3.1870 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^2 x^8} \, dx\)

Optimal. Leaf size=68 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{7/2}}+\frac{5 a}{2 b^3 x}+\frac{1}{2 b x^3 \left (a x^2+b\right )}-\frac{5}{6 b^2 x^3} \]

[Out]

-5/(6*b^2*x^3) + (5*a)/(2*b^3*x) + 1/(2*b*x^3*(b + a*x^2)) + (5*a^(3/2)*ArcTan[(
Sqrt[a]*x)/Sqrt[b]])/(2*b^(7/2))

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Rubi [A]  time = 0.082769, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{7/2}}+\frac{5 a}{2 b^3 x}+\frac{1}{2 b x^3 \left (a x^2+b\right )}-\frac{5}{6 b^2 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^2*x^8),x]

[Out]

-5/(6*b^2*x^3) + (5*a)/(2*b^3*x) + 1/(2*b*x^3*(b + a*x^2)) + (5*a^(3/2)*ArcTan[(
Sqrt[a]*x)/Sqrt[b]])/(2*b^(7/2))

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Rubi in Sympy [A]  time = 14.9566, size = 61, normalized size = 0.9 \[ \frac{5 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{2 b^{\frac{7}{2}}} + \frac{5 a}{2 b^{3} x} + \frac{1}{2 b x^{3} \left (a x^{2} + b\right )} - \frac{5}{6 b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*a**(3/2)*atan(sqrt(a)*x/sqrt(b))/(2*b**(7/2)) + 5*a/(2*b**3*x) + 1/(2*b*x**3*(
a*x**2 + b)) - 5/(6*b**2*x**3)

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Mathematica [A]  time = 0.0730313, size = 67, normalized size = 0.99 \[ \frac{5 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 b^{7/2}}+\frac{a^2 x}{2 b^3 \left (a x^2+b\right )}+\frac{2 a}{b^3 x}-\frac{1}{3 b^2 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 59, normalized size = 0.9 \[ -{\frac{1}{3\,{b}^{2}{x}^{3}}}+2\,{\frac{a}{{b}^{3}x}}+{\frac{x{a}^{2}}{2\,{b}^{3} \left ( a{x}^{2}+b \right ) }}+{\frac{5\,{a}^{2}}{2\,{b}^{3}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^2/x^8,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240285, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, a^{2} x^{4} + 20 \, a b x^{2} + 15 \,{\left (a^{2} x^{5} + a b x^{3}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) - 4 \, b^{2}}{12 \,{\left (a b^{3} x^{5} + b^{4} x^{3}\right )}}, \frac{15 \, a^{2} x^{4} + 10 \, a b x^{2} + 15 \,{\left (a^{2} x^{5} + a b x^{3}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{a x}{b \sqrt{\frac{a}{b}}}\right ) - 2 \, b^{2}}{6 \,{\left (a b^{3} x^{5} + b^{4} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(30*a^2*x^4 + 20*a*b*x^2 + 15*(a^2*x^5 + a*b*x^3)*sqrt(-a/b)*log((a*x^2 +
2*b*x*sqrt(-a/b) - b)/(a*x^2 + b)) - 4*b^2)/(a*b^3*x^5 + b^4*x^3), 1/6*(15*a^2*x
^4 + 10*a*b*x^2 + 15*(a^2*x^5 + a*b*x^3)*sqrt(a/b)*arctan(a*x/(b*sqrt(a/b))) - 2
*b^2)/(a*b^3*x^5 + b^4*x^3)]

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Sympy [A]  time = 2.1533, size = 114, normalized size = 1.68 \[ - \frac{5 \sqrt{- \frac{a^{3}}{b^{7}}} \log{\left (x - \frac{b^{4} \sqrt{- \frac{a^{3}}{b^{7}}}}{a^{2}} \right )}}{4} + \frac{5 \sqrt{- \frac{a^{3}}{b^{7}}} \log{\left (x + \frac{b^{4} \sqrt{- \frac{a^{3}}{b^{7}}}}{a^{2}} \right )}}{4} + \frac{15 a^{2} x^{4} + 10 a b x^{2} - 2 b^{2}}{6 a b^{3} x^{5} + 6 b^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*sqrt(-a**3/b**7)*log(x - b**4*sqrt(-a**3/b**7)/a**2)/4 + 5*sqrt(-a**3/b**7)*l
og(x + b**4*sqrt(-a**3/b**7)/a**2)/4 + (15*a**2*x**4 + 10*a*b*x**2 - 2*b**2)/(6*
a*b**3*x**5 + 6*b**4*x**3)

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GIAC/XCAS [A]  time = 0.219144, size = 80, normalized size = 1.18 \[ \frac{5 \, a^{2} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{3}} + \frac{a^{2} x}{2 \,{\left (a x^{2} + b\right )} b^{3}} + \frac{6 \, a x^{2} - b}{3 \, b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/2*a^2*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*b^3) + 1/2*a^2*x/((a*x^2 + b)*b^3) + 1/
3*(6*a*x^2 - b)/(b^3*x^3)