3.1840 \(\int \frac{\left (a+\frac{b}{x^2}\right )^3}{x^6} \, dx\)

Optimal. Leaf size=43 \[ -\frac{a^3}{5 x^5}-\frac{3 a^2 b}{7 x^7}-\frac{a b^2}{3 x^9}-\frac{b^3}{11 x^{11}} \]

[Out]

-b^3/(11*x^11) - (a*b^2)/(3*x^9) - (3*a^2*b)/(7*x^7) - a^3/(5*x^5)

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Rubi [A]  time = 0.0506904, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{a^3}{5 x^5}-\frac{3 a^2 b}{7 x^7}-\frac{a b^2}{3 x^9}-\frac{b^3}{11 x^{11}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^3/x^6,x]

[Out]

-b^3/(11*x^11) - (a*b^2)/(3*x^9) - (3*a^2*b)/(7*x^7) - a^3/(5*x^5)

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Rubi in Sympy [A]  time = 8.62173, size = 39, normalized size = 0.91 \[ - \frac{a^{3}}{5 x^{5}} - \frac{3 a^{2} b}{7 x^{7}} - \frac{a b^{2}}{3 x^{9}} - \frac{b^{3}}{11 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**3/x**6,x)

[Out]

-a**3/(5*x**5) - 3*a**2*b/(7*x**7) - a*b**2/(3*x**9) - b**3/(11*x**11)

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Mathematica [A]  time = 0.0125785, size = 43, normalized size = 1. \[ -\frac{a^3}{5 x^5}-\frac{3 a^2 b}{7 x^7}-\frac{a b^2}{3 x^9}-\frac{b^3}{11 x^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^3/x^6,x]

[Out]

-b^3/(11*x^11) - (a*b^2)/(3*x^9) - (3*a^2*b)/(7*x^7) - a^3/(5*x^5)

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Maple [A]  time = 0.008, size = 36, normalized size = 0.8 \[ -{\frac{{b}^{3}}{11\,{x}^{11}}}-{\frac{a{b}^{2}}{3\,{x}^{9}}}-{\frac{3\,{a}^{2}b}{7\,{x}^{7}}}-{\frac{{a}^{3}}{5\,{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^3/x^6,x)

[Out]

-1/11*b^3/x^11-1/3*a*b^2/x^9-3/7*a^2*b/x^7-1/5*a^3/x^5

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Maxima [A]  time = 1.43475, size = 50, normalized size = 1.16 \[ -\frac{231 \, a^{3} x^{6} + 495 \, a^{2} b x^{4} + 385 \, a b^{2} x^{2} + 105 \, b^{3}}{1155 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^3/x^6,x, algorithm="maxima")

[Out]

-1/1155*(231*a^3*x^6 + 495*a^2*b*x^4 + 385*a*b^2*x^2 + 105*b^3)/x^11

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Fricas [A]  time = 0.214452, size = 50, normalized size = 1.16 \[ -\frac{231 \, a^{3} x^{6} + 495 \, a^{2} b x^{4} + 385 \, a b^{2} x^{2} + 105 \, b^{3}}{1155 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^3/x^6,x, algorithm="fricas")

[Out]

-1/1155*(231*a^3*x^6 + 495*a^2*b*x^4 + 385*a*b^2*x^2 + 105*b^3)/x^11

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Sympy [A]  time = 1.77952, size = 39, normalized size = 0.91 \[ - \frac{231 a^{3} x^{6} + 495 a^{2} b x^{4} + 385 a b^{2} x^{2} + 105 b^{3}}{1155 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**3/x**6,x)

[Out]

-(231*a**3*x**6 + 495*a**2*b*x**4 + 385*a*b**2*x**2 + 105*b**3)/(1155*x**11)

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GIAC/XCAS [A]  time = 0.220084, size = 50, normalized size = 1.16 \[ -\frac{231 \, a^{3} x^{6} + 495 \, a^{2} b x^{4} + 385 \, a b^{2} x^{2} + 105 \, b^{3}}{1155 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^3/x^6,x, algorithm="giac")

[Out]

-1/1155*(231*a^3*x^6 + 495*a^2*b*x^4 + 385*a*b^2*x^2 + 105*b^3)/x^11