3.2291 \(\int \left (a+b \sqrt [3]{x}\right )^2 \, dx\)

Optimal. Leaf size=29 \[ a^2 x+\frac{3}{2} a b x^{4/3}+\frac{3}{5} b^2 x^{5/3} \]

[Out]

a^2*x + (3*a*b*x^(4/3))/2 + (3*b^2*x^(5/3))/5

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Rubi [A]  time = 0.0435078, antiderivative size = 29, 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 \[ a^2 x+\frac{3}{2} a b x^{4/3}+\frac{3}{5} b^2 x^{5/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^2,x]

[Out]

a^2*x + (3*a*b*x^(4/3))/2 + (3*b^2*x^(5/3))/5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a b x^{\frac{4}{3}}}{2} + \frac{3 b^{2} x^{\frac{5}{3}}}{5} + \int a^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a*b*x**(4/3)/2 + 3*b**2*x**(5/3)/5 + Integral(a**2, x)

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Mathematica [A]  time = 0.00837939, size = 29, normalized size = 1. \[ a^2 x+\frac{3}{2} a b x^{4/3}+\frac{3}{5} b^2 x^{5/3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^2,x]

[Out]

a^2*x + (3*a*b*x^(4/3))/2 + (3*b^2*x^(5/3))/5

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Maple [A]  time = 0.002, size = 22, normalized size = 0.8 \[ x{a}^{2}+{\frac{3\,ab}{2}{x}^{{\frac{4}{3}}}}+{\frac{3\,{b}^{2}}{5}{x}^{{\frac{5}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^2,x)

[Out]

x*a^2+3/2*a*b*x^(4/3)+3/5*b^2*x^(5/3)

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Maxima [A]  time = 1.43597, size = 28, normalized size = 0.97 \[ \frac{3}{5} \, b^{2} x^{\frac{5}{3}} + \frac{3}{2} \, a b x^{\frac{4}{3}} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/5*b^2*x^(5/3) + 3/2*a*b*x^(4/3) + a^2*x

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Fricas [A]  time = 0.21119, size = 28, normalized size = 0.97 \[ \frac{3}{5} \, b^{2} x^{\frac{5}{3}} + \frac{3}{2} \, a b x^{\frac{4}{3}} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/5*b^2*x^(5/3) + 3/2*a*b*x^(4/3) + a^2*x

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Sympy [A]  time = 1.11379, size = 27, normalized size = 0.93 \[ a^{2} x + \frac{3 a b x^{\frac{4}{3}}}{2} + \frac{3 b^{2} x^{\frac{5}{3}}}{5} \]

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**(4/3)/2 + 3*b**2*x**(5/3)/5

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GIAC/XCAS [A]  time = 0.285884, size = 28, normalized size = 0.97 \[ \frac{3}{5} \, b^{2} x^{\frac{5}{3}} + \frac{3}{2} \, a b x^{\frac{4}{3}} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/5*b^2*x^(5/3) + 3/2*a*b*x^(4/3) + a^2*x