3.2301 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^3}{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.049095, antiderivative size = 36, 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 \[ a^3 \log (x)+9 a^2 b \sqrt [3]{x}+\frac{9}{2} a b^2 x^{2/3}+b^3 x \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^3/x,x]

[Out]

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

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

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*Integral(x, (x, x**(1/3))) +
b**3*x

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Mathematica [A]  time = 0.0141602, size = 36, normalized size = 1. \[ a^3 \log (x)+9 a^2 b \sqrt [3]{x}+\frac{9}{2} a b^2 x^{2/3}+b^3 x \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^3/x,x)

[Out]

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

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Maxima [A]  time = 1.43571, size = 41, normalized size = 1.14 \[ b^{3} x + a^{3} \log \left (x\right ) + \frac{9}{2} \, a b^{2} x^{\frac{2}{3}} + 9 \, a^{2} b x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218813, size = 45, normalized size = 1.25 \[ b^{3} x + 3 \, a^{3} \log \left (x^{\frac{1}{3}}\right ) + \frac{9}{2} \, a b^{2} x^{\frac{2}{3}} + 9 \, a^{2} b x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.635514, size = 36, normalized size = 1. \[ a^{3} \log{\left (x \right )} + 9 a^{2} b \sqrt [3]{x} + \frac{9 a b^{2} x^{\frac{2}{3}}}{2} + b^{3} x \]

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*x**(2/3)/2 + b**3*x

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GIAC/XCAS [A]  time = 0.218421, size = 42, normalized size = 1.17 \[ b^{3} x + a^{3}{\rm ln}\left ({\left | x \right |}\right ) + \frac{9}{2} \, a b^{2} x^{\frac{2}{3}} + 9 \, a^{2} b x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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