3.1881 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x^4} \, dx\)

Optimal. Leaf size=65 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2}}+\frac{x}{8 a b \left (a x^2+b\right )}-\frac{x}{4 a \left (a x^2+b\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0686584, antiderivative size = 65, 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 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2}}+\frac{x}{8 a b \left (a x^2+b\right )}-\frac{x}{4 a \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^3*x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 9.62009, size = 51, normalized size = 0.78 \[ - \frac{x}{4 a \left (a x^{2} + b\right )^{2}} + \frac{x}{8 a b \left (a x^{2} + b\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 a^{\frac{3}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**3/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0547117, size = 58, normalized size = 0.89 \[ \frac{\frac{\sqrt{a} \sqrt{b} x \left (a x^2-b\right )}{\left (a x^2+b\right )^2}+\tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{3/2} b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^3*x^4),x]

[Out]

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

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Maple [A]  time = 0.006, size = 49, normalized size = 0.8 \[{\frac{1}{ \left ( a{x}^{2}+b \right ) ^{2}} \left ({\frac{{x}^{3}}{8\,b}}-{\frac{x}{8\,a}} \right ) }+{\frac{1}{8\,ab}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^3/x^4,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234326, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \log \left (\frac{2 \, a b x +{\left (a x^{2} - b\right )} \sqrt{-a b}}{a x^{2} + b}\right ) + 2 \,{\left (a x^{3} - b x\right )} \sqrt{-a b}}{16 \,{\left (a^{3} b x^{4} + 2 \, a^{2} b^{2} x^{2} + a b^{3}\right )} \sqrt{-a b}}, \frac{{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{b}\right ) +{\left (a x^{3} - b x\right )} \sqrt{a b}}{8 \,{\left (a^{3} b x^{4} + 2 \, a^{2} b^{2} x^{2} + a b^{3}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.82177, size = 110, normalized size = 1.69 \[ - \frac{\sqrt{- \frac{1}{a^{3} b^{3}}} \log{\left (- a b^{2} \sqrt{- \frac{1}{a^{3} b^{3}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{3} b^{3}}} \log{\left (a b^{2} \sqrt{- \frac{1}{a^{3} b^{3}}} + x \right )}}{16} + \frac{a x^{3} - b x}{8 a^{3} b x^{4} + 16 a^{2} b^{2} x^{2} + 8 a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**3/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.224768, size = 68, normalized size = 1.05 \[ \frac{\arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a b} + \frac{a x^{3} - b x}{8 \,{\left (a x^{2} + b\right )}^{2} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^4),x, algorithm="giac")

[Out]

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