3.2409 \(\int \left (a+\frac{b}{\sqrt [3]{x}}\right )^3 \, dx\)

Optimal. Leaf size=36 \[ a^3 x+\frac{9}{2} a^2 b x^{2/3}+9 a b^2 \sqrt [3]{x}+b^3 \log (x) \]

[Out]

9*a*b^2*x^(1/3) + (9*a^2*b*x^(2/3))/2 + a^3*x + b^3*Log[x]

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Rubi [A]  time = 0.053992, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ a^3 x+\frac{9}{2} a^2 b x^{2/3}+9 a b^2 \sqrt [3]{x}+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^3,x]

[Out]

9*a*b^2*x^(1/3) + (9*a^2*b*x^(2/3))/2 + a^3*x + b^3*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{3} x + 9 a^{2} b \int ^{\sqrt [3]{x}} x\, dx + 9 a b^{2} \sqrt [3]{x} + 3 b^{3} \log{\left (\sqrt [3]{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x + 9*a**2*b*Integral(x, (x, x**(1/3))) + 9*a*b**2*x**(1/3) + 3*b**3*log(x*
*(1/3))

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Mathematica [A]  time = 0.0150235, size = 36, normalized size = 1. \[ a^3 x+\frac{9}{2} a^2 b x^{2/3}+9 a b^2 \sqrt [3]{x}+b^3 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^(1/3))^3,x]

[Out]

9*a*b^2*x^(1/3) + (9*a^2*b*x^(2/3))/2 + a^3*x + b^3*Log[x]

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Maple [A]  time = 0.004, size = 31, normalized size = 0.9 \[ 9\,a{b}^{2}\sqrt [3]{x}+{\frac{9\,{a}^{2}b}{2}{x}^{{\frac{2}{3}}}}+{a}^{3}x+{b}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^3,x)

[Out]

9*a*b^2*x^(1/3)+9/2*a^2*b*x^(2/3)+a^3*x+b^3*ln(x)

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Maxima [A]  time = 1.43471, size = 41, normalized size = 1.14 \[ a^{3} x + b^{3} \log \left (x\right ) + \frac{9}{2} \, a^{2} b x^{\frac{2}{3}} + 9 \, a b^{2} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^3*x + b^3*log(x) + 9/2*a^2*b*x^(2/3) + 9*a*b^2*x^(1/3)

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Fricas [A]  time = 0.222779, size = 45, normalized size = 1.25 \[ a^{3} x + 3 \, b^{3} \log \left (x^{\frac{1}{3}}\right ) + \frac{9}{2} \, a^{2} b x^{\frac{2}{3}} + 9 \, a b^{2} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^3*x + 3*b^3*log(x^(1/3)) + 9/2*a^2*b*x^(2/3) + 9*a*b^2*x^(1/3)

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Sympy [A]  time = 0.624887, size = 36, normalized size = 1. \[ a^{3} x + \frac{9 a^{2} b x^{\frac{2}{3}}}{2} + 9 a b^{2} \sqrt [3]{x} + b^{3} \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**(1/3))**3,x)

[Out]

a**3*x + 9*a**2*b*x**(2/3)/2 + 9*a*b**2*x**(1/3) + b**3*log(x)

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GIAC/XCAS [A]  time = 0.210034, size = 42, normalized size = 1.17 \[ a^{3} x + b^{3}{\rm ln}\left ({\left | x \right |}\right ) + \frac{9}{2} \, a^{2} b x^{\frac{2}{3}} + 9 \, a b^{2} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^3*x + b^3*ln(abs(x)) + 9/2*a^2*b*x^(2/3) + 9*a*b^2*x^(1/3)