3.633 \(\int x^5 (a+b x^4)^3 \, dx\)

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

[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

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Rubi [A]  time = 0.0146323, 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, Rules used = {270} \[ \frac{3}{10} a^2 b x^{10}+\frac{a^3 x^6}{6}+\frac{3}{14} a b^2 x^{14}+\frac{b^3 x^{18}}{18} \]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int x^5 \left (a+b x^4\right )^3 \, dx &=\int \left (a^3 x^5+3 a^2 b x^9+3 a b^2 x^{13}+b^3 x^{17}\right ) \, dx\\ &=\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}\\ \end{align*}

Mathematica [A]  time = 0.0020353, size = 43, normalized size = 1. \[ \frac{3}{10} a^2 b x^{10}+\frac{a^3 x^6}{6}+\frac{3}{14} a b^2 x^{14}+\frac{b^3 x^{18}}{18} \]

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

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Maple [A]  time = 0.001, size = 36, normalized size = 0.8 \begin{align*}{\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}} \end{align*}

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

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Maxima [A]  time = 0.961485, size = 47, normalized size = 1.09 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^4+a)^3,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

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Fricas [A]  time = 1.25889, size = 88, normalized size = 2.05 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^4+a)^3,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

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Sympy [A]  time = 0.08905, size = 39, normalized size = 0.91 \begin{align*} \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} \end{align*}

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

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Giac [A]  time = 1.08803, size = 47, normalized size = 1.09 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^4+a)^3,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