3.2304 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^3}{x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0555846, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{a^3}{3 x^3}-\frac{9 a^2 b}{8 x^{8/3}}-\frac{9 a b^2}{7 x^{7/3}}-\frac{b^3}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^3/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.09271, size = 44, normalized size = 0.94 \[ - \frac{a^{3}}{3 x^{3}} - \frac{9 a^{2} b}{8 x^{\frac{8}{3}}} - \frac{9 a b^{2}}{7 x^{\frac{7}{3}}} - \frac{b^{3}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**3/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.0137925, size = 41, normalized size = 0.87 \[ -\frac{56 a^3+189 a^2 b \sqrt [3]{x}+216 a b^2 x^{2/3}+84 b^3 x}{168 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^3/x^4,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^3/x^4,x)

[Out]

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

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Maxima [A]  time = 1.46555, size = 47, normalized size = 1. \[ -\frac{84 \, b^{3} x + 216 \, a b^{2} x^{\frac{2}{3}} + 189 \, a^{2} b x^{\frac{1}{3}} + 56 \, a^{3}}{168 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^3/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.213939, size = 47, normalized size = 1. \[ -\frac{84 \, b^{3} x + 216 \, a b^{2} x^{\frac{2}{3}} + 189 \, a^{2} b x^{\frac{1}{3}} + 56 \, a^{3}}{168 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^3/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.54943, size = 44, normalized size = 0.94 \[ - \frac{a^{3}}{3 x^{3}} - \frac{9 a^{2} b}{8 x^{\frac{8}{3}}} - \frac{9 a b^{2}}{7 x^{\frac{7}{3}}} - \frac{b^{3}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**3/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218806, size = 47, normalized size = 1. \[ -\frac{84 \, b^{3} x + 216 \, a b^{2} x^{\frac{2}{3}} + 189 \, a^{2} b x^{\frac{1}{3}} + 56 \, a^{3}}{168 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^3/x^4,x, algorithm="giac")

[Out]

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