3.2410 \(\int \frac{\left (a+\frac{b}{\sqrt [3]{x}}\right )^3}{x} \, dx\)

Optimal. Leaf size=39 \[ a^3 \log (x)-\frac{9 a^2 b}{\sqrt [3]{x}}-\frac{9 a b^2}{2 x^{2/3}}-\frac{b^3}{x} \]

[Out]

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

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Rubi [A]  time = 0.0601997, antiderivative size = 39, 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 \[ a^3 \log (x)-\frac{9 a^2 b}{\sqrt [3]{x}}-\frac{9 a b^2}{2 x^{2/3}}-\frac{b^3}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^3/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.86226, size = 41, normalized size = 1.05 \[ 3 a^{3} \log{\left (\sqrt [3]{x} \right )} - \frac{9 a^{2} b}{\sqrt [3]{x}} - \frac{9 a b^{2}}{2 x^{\frac{2}{3}}} - \frac{b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0358506, size = 40, normalized size = 1.03 \[ a^3 \log (x)-\frac{b \left (18 a^2 x^{2/3}+9 a b \sqrt [3]{x}+2 b^2\right )}{2 x} \]

Antiderivative was successfully verified.

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

[Out]

-(b*(2*b^2 + 9*a*b*x^(1/3) + 18*a^2*x^(2/3)))/(2*x) + a^3*Log[x]

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Maple [A]  time = 0.01, size = 34, normalized size = 0.9 \[ -{\frac{{b}^{3}}{x}}-{\frac{9\,a{b}^{2}}{2}{x}^{-{\frac{2}{3}}}}-9\,{\frac{{a}^{2}b}{\sqrt [3]{x}}}+{a}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^3/x,x)

[Out]

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

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Maxima [A]  time = 1.42827, size = 45, normalized size = 1.15 \[ a^{3} \log \left (x\right ) - \frac{9 \, a^{2} b}{x^{\frac{1}{3}}} - \frac{9 \, a b^{2}}{2 \, x^{\frac{2}{3}}} - \frac{b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22596, size = 53, normalized size = 1.36 \[ \frac{6 \, a^{3} x \log \left (x^{\frac{1}{3}}\right ) - 18 \, a^{2} b x^{\frac{2}{3}} - 9 \, a b^{2} x^{\frac{1}{3}} - 2 \, b^{3}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(6*a^3*x*log(x^(1/3)) - 18*a^2*b*x^(2/3) - 9*a*b^2*x^(1/3) - 2*b^3)/x

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Sympy [A]  time = 2.31978, size = 36, normalized size = 0.92 \[ a^{3} \log{\left (x \right )} - \frac{9 a^{2} b}{\sqrt [3]{x}} - \frac{9 a b^{2}}{2 x^{\frac{2}{3}}} - \frac{b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.214629, size = 50, normalized size = 1.28 \[ a^{3}{\rm ln}\left ({\left | x \right |}\right ) - \frac{18 \, a^{2} b x^{\frac{2}{3}} + 9 \, a b^{2} x^{\frac{1}{3}} + 2 \, b^{3}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^3*ln(abs(x)) - 1/2*(18*a^2*b*x^(2/3) + 9*a*b^2*x^(1/3) + 2*b^3)/x