3.1887 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x^{10}} \, dx\)

Optimal. Leaf size=87 \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 b^{9/2}}+\frac{35 a}{8 b^4 x}+\frac{7}{8 b^2 x^3 \left (a x^2+b\right )}+\frac{1}{4 b x^3 \left (a x^2+b\right )^2}-\frac{35}{24 b^3 x^3} \]

[Out]

-35/(24*b^3*x^3) + (35*a)/(8*b^4*x) + 1/(4*b*x^3*(b + a*x^2)^2) + 7/(8*b^2*x^3*(
b + a*x^2)) + (35*a^(3/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(8*b^(9/2))

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Rubi [A]  time = 0.116545, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 b^{9/2}}+\frac{35 a}{8 b^4 x}+\frac{7}{8 b^2 x^3 \left (a x^2+b\right )}+\frac{1}{4 b x^3 \left (a x^2+b\right )^2}-\frac{35}{24 b^3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^3*x^10),x]

[Out]

-35/(24*b^3*x^3) + (35*a)/(8*b^4*x) + 1/(4*b*x^3*(b + a*x^2)^2) + 7/(8*b^2*x^3*(
b + a*x^2)) + (35*a^(3/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(8*b^(9/2))

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Rubi in Sympy [A]  time = 19.084, size = 80, normalized size = 0.92 \[ \frac{35 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 b^{\frac{9}{2}}} + \frac{35 a}{8 b^{4} x} + \frac{1}{4 b x^{3} \left (a x^{2} + b\right )^{2}} + \frac{7}{8 b^{2} x^{3} \left (a x^{2} + b\right )} - \frac{35}{24 b^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**3/x**10,x)

[Out]

35*a**(3/2)*atan(sqrt(a)*x/sqrt(b))/(8*b**(9/2)) + 35*a/(8*b**4*x) + 1/(4*b*x**3
*(a*x**2 + b)**2) + 7/(8*b**2*x**3*(a*x**2 + b)) - 35/(24*b**3*x**3)

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Mathematica [A]  time = 0.0827297, size = 79, normalized size = 0.91 \[ \frac{35 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 b^{9/2}}+\frac{105 a^3 x^6+175 a^2 b x^4+56 a b^2 x^2-8 b^3}{24 b^4 x^3 \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^3*x^10),x]

[Out]

(-8*b^3 + 56*a*b^2*x^2 + 175*a^2*b*x^4 + 105*a^3*x^6)/(24*b^4*x^3*(b + a*x^2)^2)
+ (35*a^(3/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(8*b^(9/2))

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Maple [A]  time = 0.01, size = 79, normalized size = 0.9 \[ -{\frac{1}{3\,{b}^{3}{x}^{3}}}+3\,{\frac{a}{{b}^{4}x}}+{\frac{11\,{a}^{3}{x}^{3}}{8\,{b}^{4} \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{13\,x{a}^{2}}{8\,{b}^{3} \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{35\,{a}^{2}}{8\,{b}^{4}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^3/x^10,x)

[Out]

-1/3/b^3/x^3+3*a/b^4/x+11/8/b^4*a^3/(a*x^2+b)^2*x^3+13/8/b^3*a^2/(a*x^2+b)^2*x+3
5/8/b^4*a^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^10),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238387, size = 1, normalized size = 0.01 \[ \left [\frac{210 \, a^{3} x^{6} + 350 \, a^{2} b x^{4} + 112 \, a b^{2} x^{2} - 16 \, b^{3} + 105 \,{\left (a^{3} x^{7} + 2 \, a^{2} b x^{5} + a b^{2} x^{3}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right )}{48 \,{\left (a^{2} b^{4} x^{7} + 2 \, a b^{5} x^{5} + b^{6} x^{3}\right )}}, \frac{105 \, a^{3} x^{6} + 175 \, a^{2} b x^{4} + 56 \, a b^{2} x^{2} - 8 \, b^{3} + 105 \,{\left (a^{3} x^{7} + 2 \, a^{2} b x^{5} + a b^{2} x^{3}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{a x}{b \sqrt{\frac{a}{b}}}\right )}{24 \,{\left (a^{2} b^{4} x^{7} + 2 \, a b^{5} x^{5} + b^{6} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^10),x, algorithm="fricas")

[Out]

[1/48*(210*a^3*x^6 + 350*a^2*b*x^4 + 112*a*b^2*x^2 - 16*b^3 + 105*(a^3*x^7 + 2*a
^2*b*x^5 + a*b^2*x^3)*sqrt(-a/b)*log((a*x^2 + 2*b*x*sqrt(-a/b) - b)/(a*x^2 + b))
)/(a^2*b^4*x^7 + 2*a*b^5*x^5 + b^6*x^3), 1/24*(105*a^3*x^6 + 175*a^2*b*x^4 + 56*
a*b^2*x^2 - 8*b^3 + 105*(a^3*x^7 + 2*a^2*b*x^5 + a*b^2*x^3)*sqrt(a/b)*arctan(a*x
/(b*sqrt(a/b))))/(a^2*b^4*x^7 + 2*a*b^5*x^5 + b^6*x^3)]

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Sympy [A]  time = 3.43085, size = 138, normalized size = 1.59 \[ - \frac{35 \sqrt{- \frac{a^{3}}{b^{9}}} \log{\left (x - \frac{b^{5} \sqrt{- \frac{a^{3}}{b^{9}}}}{a^{2}} \right )}}{16} + \frac{35 \sqrt{- \frac{a^{3}}{b^{9}}} \log{\left (x + \frac{b^{5} \sqrt{- \frac{a^{3}}{b^{9}}}}{a^{2}} \right )}}{16} + \frac{105 a^{3} x^{6} + 175 a^{2} b x^{4} + 56 a b^{2} x^{2} - 8 b^{3}}{24 a^{2} b^{4} x^{7} + 48 a b^{5} x^{5} + 24 b^{6} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**3/x**10,x)

[Out]

-35*sqrt(-a**3/b**9)*log(x - b**5*sqrt(-a**3/b**9)/a**2)/16 + 35*sqrt(-a**3/b**9
)*log(x + b**5*sqrt(-a**3/b**9)/a**2)/16 + (105*a**3*x**6 + 175*a**2*b*x**4 + 56
*a*b**2*x**2 - 8*b**3)/(24*a**2*b**4*x**7 + 48*a*b**5*x**5 + 24*b**6*x**3)

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GIAC/XCAS [A]  time = 0.225292, size = 96, normalized size = 1.1 \[ \frac{35 \, a^{2} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{4}} + \frac{11 \, a^{3} x^{3} + 13 \, a^{2} b x}{8 \,{\left (a x^{2} + b\right )}^{2} b^{4}} + \frac{9 \, a x^{2} - b}{3 \, b^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^10),x, algorithm="giac")

[Out]

35/8*a^2*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*b^4) + 1/8*(11*a^3*x^3 + 13*a^2*b*x)/(
(a*x^2 + b)^2*b^4) + 1/3*(9*a*x^2 - b)/(b^4*x^3)