3.2412 \(\int \frac{\left (a+\frac{b}{\sqrt [3]{x}}\right )^3}{x^3} \, dx\)

Optimal. Leaf size=47 \[ -\frac{a^3}{2 x^2}-\frac{9 a^2 b}{7 x^{7/3}}-\frac{9 a b^2}{8 x^{8/3}}-\frac{b^3}{3 x^3} \]

[Out]

-b^3/(3*x^3) - (9*a*b^2)/(8*x^(8/3)) - (9*a^2*b)/(7*x^(7/3)) - a^3/(2*x^2)

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Rubi [A]  time = 0.0633822, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{a^3}{2 x^2}-\frac{9 a^2 b}{7 x^{7/3}}-\frac{9 a b^2}{8 x^{8/3}}-\frac{b^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^3/x^3,x]

[Out]

-b^3/(3*x^3) - (9*a*b^2)/(8*x^(8/3)) - (9*a^2*b)/(7*x^(7/3)) - a^3/(2*x^2)

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Rubi in Sympy [A]  time = 10.2856, size = 44, normalized size = 0.94 \[ - \frac{a^{3}}{2 x^{2}} - \frac{9 a^{2} b}{7 x^{\frac{7}{3}}} - \frac{9 a b^{2}}{8 x^{\frac{8}{3}}} - \frac{b^{3}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3/(2*x**2) - 9*a**2*b/(7*x**(7/3)) - 9*a*b**2/(8*x**(8/3)) - b**3/(3*x**3)

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Mathematica [A]  time = 0.0140633, size = 41, normalized size = 0.87 \[ -\frac{84 a^3 x+216 a^2 b x^{2/3}+189 a b^2 \sqrt [3]{x}+56 b^3}{168 x^3} \]

Antiderivative was successfully verified.

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

[Out]

-(56*b^3 + 189*a*b^2*x^(1/3) + 216*a^2*b*x^(2/3) + 84*a^3*x)/(168*x^3)

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Maple [A]  time = 0.008, size = 36, normalized size = 0.8 \[ -{\frac{{b}^{3}}{3\,{x}^{3}}}-{\frac{9\,a{b}^{2}}{8}{x}^{-{\frac{8}{3}}}}-{\frac{9\,{a}^{2}b}{7}{x}^{-{\frac{7}{3}}}}-{\frac{{a}^{3}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^3/x^3,x)

[Out]

-1/3*b^3/x^3-9/8*a*b^2/x^(8/3)-9/7*a^2*b/x^(7/3)-1/2*a^3/x^2

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Maxima [A]  time = 1.43185, size = 132, normalized size = 2.81 \[ -\frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{9}}{3 \, b^{6}} + \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{8} a}{8 \, b^{6}} - \frac{30 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{7} a^{2}}{7 \, b^{6}} + \frac{5 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{6} a^{3}}{b^{6}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5} a^{4}}{b^{6}} + \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a^{5}}{4 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^3/x^3,x, algorithm="maxima")

[Out]

-1/3*(a + b/x^(1/3))^9/b^6 + 15/8*(a + b/x^(1/3))^8*a/b^6 - 30/7*(a + b/x^(1/3))
^7*a^2/b^6 + 5*(a + b/x^(1/3))^6*a^3/b^6 - 3*(a + b/x^(1/3))^5*a^4/b^6 + 3/4*(a
+ b/x^(1/3))^4*a^5/b^6

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Fricas [A]  time = 0.227734, size = 47, normalized size = 1. \[ -\frac{84 \, a^{3} x + 216 \, a^{2} b x^{\frac{2}{3}} + 189 \, a b^{2} x^{\frac{1}{3}} + 56 \, b^{3}}{168 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/168*(84*a^3*x + 216*a^2*b*x^(2/3) + 189*a*b^2*x^(1/3) + 56*b^3)/x^3

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Sympy [A]  time = 7.59288, size = 44, normalized size = 0.94 \[ - \frac{a^{3}}{2 x^{2}} - \frac{9 a^{2} b}{7 x^{\frac{7}{3}}} - \frac{9 a b^{2}}{8 x^{\frac{8}{3}}} - \frac{b^{3}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3/(2*x**2) - 9*a**2*b/(7*x**(7/3)) - 9*a*b**2/(8*x**(8/3)) - b**3/(3*x**3)

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GIAC/XCAS [A]  time = 0.211153, size = 47, normalized size = 1. \[ -\frac{84 \, a^{3} x + 216 \, a^{2} b x^{\frac{2}{3}} + 189 \, a b^{2} x^{\frac{1}{3}} + 56 \, b^{3}}{168 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/168*(84*a^3*x + 216*a^2*b*x^(2/3) + 189*a*b^2*x^(1/3) + 56*b^3)/x^3