3.2398 \(\int \left (a+\frac{b}{\sqrt [3]{x}}\right )^2 x^2 \, dx\)

Optimal. Leaf size=34 \[ \frac{a^2 x^3}{3}+\frac{3}{4} a b x^{8/3}+\frac{3}{7} b^2 x^{7/3} \]

[Out]

(3*b^2*x^(7/3))/7 + (3*a*b*x^(8/3))/4 + (a^2*x^3)/3

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Rubi [A]  time = 0.064753, antiderivative size = 34, 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 \[ \frac{a^2 x^3}{3}+\frac{3}{4} a b x^{8/3}+\frac{3}{7} b^2 x^{7/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^2*x^2,x]

[Out]

(3*b^2*x^(7/3))/7 + (3*a*b*x^(8/3))/4 + (a^2*x^3)/3

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Rubi in Sympy [A]  time = 9.85524, size = 31, normalized size = 0.91 \[ \frac{a^{2} x^{3}}{3} + \frac{3 a b x^{\frac{8}{3}}}{4} + \frac{3 b^{2} x^{\frac{7}{3}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**3/3 + 3*a*b*x**(8/3)/4 + 3*b**2*x**(7/3)/7

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Mathematica [A]  time = 0.0157086, size = 34, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{3}{4} a b x^{8/3}+\frac{3}{7} b^2 x^{7/3} \]

Antiderivative was successfully verified.

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

[Out]

(3*b^2*x^(7/3))/7 + (3*a*b*x^(8/3))/4 + (a^2*x^3)/3

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Maple [A]  time = 0.001, size = 25, normalized size = 0.7 \[{\frac{3\,{b}^{2}}{7}{x}^{{\frac{7}{3}}}}+{\frac{3\,ab}{4}{x}^{{\frac{8}{3}}}}+{\frac{{x}^{3}{a}^{2}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^2*x^2,x)

[Out]

3/7*b^2*x^(7/3)+3/4*a*b*x^(8/3)+1/3*x^3*a^2

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Maxima [A]  time = 1.43734, size = 35, normalized size = 1.03 \[ \frac{1}{84} \,{\left (28 \, a^{2} + \frac{63 \, a b}{x^{\frac{1}{3}}} + \frac{36 \, b^{2}}{x^{\frac{2}{3}}}\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/84*(28*a^2 + 63*a*b/x^(1/3) + 36*b^2/x^(2/3))*x^3

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Fricas [A]  time = 0.219427, size = 32, normalized size = 0.94 \[ \frac{1}{3} \, a^{2} x^{3} + \frac{3}{4} \, a b x^{\frac{8}{3}} + \frac{3}{7} \, b^{2} x^{\frac{7}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^2*x^3 + 3/4*a*b*x^(8/3) + 3/7*b^2*x^(7/3)

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Sympy [A]  time = 2.39865, size = 31, normalized size = 0.91 \[ \frac{a^{2} x^{3}}{3} + \frac{3 a b x^{\frac{8}{3}}}{4} + \frac{3 b^{2} x^{\frac{7}{3}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**3/3 + 3*a*b*x**(8/3)/4 + 3*b**2*x**(7/3)/7

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GIAC/XCAS [A]  time = 0.21014, size = 32, normalized size = 0.94 \[ \frac{1}{3} \, a^{2} x^{3} + \frac{3}{4} \, a b x^{\frac{8}{3}} + \frac{3}{7} \, b^{2} x^{\frac{7}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^2*x^3 + 3/4*a*b*x^(8/3) + 3/7*b^2*x^(7/3)