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

Optimal. Leaf size=34 \[ \frac{\left (a+b x^3\right )^5}{15 b^2}-\frac{a \left (a+b x^3\right )^4}{12 b^2} \]

[Out]

-(a*(a + b*x^3)^4)/(12*b^2) + (a + b*x^3)^5/(15*b^2)

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Rubi [A]  time = 0.0814258, antiderivative size = 34, 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 \[ \frac{\left (a+b x^3\right )^5}{15 b^2}-\frac{a \left (a+b x^3\right )^4}{12 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x^3)^3,x]

[Out]

-(a*(a + b*x^3)^4)/(12*b^2) + (a + b*x^3)^5/(15*b^2)

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Rubi in Sympy [A]  time = 8.36067, size = 27, normalized size = 0.79 \[ - \frac{a \left (a + b x^{3}\right )^{4}}{12 b^{2}} + \frac{\left (a + b x^{3}\right )^{5}}{15 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(a + b*x**3)**4/(12*b**2) + (a + b*x**3)**5/(15*b**2)

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Mathematica [A]  time = 0.00291824, size = 43, normalized size = 1.26 \[ \frac{a^3 x^6}{6}+\frac{1}{3} a^2 b x^9+\frac{1}{4} a b^2 x^{12}+\frac{b^3 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x^3)^3,x]

[Out]

(a^3*x^6)/6 + (a^2*b*x^9)/3 + (a*b^2*x^12)/4 + (b^3*x^15)/15

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Maple [A]  time = 0.001, size = 36, normalized size = 1.1 \[{\frac{{b}^{3}{x}^{15}}{15}}+{\frac{a{b}^{2}{x}^{12}}{4}}+{\frac{{a}^{2}b{x}^{9}}{3}}+{\frac{{a}^{3}{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^3+a)^3,x)

[Out]

1/15*b^3*x^15+1/4*a*b^2*x^12+1/3*a^2*b*x^9+1/6*a^3*x^6

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Maxima [A]  time = 1.44424, size = 47, normalized size = 1.38 \[ \frac{1}{15} \, b^{3} x^{15} + \frac{1}{4} \, a b^{2} x^{12} + \frac{1}{3} \, a^{2} b x^{9} + \frac{1}{6} \, a^{3} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*b^3*x^15 + 1/4*a*b^2*x^12 + 1/3*a^2*b*x^9 + 1/6*a^3*x^6

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Fricas [A]  time = 0.191886, size = 1, normalized size = 0.03 \[ \frac{1}{15} x^{15} b^{3} + \frac{1}{4} x^{12} b^{2} a + \frac{1}{3} x^{9} b a^{2} + \frac{1}{6} x^{6} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*x^15*b^3 + 1/4*x^12*b^2*a + 1/3*x^9*b*a^2 + 1/6*x^6*a^3

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Sympy [A]  time = 0.096421, size = 36, normalized size = 1.06 \[ \frac{a^{3} x^{6}}{6} + \frac{a^{2} b x^{9}}{3} + \frac{a b^{2} x^{12}}{4} + \frac{b^{3} x^{15}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**3+a)**3,x)

[Out]

a**3*x**6/6 + a**2*b*x**9/3 + a*b**2*x**12/4 + b**3*x**15/15

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GIAC/XCAS [A]  time = 0.224984, size = 47, normalized size = 1.38 \[ \frac{1}{15} \, b^{3} x^{15} + \frac{1}{4} \, a b^{2} x^{12} + \frac{1}{3} \, a^{2} b x^{9} + \frac{1}{6} \, a^{3} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*b^3*x^15 + 1/4*a*b^2*x^12 + 1/3*a^2*b*x^9 + 1/6*a^3*x^6