3.2399 \(\int \left (a+\frac{b}{\sqrt [3]{x}}\right )^2 x \, dx\)

Optimal. Leaf size=34 \[ \frac{a^2 x^2}{2}+\frac{6}{5} a b x^{5/3}+\frac{3}{4} b^2 x^{4/3} \]

[Out]

(3*b^2*x^(4/3))/4 + (6*a*b*x^(5/3))/5 + (a^2*x^2)/2

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Rubi [A]  time = 0.0587188, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a^2 x^2}{2}+\frac{6}{5} a b x^{5/3}+\frac{3}{4} b^2 x^{4/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^2*x,x]

[Out]

(3*b^2*x^(4/3))/4 + (6*a*b*x^(5/3))/5 + (a^2*x^2)/2

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Rubi in Sympy [A]  time = 8.84955, size = 31, normalized size = 0.91 \[ \frac{a^{2} x^{2}}{2} + \frac{6 a b x^{\frac{5}{3}}}{5} + \frac{3 b^{2} x^{\frac{4}{3}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**2/2 + 6*a*b*x**(5/3)/5 + 3*b**2*x**(4/3)/4

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Mathematica [A]  time = 0.0111799, size = 34, normalized size = 1. \[ \frac{a^2 x^2}{2}+\frac{6}{5} a b x^{5/3}+\frac{3}{4} b^2 x^{4/3} \]

Antiderivative was successfully verified.

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

[Out]

(3*b^2*x^(4/3))/4 + (6*a*b*x^(5/3))/5 + (a^2*x^2)/2

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Maple [A]  time = 0.001, size = 25, normalized size = 0.7 \[{\frac{3\,{b}^{2}}{4}{x}^{{\frac{4}{3}}}}+{\frac{6\,ab}{5}{x}^{{\frac{5}{3}}}}+{\frac{{a}^{2}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^2*x,x)

[Out]

3/4*b^2*x^(4/3)+6/5*a*b*x^(5/3)+1/2*a^2*x^2

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Maxima [A]  time = 1.43608, size = 35, normalized size = 1.03 \[ \frac{1}{20} \,{\left (10 \, a^{2} + \frac{24 \, a b}{x^{\frac{1}{3}}} + \frac{15 \, b^{2}}{x^{\frac{2}{3}}}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(10*a^2 + 24*a*b/x^(1/3) + 15*b^2/x^(2/3))*x^2

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Fricas [A]  time = 0.221071, size = 32, normalized size = 0.94 \[ \frac{1}{2} \, a^{2} x^{2} + \frac{6}{5} \, a b x^{\frac{5}{3}} + \frac{3}{4} \, b^{2} x^{\frac{4}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a^2*x^2 + 6/5*a*b*x^(5/3) + 3/4*b^2*x^(4/3)

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Sympy [A]  time = 1.9747, size = 31, normalized size = 0.91 \[ \frac{a^{2} x^{2}}{2} + \frac{6 a b x^{\frac{5}{3}}}{5} + \frac{3 b^{2} x^{\frac{4}{3}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**2/2 + 6*a*b*x**(5/3)/5 + 3*b**2*x**(4/3)/4

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GIAC/XCAS [A]  time = 0.211286, size = 32, normalized size = 0.94 \[ \frac{1}{2} \, a^{2} x^{2} + \frac{6}{5} \, a b x^{\frac{5}{3}} + \frac{3}{4} \, b^{2} x^{\frac{4}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a^2*x^2 + 6/5*a*b*x^(5/3) + 3/4*b^2*x^(4/3)