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

Optimal. Leaf size=39 \[ -\frac{a^3}{5 x^5}-\frac{3 a^2 b}{2 x^2}+3 a b^2 x+\frac{b^3 x^4}{4} \]

[Out]

-a^3/(5*x^5) - (3*a^2*b)/(2*x^2) + 3*a*b^2*x + (b^3*x^4)/4

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Rubi [A]  time = 0.0376578, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{a^3}{5 x^5}-\frac{3 a^2 b}{2 x^2}+3 a b^2 x+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^3/x^6,x]

[Out]

-a^3/(5*x^5) - (3*a^2*b)/(2*x^2) + 3*a*b^2*x + (b^3*x^4)/4

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Rubi in Sympy [A]  time = 6.71569, size = 36, normalized size = 0.92 \[ - \frac{a^{3}}{5 x^{5}} - \frac{3 a^{2} b}{2 x^{2}} + 3 a b^{2} x + \frac{b^{3} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3/(5*x**5) - 3*a**2*b/(2*x**2) + 3*a*b**2*x + b**3*x**4/4

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Mathematica [A]  time = 0.0112957, size = 39, normalized size = 1. \[ -\frac{a^3}{5 x^5}-\frac{3 a^2 b}{2 x^2}+3 a b^2 x+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^3/x^6,x]

[Out]

-a^3/(5*x^5) - (3*a^2*b)/(2*x^2) + 3*a*b^2*x + (b^3*x^4)/4

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Maple [A]  time = 0.009, size = 34, normalized size = 0.9 \[ -{\frac{{a}^{3}}{5\,{x}^{5}}}-{\frac{3\,{a}^{2}b}{2\,{x}^{2}}}+3\,a{b}^{2}x+{\frac{{b}^{3}{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^3/x^6,x)

[Out]

-1/5*a^3/x^5-3/2*a^2*b/x^2+3*a*b^2*x+1/4*b^3*x^4

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Maxima [A]  time = 1.42654, size = 49, normalized size = 1.26 \[ \frac{1}{4} \, b^{3} x^{4} + 3 \, a b^{2} x - \frac{15 \, a^{2} b x^{3} + 2 \, a^{3}}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*b^3*x^4 + 3*a*b^2*x - 1/10*(15*a^2*b*x^3 + 2*a^3)/x^5

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Fricas [A]  time = 0.201744, size = 50, normalized size = 1.28 \[ \frac{5 \, b^{3} x^{9} + 60 \, a b^{2} x^{6} - 30 \, a^{2} b x^{3} - 4 \, a^{3}}{20 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(5*b^3*x^9 + 60*a*b^2*x^6 - 30*a^2*b*x^3 - 4*a^3)/x^5

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Sympy [A]  time = 1.27288, size = 36, normalized size = 0.92 \[ 3 a b^{2} x + \frac{b^{3} x^{4}}{4} - \frac{2 a^{3} + 15 a^{2} b x^{3}}{10 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**3/x**6,x)

[Out]

3*a*b**2*x + b**3*x**4/4 - (2*a**3 + 15*a**2*b*x**3)/(10*x**5)

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GIAC/XCAS [A]  time = 0.229381, size = 49, normalized size = 1.26 \[ \frac{1}{4} \, b^{3} x^{4} + 3 \, a b^{2} x - \frac{15 \, a^{2} b x^{3} + 2 \, a^{3}}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*b^3*x^4 + 3*a*b^2*x - 1/10*(15*a^2*b*x^3 + 2*a^3)/x^5