3.221 \(\int x \left (a+b x^3\right )^2 \, dx\)

Optimal. Leaf size=30 \[ \frac{a^2 x^2}{2}+\frac{2}{5} a b x^5+\frac{b^2 x^8}{8} \]

[Out]

(a^2*x^2)/2 + (2*a*b*x^5)/5 + (b^2*x^8)/8

_______________________________________________________________________________________

Rubi [A]  time = 0.0300272, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a^2 x^2}{2}+\frac{2}{5} a b x^5+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^3)^2,x]

[Out]

(a^2*x^2)/2 + (2*a*b*x^5)/5 + (b^2*x^8)/8

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{2} \int x\, dx + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*Integral(x, x) + 2*a*b*x**5/5 + b**2*x**8/8

_______________________________________________________________________________________

Mathematica [A]  time = 0.00237939, size = 30, normalized size = 1. \[ \frac{a^2 x^2}{2}+\frac{2}{5} a b x^5+\frac{b^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x^3)^2,x]

[Out]

(a^2*x^2)/2 + (2*a*b*x^5)/5 + (b^2*x^8)/8

_______________________________________________________________________________________

Maple [A]  time = 0., size = 25, normalized size = 0.8 \[{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{2\,{x}^{5}ab}{5}}+{\frac{{b}^{2}{x}^{8}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^3+a)^2,x)

[Out]

1/2*a^2*x^2+2/5*x^5*a*b+1/8*b^2*x^8

_______________________________________________________________________________________

Maxima [A]  time = 1.4479, size = 32, normalized size = 1.07 \[ \frac{1}{8} \, b^{2} x^{8} + \frac{2}{5} \, a b x^{5} + \frac{1}{2} \, a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*b^2*x^8 + 2/5*a*b*x^5 + 1/2*a^2*x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.213689, size = 1, normalized size = 0.03 \[ \frac{1}{8} x^{8} b^{2} + \frac{2}{5} x^{5} b a + \frac{1}{2} x^{2} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*x^8*b^2 + 2/5*x^5*b*a + 1/2*x^2*a^2

_______________________________________________________________________________________

Sympy [A]  time = 0.083848, size = 26, normalized size = 0.87 \[ \frac{a^{2} x^{2}}{2} + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x**3+a)**2,x)

[Out]

a**2*x**2/2 + 2*a*b*x**5/5 + b**2*x**8/8

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217683, size = 32, normalized size = 1.07 \[ \frac{1}{8} \, b^{2} x^{8} + \frac{2}{5} \, a b x^{5} + \frac{1}{2} \, a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*b^2*x^8 + 2/5*a*b*x^5 + 1/2*a^2*x^2