3.2400 \(\int \left (a+\frac{b}{\sqrt [3]{x}}\right )^2 \, dx\)

Optimal. Leaf size=16 \[ \frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )^3}{a} \]

[Out]

((a + b/x^(1/3))^3*x)/a

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Rubi [A]  time = 0.0252124, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )^3}{a} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^2,x]

[Out]

((a + b/x^(1/3))^3*x)/a

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Rubi in Sympy [A]  time = 1.28571, size = 12, normalized size = 0.75 \[ \frac{x \left (a + \frac{b}{\sqrt [3]{x}}\right )^{3}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**(1/3))**2,x)

[Out]

x*(a + b/x**(1/3))**3/a

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Mathematica [A]  time = 0.007462, size = 25, normalized size = 1.56 \[ a^2 x+3 a b x^{2/3}+3 b^2 \sqrt [3]{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^(1/3))^2,x]

[Out]

3*b^2*x^(1/3) + 3*a*b*x^(2/3) + a^2*x

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Maple [A]  time = 0.002, size = 14, normalized size = 0.9 \[{\frac{1}{a} \left ( b+a\sqrt [3]{x} \right ) ^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^2,x)

[Out]

(b+a*x^(1/3))^3/a

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Maxima [A]  time = 1.441, size = 28, normalized size = 1.75 \[ a^{2} x + 3 \, a b x^{\frac{2}{3}} + 3 \, b^{2} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^2,x, algorithm="maxima")

[Out]

a^2*x + 3*a*b*x^(2/3) + 3*b^2*x^(1/3)

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Fricas [A]  time = 0.217698, size = 28, normalized size = 1.75 \[ a^{2} x + 3 \, a b x^{\frac{2}{3}} + 3 \, b^{2} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^2,x, algorithm="fricas")

[Out]

a^2*x + 3*a*b*x^(2/3) + 3*b^2*x^(1/3)

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Sympy [A]  time = 0.421584, size = 24, normalized size = 1.5 \[ a^{2} x + 3 a b x^{\frac{2}{3}} + 3 b^{2} \sqrt [3]{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**(1/3))**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.211036, size = 28, normalized size = 1.75 \[ a^{2} x + 3 \, a b x^{\frac{2}{3}} + 3 \, b^{2} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^(1/3))^2,x, algorithm="giac")

[Out]

a^2*x + 3*a*b*x^(2/3) + 3*b^2*x^(1/3)