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

Optimal. Leaf size=43 \[ \frac{a^3 x^2}{2}+\frac{3}{5} a^2 b x^5+\frac{3}{8} a b^2 x^8+\frac{b^3 x^{11}}{11} \]

[Out]

(a^3*x^2)/2 + (3*a^2*b*x^5)/5 + (3*a*b^2*x^8)/8 + (b^3*x^11)/11

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Rubi [A]  time = 0.0386325, antiderivative size = 43, 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 \[ \frac{a^3 x^2}{2}+\frac{3}{5} a^2 b x^5+\frac{3}{8} a b^2 x^8+\frac{b^3 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^3)^3,x]

[Out]

(a^3*x^2)/2 + (3*a^2*b*x^5)/5 + (3*a*b^2*x^8)/8 + (b^3*x^11)/11

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{3} \int x\, dx + \frac{3 a^{2} b x^{5}}{5} + \frac{3 a b^{2} x^{8}}{8} + \frac{b^{3} x^{11}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**3*Integral(x, x) + 3*a**2*b*x**5/5 + 3*a*b**2*x**8/8 + b**3*x**11/11

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Mathematica [A]  time = 0.0026725, size = 43, normalized size = 1. \[ \frac{a^3 x^2}{2}+\frac{3}{5} a^2 b x^5+\frac{3}{8} a b^2 x^8+\frac{b^3 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x^3)^3,x]

[Out]

(a^3*x^2)/2 + (3*a^2*b*x^5)/5 + (3*a*b^2*x^8)/8 + (b^3*x^11)/11

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Maple [A]  time = 0.001, size = 36, normalized size = 0.8 \[{\frac{{x}^{2}{a}^{3}}{2}}+{\frac{3\,{a}^{2}b{x}^{5}}{5}}+{\frac{3\,a{b}^{2}{x}^{8}}{8}}+{\frac{{b}^{3}{x}^{11}}{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^3+a)^3,x)

[Out]

1/2*x^2*a^3+3/5*a^2*b*x^5+3/8*a*b^2*x^8+1/11*b^3*x^11

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Maxima [A]  time = 1.41354, size = 47, normalized size = 1.09 \[ \frac{1}{11} \, b^{3} x^{11} + \frac{3}{8} \, a b^{2} x^{8} + \frac{3}{5} \, a^{2} b x^{5} + \frac{1}{2} \, a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*b^3*x^11 + 3/8*a*b^2*x^8 + 3/5*a^2*b*x^5 + 1/2*a^3*x^2

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Fricas [A]  time = 0.18834, size = 1, normalized size = 0.02 \[ \frac{1}{11} x^{11} b^{3} + \frac{3}{8} x^{8} b^{2} a + \frac{3}{5} x^{5} b a^{2} + \frac{1}{2} x^{2} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*x^11*b^3 + 3/8*x^8*b^2*a + 3/5*x^5*b*a^2 + 1/2*x^2*a^3

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Sympy [A]  time = 0.095663, size = 39, normalized size = 0.91 \[ \frac{a^{3} x^{2}}{2} + \frac{3 a^{2} b x^{5}}{5} + \frac{3 a b^{2} x^{8}}{8} + \frac{b^{3} x^{11}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x**3+a)**3,x)

[Out]

a**3*x**2/2 + 3*a**2*b*x**5/5 + 3*a*b**2*x**8/8 + b**3*x**11/11

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GIAC/XCAS [A]  time = 0.213947, size = 47, normalized size = 1.09 \[ \frac{1}{11} \, b^{3} x^{11} + \frac{3}{8} \, a b^{2} x^{8} + \frac{3}{5} \, a^{2} b x^{5} + \frac{1}{2} \, a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*b^3*x^11 + 3/8*a*b^2*x^8 + 3/5*a^2*b*x^5 + 1/2*a^3*x^2