3.1885 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x^8} \, dx\)

Optimal. Leaf size=76 \[ -\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 b^{7/2}}+\frac{5}{8 b^2 x \left (a x^2+b\right )}+\frac{1}{4 b x \left (a x^2+b\right )^2}-\frac{15}{8 b^3 x} \]

[Out]

-15/(8*b^3*x) + 1/(4*b*x*(b + a*x^2)^2) + 5/(8*b^2*x*(b + a*x^2)) - (15*Sqrt[a]*
ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(8*b^(7/2))

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Rubi [A]  time = 0.0885268, antiderivative size = 76, 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 \[ -\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 b^{7/2}}+\frac{5}{8 b^2 x \left (a x^2+b\right )}+\frac{1}{4 b x \left (a x^2+b\right )^2}-\frac{15}{8 b^3 x} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^3*x^8),x]

[Out]

-15/(8*b^3*x) + 1/(4*b*x*(b + a*x^2)^2) + 5/(8*b^2*x*(b + a*x^2)) - (15*Sqrt[a]*
ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(8*b^(7/2))

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Rubi in Sympy [A]  time = 14.834, size = 65, normalized size = 0.86 \[ - \frac{15 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 b^{\frac{7}{2}}} + \frac{1}{4 b x \left (a x^{2} + b\right )^{2}} + \frac{5}{8 b^{2} x \left (a x^{2} + b\right )} - \frac{15}{8 b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**3/x**8,x)

[Out]

-15*sqrt(a)*atan(sqrt(a)*x/sqrt(b))/(8*b**(7/2)) + 1/(4*b*x*(a*x**2 + b)**2) + 5
/(8*b**2*x*(a*x**2 + b)) - 15/(8*b**3*x)

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Mathematica [A]  time = 0.0822347, size = 68, normalized size = 0.89 \[ -\frac{15 a^2 x^4+25 a b x^2+8 b^2}{8 b^3 x \left (a x^2+b\right )^2}-\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^3*x^8),x]

[Out]

-(8*b^2 + 25*a*b*x^2 + 15*a^2*x^4)/(8*b^3*x*(b + a*x^2)^2) - (15*Sqrt[a]*ArcTan[
(Sqrt[a]*x)/Sqrt[b]])/(8*b^(7/2))

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Maple [A]  time = 0.01, size = 66, normalized size = 0.9 \[ -{\frac{1}{{b}^{3}x}}-{\frac{7\,{x}^{3}{a}^{2}}{8\,{b}^{3} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{9\,ax}{8\,{b}^{2} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{15\,a}{8\,{b}^{3}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^3/x^8,x)

[Out]

-1/b^3/x-7/8/b^3*a^2/(a*x^2+b)^2*x^3-9/8/b^2*a/(a*x^2+b)^2*x-15/8/b^3*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^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244151, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \, a^{2} x^{4} + 50 \, a b x^{2} - 15 \,{\left (a^{2} x^{5} + 2 \, a b x^{3} + b^{2} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) + 16 \, b^{2}}{16 \,{\left (a^{2} b^{3} x^{5} + 2 \, a b^{4} x^{3} + b^{5} x\right )}}, -\frac{15 \, a^{2} x^{4} + 25 \, a b x^{2} + 15 \,{\left (a^{2} x^{5} + 2 \, a b x^{3} + b^{2} x\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{a x}{b \sqrt{\frac{a}{b}}}\right ) + 8 \, b^{2}}{8 \,{\left (a^{2} b^{3} x^{5} + 2 \, a b^{4} x^{3} + b^{5} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^8),x, algorithm="fricas")

[Out]

[-1/16*(30*a^2*x^4 + 50*a*b*x^2 - 15*(a^2*x^5 + 2*a*b*x^3 + b^2*x)*sqrt(-a/b)*lo
g((a*x^2 - 2*b*x*sqrt(-a/b) - b)/(a*x^2 + b)) + 16*b^2)/(a^2*b^3*x^5 + 2*a*b^4*x
^3 + b^5*x), -1/8*(15*a^2*x^4 + 25*a*b*x^2 + 15*(a^2*x^5 + 2*a*b*x^3 + b^2*x)*sq
rt(a/b)*arctan(a*x/(b*sqrt(a/b))) + 8*b^2)/(a^2*b^3*x^5 + 2*a*b^4*x^3 + b^5*x)]

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Sympy [A]  time = 2.50179, size = 114, normalized size = 1.5 \[ \frac{15 \sqrt{- \frac{a}{b^{7}}} \log{\left (x - \frac{b^{4} \sqrt{- \frac{a}{b^{7}}}}{a} \right )}}{16} - \frac{15 \sqrt{- \frac{a}{b^{7}}} \log{\left (x + \frac{b^{4} \sqrt{- \frac{a}{b^{7}}}}{a} \right )}}{16} - \frac{15 a^{2} x^{4} + 25 a b x^{2} + 8 b^{2}}{8 a^{2} b^{3} x^{5} + 16 a b^{4} x^{3} + 8 b^{5} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**3/x**8,x)

[Out]

15*sqrt(-a/b**7)*log(x - b**4*sqrt(-a/b**7)/a)/16 - 15*sqrt(-a/b**7)*log(x + b**
4*sqrt(-a/b**7)/a)/16 - (15*a**2*x**4 + 25*a*b*x**2 + 8*b**2)/(8*a**2*b**3*x**5
+ 16*a*b**4*x**3 + 8*b**5*x)

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GIAC/XCAS [A]  time = 0.231957, size = 77, normalized size = 1.01 \[ -\frac{15 \, a \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{3}} - \frac{7 \, a^{2} x^{3} + 9 \, a b x}{8 \,{\left (a x^{2} + b\right )}^{2} b^{3}} - \frac{1}{b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^3*x^8),x, algorithm="giac")

[Out]

-15/8*a*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*b^3) - 1/8*(7*a^2*x^3 + 9*a*b*x)/((a*x^
2 + b)^2*b^3) - 1/(b^3*x)