3.2397 \(\int \left (a+\frac{b}{\sqrt [3]{x}}\right )^2 x^3 \, dx\)

Optimal. Leaf size=34 \[ \frac{a^2 x^4}{4}+\frac{6}{11} a b x^{11/3}+\frac{3}{10} b^2 x^{10/3} \]

[Out]

(3*b^2*x^(10/3))/10 + (6*a*b*x^(11/3))/11 + (a^2*x^4)/4

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Rubi [A]  time = 0.0734771, 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 \[ \frac{a^2 x^4}{4}+\frac{6}{11} a b x^{11/3}+\frac{3}{10} b^2 x^{10/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^2*x^3,x]

[Out]

(3*b^2*x^(10/3))/10 + (6*a*b*x^(11/3))/11 + (a^2*x^4)/4

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Rubi in Sympy [A]  time = 11.4955, size = 31, normalized size = 0.91 \[ \frac{a^{2} x^{4}}{4} + \frac{6 a b x^{\frac{11}{3}}}{11} + \frac{3 b^{2} x^{\frac{10}{3}}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**4/4 + 6*a*b*x**(11/3)/11 + 3*b**2*x**(10/3)/10

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Mathematica [A]  time = 0.0090696, size = 34, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{6}{11} a b x^{11/3}+\frac{3}{10} b^2 x^{10/3} \]

Antiderivative was successfully verified.

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

[Out]

(3*b^2*x^(10/3))/10 + (6*a*b*x^(11/3))/11 + (a^2*x^4)/4

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Maple [A]  time = 0.002, size = 25, normalized size = 0.7 \[{\frac{3\,{b}^{2}}{10}{x}^{{\frac{10}{3}}}}+{\frac{6\,ab}{11}{x}^{{\frac{11}{3}}}}+{\frac{{x}^{4}{a}^{2}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^2*x^3,x)

[Out]

3/10*b^2*x^(10/3)+6/11*a*b*x^(11/3)+1/4*x^4*a^2

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Maxima [A]  time = 1.43193, size = 35, normalized size = 1.03 \[ \frac{1}{220} \,{\left (55 \, a^{2} + \frac{120 \, a b}{x^{\frac{1}{3}}} + \frac{66 \, b^{2}}{x^{\frac{2}{3}}}\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/220*(55*a^2 + 120*a*b/x^(1/3) + 66*b^2/x^(2/3))*x^4

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Fricas [A]  time = 0.219807, size = 32, normalized size = 0.94 \[ \frac{1}{4} \, a^{2} x^{4} + \frac{6}{11} \, a b x^{\frac{11}{3}} + \frac{3}{10} \, b^{2} x^{\frac{10}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*a^2*x^4 + 6/11*a*b*x^(11/3) + 3/10*b^2*x^(10/3)

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Sympy [A]  time = 5.70381, size = 31, normalized size = 0.91 \[ \frac{a^{2} x^{4}}{4} + \frac{6 a b x^{\frac{11}{3}}}{11} + \frac{3 b^{2} x^{\frac{10}{3}}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**4/4 + 6*a*b*x**(11/3)/11 + 3*b**2*x**(10/3)/10

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GIAC/XCAS [A]  time = 0.209805, size = 32, normalized size = 0.94 \[ \frac{1}{4} \, a^{2} x^{4} + \frac{6}{11} \, a b x^{\frac{11}{3}} + \frac{3}{10} \, b^{2} x^{\frac{10}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*a^2*x^4 + 6/11*a*b*x^(11/3) + 3/10*b^2*x^(10/3)