3.1838 \(\int \frac{\left (a+\frac{b}{x^2}\right )^3}{x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0514414, 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 \[ -\frac{a^3}{3 x^3}-\frac{3 a^2 b}{5 x^5}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{9 x^9} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)^3/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.56886, size = 41, normalized size = 0.95 \[ - \frac{a^{3}}{3 x^{3}} - \frac{3 a^{2} b}{5 x^{5}} - \frac{3 a b^{2}}{7 x^{7}} - \frac{b^{3}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**3/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.00722554, size = 43, normalized size = 1. \[ -\frac{a^3}{3 x^3}-\frac{3 a^2 b}{5 x^5}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{9 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)^3/x^4,x]

[Out]

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

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Maple [A]  time = 0.007, size = 36, normalized size = 0.8 \[ -{\frac{{b}^{3}}{9\,{x}^{9}}}-{\frac{3\,a{b}^{2}}{7\,{x}^{7}}}-{\frac{3\,{a}^{2}b}{5\,{x}^{5}}}-{\frac{{a}^{3}}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)^3/x^4,x)

[Out]

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

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Maxima [A]  time = 1.43293, size = 50, normalized size = 1.16 \[ -\frac{105 \, a^{3} x^{6} + 189 \, a^{2} b x^{4} + 135 \, a b^{2} x^{2} + 35 \, b^{3}}{315 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^3/x^4,x, algorithm="maxima")

[Out]

-1/315*(105*a^3*x^6 + 189*a^2*b*x^4 + 135*a*b^2*x^2 + 35*b^3)/x^9

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Fricas [A]  time = 0.217135, size = 50, normalized size = 1.16 \[ -\frac{105 \, a^{3} x^{6} + 189 \, a^{2} b x^{4} + 135 \, a b^{2} x^{2} + 35 \, b^{3}}{315 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^3/x^4,x, algorithm="fricas")

[Out]

-1/315*(105*a^3*x^6 + 189*a^2*b*x^4 + 135*a*b^2*x^2 + 35*b^3)/x^9

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Sympy [A]  time = 1.66424, size = 39, normalized size = 0.91 \[ - \frac{105 a^{3} x^{6} + 189 a^{2} b x^{4} + 135 a b^{2} x^{2} + 35 b^{3}}{315 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)**3/x**4,x)

[Out]

-(105*a**3*x**6 + 189*a**2*b*x**4 + 135*a*b**2*x**2 + 35*b**3)/(315*x**9)

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GIAC/XCAS [A]  time = 0.227055, size = 50, normalized size = 1.16 \[ -\frac{105 \, a^{3} x^{6} + 189 \, a^{2} b x^{4} + 135 \, a b^{2} x^{2} + 35 \, b^{3}}{315 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)^3/x^4,x, algorithm="giac")

[Out]

-1/315*(105*a^3*x^6 + 189*a^2*b*x^4 + 135*a*b^2*x^2 + 35*b^3)/x^9