3.237 \(\int x^{11} (a+b x^3)^3 \, dx\)

Optimal. Leaf size=43 \[ \frac{1}{5} a^2 b x^{15}+\frac{a^3 x^{12}}{12}+\frac{1}{6} a b^2 x^{18}+\frac{b^3 x^{21}}{21} \]

[Out]

(a^3*x^12)/12 + (a^2*b*x^15)/5 + (a*b^2*x^18)/6 + (b^3*x^21)/21

________________________________________________________________________________________

Rubi [A]  time = 0.0248982, 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, Rules used = {266, 43} \[ \frac{1}{5} a^2 b x^{15}+\frac{a^3 x^{12}}{12}+\frac{1}{6} a b^2 x^{18}+\frac{b^3 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]

Int[x^11*(a + b*x^3)^3,x]

[Out]

(a^3*x^12)/12 + (a^2*b*x^15)/5 + (a*b^2*x^18)/6 + (b^3*x^21)/21

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^{11} \left (a+b x^3\right )^3 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^3 (a+b x)^3 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (a^3 x^3+3 a^2 b x^4+3 a b^2 x^5+b^3 x^6\right ) \, dx,x,x^3\right )\\ &=\frac{a^3 x^{12}}{12}+\frac{1}{5} a^2 b x^{15}+\frac{1}{6} a b^2 x^{18}+\frac{b^3 x^{21}}{21}\\ \end{align*}

Mathematica [A]  time = 0.0018367, size = 43, normalized size = 1. \[ \frac{1}{5} a^2 b x^{15}+\frac{a^3 x^{12}}{12}+\frac{1}{6} a b^2 x^{18}+\frac{b^3 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11*(a + b*x^3)^3,x]

[Out]

(a^3*x^12)/12 + (a^2*b*x^15)/5 + (a*b^2*x^18)/6 + (b^3*x^21)/21

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 36, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}{x}^{12}}{12}}+{\frac{{a}^{2}b{x}^{15}}{5}}+{\frac{a{b}^{2}{x}^{18}}{6}}+{\frac{{b}^{3}{x}^{21}}{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(b*x^3+a)^3,x)

[Out]

1/12*a^3*x^12+1/5*a^2*b*x^15+1/6*a*b^2*x^18+1/21*b^3*x^21

________________________________________________________________________________________

Maxima [A]  time = 0.954358, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{21} \, b^{3} x^{21} + \frac{1}{6} \, a b^{2} x^{18} + \frac{1}{5} \, a^{2} b x^{15} + \frac{1}{12} \, a^{3} x^{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/21*b^3*x^21 + 1/6*a*b^2*x^18 + 1/5*a^2*b*x^15 + 1/12*a^3*x^12

________________________________________________________________________________________

Fricas [A]  time = 1.54268, size = 88, normalized size = 2.05 \begin{align*} \frac{1}{21} x^{21} b^{3} + \frac{1}{6} x^{18} b^{2} a + \frac{1}{5} x^{15} b a^{2} + \frac{1}{12} x^{12} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/21*x^21*b^3 + 1/6*x^18*b^2*a + 1/5*x^15*b*a^2 + 1/12*x^12*a^3

________________________________________________________________________________________

Sympy [A]  time = 0.072994, size = 36, normalized size = 0.84 \begin{align*} \frac{a^{3} x^{12}}{12} + \frac{a^{2} b x^{15}}{5} + \frac{a b^{2} x^{18}}{6} + \frac{b^{3} x^{21}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(b*x**3+a)**3,x)

[Out]

a**3*x**12/12 + a**2*b*x**15/5 + a*b**2*x**18/6 + b**3*x**21/21

________________________________________________________________________________________

Giac [A]  time = 1.13012, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{21} \, b^{3} x^{21} + \frac{1}{6} \, a b^{2} x^{18} + \frac{1}{5} \, a^{2} b x^{15} + \frac{1}{12} \, a^{3} x^{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/21*b^3*x^21 + 1/6*a*b^2*x^18 + 1/5*a^2*b*x^15 + 1/12*a^3*x^12