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

Optimal. Leaf size=43 \[ \frac{3}{8} a^2 b x^8+\frac{a^3 x^5}{5}+\frac{3}{11} a b^2 x^{11}+\frac{b^3 x^{14}}{14} \]

[Out]

(a^3*x^5)/5 + (3*a^2*b*x^8)/8 + (3*a*b^2*x^11)/11 + (b^3*x^14)/14

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Rubi [A]  time = 0.0151228, 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} \[ \frac{3}{8} a^2 b x^8+\frac{a^3 x^5}{5}+\frac{3}{11} a b^2 x^{11}+\frac{b^3 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^3)^3,x]

[Out]

(a^3*x^5)/5 + (3*a^2*b*x^8)/8 + (3*a*b^2*x^11)/11 + (b^3*x^14)/14

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^4 \left (a+b x^3\right )^3 \, dx &=\int \left (a^3 x^4+3 a^2 b x^7+3 a b^2 x^{10}+b^3 x^{13}\right ) \, dx\\ &=\frac{a^3 x^5}{5}+\frac{3}{8} a^2 b x^8+\frac{3}{11} a b^2 x^{11}+\frac{b^3 x^{14}}{14}\\ \end{align*}

Mathematica [A]  time = 0.0018141, size = 43, normalized size = 1. \[ \frac{3}{8} a^2 b x^8+\frac{a^3 x^5}{5}+\frac{3}{11} a b^2 x^{11}+\frac{b^3 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^3)^3,x]

[Out]

(a^3*x^5)/5 + (3*a^2*b*x^8)/8 + (3*a*b^2*x^11)/11 + (b^3*x^14)/14

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Maple [A]  time = 0.001, size = 36, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}{x}^{5}}{5}}+{\frac{3\,{a}^{2}b{x}^{8}}{8}}+{\frac{3\,a{b}^{2}{x}^{11}}{11}}+{\frac{{b}^{3}{x}^{14}}{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^3+a)^3,x)

[Out]

1/5*a^3*x^5+3/8*a^2*b*x^8+3/11*a*b^2*x^11+1/14*b^3*x^14

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Maxima [A]  time = 0.985635, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{14} \, b^{3} x^{14} + \frac{3}{11} \, a b^{2} x^{11} + \frac{3}{8} \, a^{2} b x^{8} + \frac{1}{5} \, a^{3} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*b^3*x^14 + 3/11*a*b^2*x^11 + 3/8*a^2*b*x^8 + 1/5*a^3*x^5

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Fricas [A]  time = 1.4973, size = 85, normalized size = 1.98 \begin{align*} \frac{1}{14} x^{14} b^{3} + \frac{3}{11} x^{11} b^{2} a + \frac{3}{8} x^{8} b a^{2} + \frac{1}{5} x^{5} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*x^14*b^3 + 3/11*x^11*b^2*a + 3/8*x^8*b*a^2 + 1/5*x^5*a^3

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Sympy [A]  time = 0.096077, size = 39, normalized size = 0.91 \begin{align*} \frac{a^{3} x^{5}}{5} + \frac{3 a^{2} b x^{8}}{8} + \frac{3 a b^{2} x^{11}}{11} + \frac{b^{3} x^{14}}{14} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**5/5 + 3*a**2*b*x**8/8 + 3*a*b**2*x**11/11 + b**3*x**14/14

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Giac [A]  time = 1.11495, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{14} \, b^{3} x^{14} + \frac{3}{11} \, a b^{2} x^{11} + \frac{3}{8} \, a^{2} b x^{8} + \frac{1}{5} \, a^{3} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*b^3*x^14 + 3/11*a*b^2*x^11 + 3/8*a^2*b*x^8 + 1/5*a^3*x^5