3.2299 \(\int \left (a+b \sqrt [3]{x}\right )^3 x \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0680476, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{a^3 x^2}{2}+\frac{9}{7} a^2 b x^{7/3}+\frac{9}{8} a b^2 x^{8/3}+\frac{b^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^3*x,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.3657, size = 42, normalized size = 0.89 \[ \frac{a^{3} x^{2}}{2} + \frac{9 a^{2} b x^{\frac{7}{3}}}{7} + \frac{9 a b^{2} x^{\frac{8}{3}}}{8} + \frac{b^{3} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0109354, size = 47, normalized size = 1. \[ \frac{a^3 x^2}{2}+\frac{9}{7} a^2 b x^{7/3}+\frac{9}{8} a b^2 x^{8/3}+\frac{b^3 x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 36, normalized size = 0.8 \[{\frac{{x}^{2}{a}^{3}}{2}}+{\frac{9\,{a}^{2}b}{7}{x}^{{\frac{7}{3}}}}+{\frac{9\,a{b}^{2}}{8}{x}^{{\frac{8}{3}}}}+{\frac{{b}^{3}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^3*x,x)

[Out]

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

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Maxima [A]  time = 1.42071, size = 132, normalized size = 2.81 \[ \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{9}}{3 \, b^{6}} - \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{8} a}{8 \, b^{6}} + \frac{30 \,{\left (b x^{\frac{1}{3}} + a\right )}^{7} a^{2}}{7 \, b^{6}} - \frac{5 \,{\left (b x^{\frac{1}{3}} + a\right )}^{6} a^{3}}{b^{6}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{5} a^{4}}{b^{6}} - \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4} a^{5}}{4 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b*x^(1/3) + a)^9/b^6 - 15/8*(b*x^(1/3) + a)^8*a/b^6 + 30/7*(b*x^(1/3) + a)^
7*a^2/b^6 - 5*(b*x^(1/3) + a)^6*a^3/b^6 + 3*(b*x^(1/3) + a)^5*a^4/b^6 - 3/4*(b*x
^(1/3) + a)^4*a^5/b^6

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Fricas [A]  time = 0.210735, size = 47, normalized size = 1. \[ \frac{1}{3} \, b^{3} x^{3} + \frac{9}{8} \, a b^{2} x^{\frac{8}{3}} + \frac{9}{7} \, a^{2} b x^{\frac{7}{3}} + \frac{1}{2} \, a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.41488, size = 42, normalized size = 0.89 \[ \frac{a^{3} x^{2}}{2} + \frac{9 a^{2} b x^{\frac{7}{3}}}{7} + \frac{9 a b^{2} x^{\frac{8}{3}}}{8} + \frac{b^{3} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.221367, size = 47, normalized size = 1. \[ \frac{1}{3} \, b^{3} x^{3} + \frac{9}{8} \, a b^{2} x^{\frac{8}{3}} + \frac{9}{7} \, a^{2} b x^{\frac{7}{3}} + \frac{1}{2} \, a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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