3.31 \(\int x^5 \left (a+b x^2\right )^3 \, dx\)

Optimal. Leaf size=43 \[ \frac{a^3 x^6}{6}+\frac{3}{8} a^2 b x^8+\frac{3}{10} a b^2 x^{10}+\frac{b^3 x^{12}}{12} \]

[Out]

(a^3*x^6)/6 + (3*a^2*b*x^8)/8 + (3*a*b^2*x^10)/10 + (b^3*x^12)/12

_______________________________________________________________________________________

Rubi [A]  time = 0.0754728, antiderivative size = 43, 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 \[ \frac{a^3 x^6}{6}+\frac{3}{8} a^2 b x^8+\frac{3}{10} a b^2 x^{10}+\frac{b^3 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^2)^3,x]

[Out]

(a^3*x^6)/6 + (3*a^2*b*x^8)/8 + (3*a*b^2*x^10)/10 + (b^3*x^12)/12

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.0558, size = 39, normalized size = 0.91 \[ \frac{a^{3} x^{6}}{6} + \frac{3 a^{2} b x^{8}}{8} + \frac{3 a b^{2} x^{10}}{10} + \frac{b^{3} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x**2+a)**3,x)

[Out]

a**3*x**6/6 + 3*a**2*b*x**8/8 + 3*a*b**2*x**10/10 + b**3*x**12/12

_______________________________________________________________________________________

Mathematica [A]  time = 0.00302224, size = 43, normalized size = 1. \[ \frac{a^3 x^6}{6}+\frac{3}{8} a^2 b x^8+\frac{3}{10} a b^2 x^{10}+\frac{b^3 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x^2)^3,x]

[Out]

(a^3*x^6)/6 + (3*a^2*b*x^8)/8 + (3*a*b^2*x^10)/10 + (b^3*x^12)/12

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 36, normalized size = 0.8 \[{\frac{{a}^{3}{x}^{6}}{6}}+{\frac{3\,{a}^{2}b{x}^{8}}{8}}+{\frac{3\,a{b}^{2}{x}^{10}}{10}}+{\frac{{b}^{3}{x}^{12}}{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^2+a)^3,x)

[Out]

1/6*a^3*x^6+3/8*a^2*b*x^8+3/10*a*b^2*x^10+1/12*b^3*x^12

_______________________________________________________________________________________

Maxima [A]  time = 1.35017, size = 47, normalized size = 1.09 \[ \frac{1}{12} \, b^{3} x^{12} + \frac{3}{10} \, a b^{2} x^{10} + \frac{3}{8} \, a^{2} b x^{8} + \frac{1}{6} \, a^{3} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3*x^5,x, algorithm="maxima")

[Out]

1/12*b^3*x^12 + 3/10*a*b^2*x^10 + 3/8*a^2*b*x^8 + 1/6*a^3*x^6

_______________________________________________________________________________________

Fricas [A]  time = 0.179869, size = 1, normalized size = 0.02 \[ \frac{1}{12} x^{12} b^{3} + \frac{3}{10} x^{10} b^{2} a + \frac{3}{8} x^{8} b a^{2} + \frac{1}{6} x^{6} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3*x^5,x, algorithm="fricas")

[Out]

1/12*x^12*b^3 + 3/10*x^10*b^2*a + 3/8*x^8*b*a^2 + 1/6*x^6*a^3

_______________________________________________________________________________________

Sympy [A]  time = 0.098994, size = 39, normalized size = 0.91 \[ \frac{a^{3} x^{6}}{6} + \frac{3 a^{2} b x^{8}}{8} + \frac{3 a b^{2} x^{10}}{10} + \frac{b^{3} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**2+a)**3,x)

[Out]

a**3*x**6/6 + 3*a**2*b*x**8/8 + 3*a*b**2*x**10/10 + b**3*x**12/12

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.206851, size = 47, normalized size = 1.09 \[ \frac{1}{12} \, b^{3} x^{12} + \frac{3}{10} \, a b^{2} x^{10} + \frac{3}{8} \, a^{2} b x^{8} + \frac{1}{6} \, a^{3} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3*x^5,x, algorithm="giac")

[Out]

1/12*b^3*x^12 + 3/10*a*b^2*x^10 + 3/8*a^2*b*x^8 + 1/6*a^3*x^6