^2+b\right )}+\frac{x}{4 b \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^3*x^6),x]

[Out]

x/(4*b*(b + a*x^2)^2) + (3*x)/(8*b^2*(b + a*x^2)) + (3*ArcTan[(Sqrt[a]*x)/Sqrt[b
]])/(8*Sqrt[a]*b^(5/2))

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Rubi in Sympy [A]  time = 7.14342, size = 54, normalized size = 0.87 \[ \frac{x}{4 b \left (a x^{2} + b\right )^{2}} + \frac{3 x}{8 b^{2} \left (a x^{2} + b\right )} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 \sqrt{a} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**3/x**6,x)

[Out]

x/(4*b*(a*x**2 + b)**2) + 3*x/(8*b**2*(a*x**2 + b)) + 3*atan(sqrt(a)*x/sqrt(b))/
(8*sqrt(a)*b**(5/2))

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Mathematica [A]  time = 0.0664809, size = 55, normalized size = 0.89 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 \sqrt{a} b^{5/2}}+\frac{3 a x^3+5 b x}{8 b^2 \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^3*x^6),x]

[Out]

(5*b*x + 3*a*x^3)/(8*b^2*(b + a*x^2)^2) + (3*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(8*Sqr
t[a]*b^(5/2))

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Maple [A]  time = 0.005, size = 51, normalized size = 0.8 \[{\frac{x}{4\,b \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{3\,x}{8\,{b}^{2} \left ( a{x}^{2}+b \right ) }}+{\frac{3}{8\,{b}^{2}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^3/x^6,x)

[Out]

1/4*x/b/(a*x^2+b)^2+3/8*x/b^2/(a*x^2+b)+3/8/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)^3*x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237404, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \log \left (\frac{2 \, a b x +{\left (a x^{2} - b\right )} \sqrt{-a b}}{a x^{2} + b}\right ) + 2 \,{\left (3 \, a x^{3} + 5 \, b x\right )} \sqrt{-a b}}{16 \,{\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )} \sqrt{-a b}}, \frac{3 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{b}\right ) +{\left (3 \, a x^{3} + 5 \, b x\right )} \sqrt{a b}}{8 \,{\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^6),x, algorithm="fricas")

[Out]

[1/16*(3*(a^2*x^4 + 2*a*b*x^2 + b^2)*log((2*a*b*x + (a*x^2 - b)*sqrt(-a*b))/(a*x
^2 + b)) + 2*(3*a*x^3 + 5*b*x)*sqrt(-a*b))/((a^2*b^2*x^4 + 2*a*b^3*x^2 + b^4)*sq
rt(-a*b)), 1/8*(3*(a^2*x^4 + 2*a*b*x^2 + b^2)*arctan(sqrt(a*b)*x/b) + (3*a*x^3 +
5*b*x)*sqrt(a*b))/((a^2*b^2*x^4 + 2*a*b^3*x^2 + b^4)*sqrt(a*b))]

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Sympy [A]  time = 1.89881, size = 105, normalized size = 1.69 \[ - \frac{3 \sqrt{- \frac{1}{a b^{5}}} \log{\left (- b^{3} \sqrt{- \frac{1}{a b^{5}}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a b^{5}}} \log{\left (b^{3} \sqrt{- \frac{1}{a b^{5}}} + x \right )}}{16} + \frac{3 a x^{3} + 5 b x}{8 a^{2} b^{2} x^{4} + 16 a b^{3} x^{2} + 8 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**3/x**6,x)

[Out]

-3*sqrt(-1/(a*b**5))*log(-b**3*sqrt(-1/(a*b**5)) + x)/16 + 3*sqrt(-1/(a*b**5))*l
og(b**3*sqrt(-1/(a*b**5)) + x)/16 + (3*a*x**3 + 5*b*x)/(8*a**2*b**2*x**4 + 16*a*
b**3*x**2 + 8*b**4)

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GIAC/XCAS [A]  time = 0.228677, size = 61, normalized size = 0.98 \[ \frac{3 \, \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{2}} + \frac{3 \, a x^{3} + 5 \, b x}{8 \,{\left (a x^{2} + b\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^6),x, algorithm="giac")

[Out]

3/8*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*b^2) + 1/8*(3*a*x^3 + 5*b*x)/((a*x^2 + b)^2
*b^2)