3.2300 \(\int \left (a+b \sqrt [3]{x}\right )^3 \, dx\)

Optimal. Leaf size=42 \[ a^3 x+\frac{9}{4} a^2 b x^{4/3}+\frac{9}{5} a b^2 x^{5/3}+\frac{b^3 x^2}{2} \]

[Out]

a^3*x + (9*a^2*b*x^(4/3))/4 + (9*a*b^2*x^(5/3))/5 + (b^3*x^2)/2

_______________________________________________________________________________________

Rubi [A]  time = 0.0607792, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ a^3 x+\frac{9}{4} a^2 b x^{4/3}+\frac{9}{5} a b^2 x^{5/3}+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^3,x]

[Out]

a^3*x + (9*a^2*b*x^(4/3))/4 + (9*a*b^2*x^(5/3))/5 + (b^3*x^2)/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.2966, size = 39, normalized size = 0.93 \[ a^{3} x + \frac{9 a^{2} b x^{\frac{4}{3}}}{4} + \frac{9 a b^{2} x^{\frac{5}{3}}}{5} + \frac{b^{3} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*x + 9*a**2*b*x**(4/3)/4 + 9*a*b**2*x**(5/3)/5 + b**3*x**2/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.00924879, size = 42, normalized size = 1. \[ a^3 x+\frac{9}{4} a^2 b x^{4/3}+\frac{9}{5} a b^2 x^{5/3}+\frac{b^3 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^3,x]

[Out]

a^3*x + (9*a^2*b*x^(4/3))/4 + (9*a*b^2*x^(5/3))/5 + (b^3*x^2)/2

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 33, normalized size = 0.8 \[{a}^{3}x+{\frac{9\,{a}^{2}b}{4}{x}^{{\frac{4}{3}}}}+{\frac{9\,a{b}^{2}}{5}{x}^{{\frac{5}{3}}}}+{\frac{{b}^{3}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^3,x)

[Out]

a^3*x+9/4*a^2*b*x^(4/3)+9/5*a*b^2*x^(5/3)+1/2*b^3*x^2

_______________________________________________________________________________________

Maxima [A]  time = 1.4467, size = 43, normalized size = 1.02 \[ \frac{1}{2} \, b^{3} x^{2} + \frac{9}{5} \, a b^{2} x^{\frac{5}{3}} + \frac{9}{4} \, a^{2} b x^{\frac{4}{3}} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^3*x^2 + 9/5*a*b^2*x^(5/3) + 9/4*a^2*b*x^(4/3) + a^3*x

_______________________________________________________________________________________

Fricas [A]  time = 0.214309, size = 43, normalized size = 1.02 \[ \frac{1}{2} \, b^{3} x^{2} + \frac{9}{5} \, a b^{2} x^{\frac{5}{3}} + \frac{9}{4} \, a^{2} b x^{\frac{4}{3}} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^3*x^2 + 9/5*a*b^2*x^(5/3) + 9/4*a^2*b*x^(4/3) + a^3*x

_______________________________________________________________________________________

Sympy [A]  time = 1.18059, size = 39, normalized size = 0.93 \[ a^{3} x + \frac{9 a^{2} b x^{\frac{4}{3}}}{4} + \frac{9 a b^{2} x^{\frac{5}{3}}}{5} + \frac{b^{3} x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**3,x)

[Out]

a**3*x + 9*a**2*b*x**(4/3)/4 + 9*a*b**2*x**(5/3)/5 + b**3*x**2/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212922, size = 43, normalized size = 1.02 \[ \frac{1}{2} \, b^{3} x^{2} + \frac{9}{5} \, a b^{2} x^{\frac{5}{3}} + \frac{9}{4} \, a^{2} b x^{\frac{4}{3}} + a^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b^3*x^2 + 9/5*a*b^2*x^(5/3) + 9/4*a^2*b*x^(4/3) + a^3*x