3.2406 \(\int \left (a+\frac{b}{\sqrt [3]{x}}\right )^3 x^3 \, dx\)

Optimal. Leaf size=47 \[ \frac{a^3 x^4}{4}+\frac{9}{11} a^2 b x^{11/3}+\frac{9}{10} a b^2 x^{10/3}+\frac{b^3 x^3}{3} \]

[Out]

(b^3*x^3)/3 + (9*a*b^2*x^(10/3))/10 + (9*a^2*b*x^(11/3))/11 + (a^3*x^4)/4

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Rubi [A]  time = 0.0838871, antiderivative size = 47, 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 \[ \frac{a^3 x^4}{4}+\frac{9}{11} a^2 b x^{11/3}+\frac{9}{10} a b^2 x^{10/3}+\frac{b^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^3*x^3,x]

[Out]

(b^3*x^3)/3 + (9*a*b^2*x^(10/3))/10 + (9*a^2*b*x^(11/3))/11 + (a^3*x^4)/4

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Rubi in Sympy [A]  time = 13.1242, size = 42, normalized size = 0.89 \[ \frac{a^{3} x^{4}}{4} + \frac{9 a^{2} b x^{\frac{11}{3}}}{11} + \frac{9 a b^{2} x^{\frac{10}{3}}}{10} + \frac{b^{3} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**4/4 + 9*a**2*b*x**(11/3)/11 + 9*a*b**2*x**(10/3)/10 + b**3*x**3/3

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Mathematica [A]  time = 0.0121181, size = 41, normalized size = 0.87 \[ \frac{1}{660} x^3 \left (165 a^3 x+540 a^2 b x^{2/3}+594 a b^2 \sqrt [3]{x}+220 b^3\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(220*b^3 + 594*a*b^2*x^(1/3) + 540*a^2*b*x^(2/3) + 165*a^3*x))/660

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Maple [A]  time = 0.003, size = 36, normalized size = 0.8 \[{\frac{{b}^{3}{x}^{3}}{3}}+{\frac{9\,a{b}^{2}}{10}{x}^{{\frac{10}{3}}}}+{\frac{9\,{a}^{2}b}{11}{x}^{{\frac{11}{3}}}}+{\frac{{a}^{3}{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^3*x^3,x)

[Out]

1/3*b^3*x^3+9/10*a*b^2*x^(10/3)+9/11*a^2*b*x^(11/3)+1/4*a^3*x^4

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Maxima [A]  time = 1.43766, size = 50, normalized size = 1.06 \[ \frac{1}{660} \,{\left (165 \, a^{3} + \frac{540 \, a^{2} b}{x^{\frac{1}{3}}} + \frac{594 \, a b^{2}}{x^{\frac{2}{3}}} + \frac{220 \, b^{3}}{x}\right )} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/660*(165*a^3 + 540*a^2*b/x^(1/3) + 594*a*b^2/x^(2/3) + 220*b^3/x)*x^4

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Fricas [A]  time = 0.218987, size = 47, normalized size = 1. \[ \frac{1}{4} \, a^{3} x^{4} + \frac{9}{11} \, a^{2} b x^{\frac{11}{3}} + \frac{9}{10} \, a b^{2} x^{\frac{10}{3}} + \frac{1}{3} \, b^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*a^3*x^4 + 9/11*a^2*b*x^(11/3) + 9/10*a*b^2*x^(10/3) + 1/3*b^3*x^3

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Sympy [A]  time = 7.31447, size = 42, normalized size = 0.89 \[ \frac{a^{3} x^{4}}{4} + \frac{9 a^{2} b x^{\frac{11}{3}}}{11} + \frac{9 a b^{2} x^{\frac{10}{3}}}{10} + \frac{b^{3} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**4/4 + 9*a**2*b*x**(11/3)/11 + 9*a*b**2*x**(10/3)/10 + b**3*x**3/3

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GIAC/XCAS [A]  time = 0.212023, size = 47, normalized size = 1. \[ \frac{1}{4} \, a^{3} x^{4} + \frac{9}{11} \, a^{2} b x^{\frac{11}{3}} + \frac{9}{10} \, a b^{2} x^{\frac{10}{3}} + \frac{1}{3} \, b^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*a^3*x^4 + 9/11*a^2*b*x^(11/3) + 9/10*a*b^2*x^(10/3) + 1/3*b^3*x^3