3.631 \(\int x^5 \left (a+b x^4\right )^3 \, dx\)

Optimal. Leaf size=43 \[ \frac{a^3 x^6}{6}+\frac{3}{10} a^2 b x^{10}+\frac{3}{14} a b^2 x^{14}+\frac{b^3 x^{18}}{18} \]

[Out]

(a^3*x^6)/6 + (3*a^2*b*x^10)/10 + (3*a*b^2*x^14)/14 + (b^3*x^18)/18

_______________________________________________________________________________________

Rubi [A]  time = 0.0439554, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{a^3 x^6}{6}+\frac{3}{10} a^2 b x^{10}+\frac{3}{14} a b^2 x^{14}+\frac{b^3 x^{18}}{18} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^4)^3,x]

[Out]

(a^3*x^6)/6 + (3*a^2*b*x^10)/10 + (3*a*b^2*x^14)/14 + (b^3*x^18)/18

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.89522, size = 39, normalized size = 0.91 \[ \frac{a^{3} x^{6}}{6} + \frac{3 a^{2} b x^{10}}{10} + \frac{3 a b^{2} x^{14}}{14} + \frac{b^{3} x^{18}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x**4+a)**3,x)

[Out]

a**3*x**6/6 + 3*a**2*b*x**10/10 + 3*a*b**2*x**14/14 + b**3*x**18/18

_______________________________________________________________________________________

Mathematica [A]  time = 0.00389707, size = 43, normalized size = 1. \[ \frac{a^3 x^6}{6}+\frac{3}{10} a^2 b x^{10}+\frac{3}{14} a b^2 x^{14}+\frac{b^3 x^{18}}{18} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x^4)^3,x]

[Out]

(a^3*x^6)/6 + (3*a^2*b*x^10)/10 + (3*a*b^2*x^14)/14 + (b^3*x^18)/18

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 36, normalized size = 0.8 \[{\frac{{a}^{3}{x}^{6}}{6}}+{\frac{3\,{a}^{2}b{x}^{10}}{10}}+{\frac{3\,a{b}^{2}{x}^{14}}{14}}+{\frac{{b}^{3}{x}^{18}}{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^4+a)^3,x)

[Out]

1/6*a^3*x^6+3/10*a^2*b*x^10+3/14*a*b^2*x^14+1/18*b^3*x^18

_______________________________________________________________________________________

Maxima [A]  time = 1.42456, size = 47, normalized size = 1.09 \[ \frac{1}{18} \, b^{3} x^{18} + \frac{3}{14} \, a b^{2} x^{14} + \frac{3}{10} \, a^{2} b x^{10} + \frac{1}{6} \, a^{3} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^3*x^5,x, algorithm="maxima")

[Out]

1/18*b^3*x^18 + 3/14*a*b^2*x^14 + 3/10*a^2*b*x^10 + 1/6*a^3*x^6

_______________________________________________________________________________________

Fricas [A]  time = 0.199748, size = 1, normalized size = 0.02 \[ \frac{1}{18} x^{18} b^{3} + \frac{3}{14} x^{14} b^{2} a + \frac{3}{10} x^{10} b a^{2} + \frac{1}{6} x^{6} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^3*x^5,x, algorithm="fricas")

[Out]

1/18*x^18*b^3 + 3/14*x^14*b^2*a + 3/10*x^10*b*a^2 + 1/6*x^6*a^3

_______________________________________________________________________________________

Sympy [A]  time = 0.098465, size = 39, normalized size = 0.91 \[ \frac{a^{3} x^{6}}{6} + \frac{3 a^{2} b x^{10}}{10} + \frac{3 a b^{2} x^{14}}{14} + \frac{b^{3} x^{18}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**4+a)**3,x)

[Out]

a**3*x**6/6 + 3*a**2*b*x**10/10 + 3*a*b**2*x**14/14 + b**3*x**18/18

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.218582, size = 47, normalized size = 1.09 \[ \frac{1}{18} \, b^{3} x^{18} + \frac{3}{14} \, a b^{2} x^{14} + \frac{3}{10} \, a^{2} b x^{10} + \frac{1}{6} \, a^{3} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^3*x^5,x, algorithm="giac")

[Out]

1/18*b^3*x^18 + 3/14*a*b^2*x^14 + 3/10*a^2*b*x^10 + 1/6*a^3*x^6