3.2407 \(\int \left (a+\frac{b}{\sqrt [3]{x}}\right )^3 x^2 \, dx\)

Optimal. Leaf size=47 \[ \frac{a^3 x^3}{3}+\frac{9}{8} a^2 b x^{8/3}+\frac{9}{7} a b^2 x^{7/3}+\frac{b^3 x^2}{2} \]

[Out]

(b^3*x^2)/2 + (9*a*b^2*x^(7/3))/7 + (9*a^2*b*x^(8/3))/8 + (a^3*x^3)/3

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Rubi [A]  time = 0.0766702, 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 \[ \frac{a^3 x^3}{3}+\frac{9}{8} a^2 b x^{8/3}+\frac{9}{7} a b^2 x^{7/3}+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^3*x^2,x]

[Out]

(b^3*x^2)/2 + (9*a*b^2*x^(7/3))/7 + (9*a^2*b*x^(8/3))/8 + (a^3*x^3)/3

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Rubi in Sympy [A]  time = 11.7312, size = 42, normalized size = 0.89 \[ \frac{a^{3} x^{3}}{3} + \frac{9 a^{2} b x^{\frac{8}{3}}}{8} + \frac{9 a b^{2} x^{\frac{7}{3}}}{7} + \frac{b^{3} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**3/3 + 9*a**2*b*x**(8/3)/8 + 9*a*b**2*x**(7/3)/7 + b**3*x**2/2

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Mathematica [A]  time = 0.0120192, size = 47, normalized size = 1. \[ \frac{a^3 x^3}{3}+\frac{9}{8} a^2 b x^{8/3}+\frac{9}{7} a b^2 x^{7/3}+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

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

[Out]

(b^3*x^2)/2 + (9*a*b^2*x^(7/3))/7 + (9*a^2*b*x^(8/3))/8 + (a^3*x^3)/3

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Maple [A]  time = 0.002, size = 36, normalized size = 0.8 \[{\frac{{b}^{3}{x}^{2}}{2}}+{\frac{9\,a{b}^{2}}{7}{x}^{{\frac{7}{3}}}}+{\frac{9\,{a}^{2}b}{8}{x}^{{\frac{8}{3}}}}+{\frac{{a}^{3}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^(1/3))^3*x^2,x)

[Out]

1/2*b^3*x^2+9/7*a*b^2*x^(7/3)+9/8*a^2*b*x^(8/3)+1/3*a^3*x^3

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Maxima [A]  time = 1.43676, size = 50, normalized size = 1.06 \[ \frac{1}{168} \,{\left (56 \, a^{3} + \frac{189 \, a^{2} b}{x^{\frac{1}{3}}} + \frac{216 \, a b^{2}}{x^{\frac{2}{3}}} + \frac{84 \, b^{3}}{x}\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/168*(56*a^3 + 189*a^2*b/x^(1/3) + 216*a*b^2/x^(2/3) + 84*b^3/x)*x^3

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Fricas [A]  time = 0.217441, size = 47, normalized size = 1. \[ \frac{1}{3} \, a^{3} x^{3} + \frac{9}{8} \, a^{2} b x^{\frac{8}{3}} + \frac{9}{7} \, a b^{2} x^{\frac{7}{3}} + \frac{1}{2} \, b^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^3*x^3 + 9/8*a^2*b*x^(8/3) + 9/7*a*b^2*x^(7/3) + 1/2*b^3*x^2

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Sympy [A]  time = 3.51304, size = 42, normalized size = 0.89 \[ \frac{a^{3} x^{3}}{3} + \frac{9 a^{2} b x^{\frac{8}{3}}}{8} + \frac{9 a b^{2} x^{\frac{7}{3}}}{7} + \frac{b^{3} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x**3/3 + 9*a**2*b*x**(8/3)/8 + 9*a*b**2*x**(7/3)/7 + b**3*x**2/2

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GIAC/XCAS [A]  time = 0.208947, size = 47, normalized size = 1. \[ \frac{1}{3} \, a^{3} x^{3} + \frac{9}{8} \, a^{2} b x^{\frac{8}{3}} + \frac{9}{7} \, a b^{2} x^{\frac{7}{3}} + \frac{1}{2} \, b^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*a^3*x^3 + 9/8*a^2*b*x^(8/3) + 9/7*a*b^2*x^(7/3) + 1/2*b^3*x^2