3.1877 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^3} \, dx\)

Optimal. Leaf size=74 \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{7/2}}+\frac{15 x}{8 a^3}-\frac{5 x^3}{8 a^2 \left (a x^2+b\right )}-\frac{x^5}{4 a \left (a x^2+b\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0865592, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ -\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{7/2}}+\frac{15 x}{8 a^3}-\frac{5 x^3}{8 a^2 \left (a x^2+b\right )}-\frac{x^5}{4 a \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^(-3),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.5185, size = 66, normalized size = 0.89 \[ - \frac{x^{5}}{4 a \left (a x^{2} + b\right )^{2}} - \frac{5 x^{3}}{8 a^{2} \left (a x^{2} + b\right )} + \frac{15 x}{8 a^{3}} - \frac{15 \sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )}}{8 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0928863, size = 66, normalized size = 0.89 \[ \frac{8 a^2 x^5+25 a b x^3+15 b^2 x}{8 a^3 \left (a x^2+b\right )^2}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^(-3),x]

[Out]

(15*b^2*x + 25*a*b*x^3 + 8*a^2*x^5)/(8*a^3*(b + a*x^2)^2) - (15*Sqrt[b]*ArcTan[(
Sqrt[a]*x)/Sqrt[b]])/(8*a^(7/2))

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Maple [A]  time = 0.008, size = 63, normalized size = 0.9 \[{\frac{x}{{a}^{3}}}+{\frac{9\,b{x}^{3}}{8\,{a}^{2} \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{7\,{b}^{2}x}{8\,{a}^{3} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{15\,b}{8\,{a}^{3}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229005, size = 1, normalized size = 0.01 \[ \left [\frac{16 \, a^{2} x^{5} + 50 \, a b x^{3} + 30 \, b^{2} x + 15 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right )}{16 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}}, \frac{8 \, a^{2} x^{5} + 25 \, a b x^{3} + 15 \, b^{2} x - 15 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{x}{\sqrt{\frac{b}{a}}}\right )}{8 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.20351, size = 107, normalized size = 1.45 \[ \frac{15 \sqrt{- \frac{b}{a^{7}}} \log{\left (- a^{3} \sqrt{- \frac{b}{a^{7}}} + x \right )}}{16} - \frac{15 \sqrt{- \frac{b}{a^{7}}} \log{\left (a^{3} \sqrt{- \frac{b}{a^{7}}} + x \right )}}{16} + \frac{9 a b x^{3} + 7 b^{2} x}{8 a^{5} x^{4} + 16 a^{4} b x^{2} + 8 a^{3} b^{2}} + \frac{x}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**3,x)

[Out]

15*sqrt(-b/a**7)*log(-a**3*sqrt(-b/a**7) + x)/16 - 15*sqrt(-b/a**7)*log(a**3*sqr
t(-b/a**7) + x)/16 + (9*a*b*x**3 + 7*b**2*x)/(8*a**5*x**4 + 16*a**4*b*x**2 + 8*a
**3*b**2) + x/a**3

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GIAC/XCAS [A]  time = 0.22085, size = 73, normalized size = 0.99 \[ -\frac{15 \, b \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{3}} + \frac{x}{a^{3}} + \frac{9 \, a b x^{3} + 7 \, b^{2} x}{8 \,{\left (a x^{2} + b\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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