3.236 \(\int x^{14} (a+b x^3)^3 \, dx\)

Optimal. Leaf size=43 \[ \frac{1}{6} a^2 b x^{18}+\frac{a^3 x^{15}}{15}+\frac{1}{7} a b^2 x^{21}+\frac{b^3 x^{24}}{24} \]

[Out]

(a^3*x^15)/15 + (a^2*b*x^18)/6 + (a*b^2*x^21)/7 + (b^3*x^24)/24

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Rubi [A]  time = 0.0273662, 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} \[ \frac{1}{6} a^2 b x^{18}+\frac{a^3 x^{15}}{15}+\frac{1}{7} a b^2 x^{21}+\frac{b^3 x^{24}}{24} \]

Antiderivative was successfully verified.

[In]

Int[x^14*(a + b*x^3)^3,x]

[Out]

(a^3*x^15)/15 + (a^2*b*x^18)/6 + (a*b^2*x^21)/7 + (b^3*x^24)/24

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^{14} \left (a+b x^3\right )^3 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^4 (a+b x)^3 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (a^3 x^4+3 a^2 b x^5+3 a b^2 x^6+b^3 x^7\right ) \, dx,x,x^3\right )\\ &=\frac{a^3 x^{15}}{15}+\frac{1}{6} a^2 b x^{18}+\frac{1}{7} a b^2 x^{21}+\frac{b^3 x^{24}}{24}\\ \end{align*}

Mathematica [A]  time = 0.0021191, size = 43, normalized size = 1. \[ \frac{1}{6} a^2 b x^{18}+\frac{a^3 x^{15}}{15}+\frac{1}{7} a b^2 x^{21}+\frac{b^3 x^{24}}{24} \]

Antiderivative was successfully verified.

[In]

Integrate[x^14*(a + b*x^3)^3,x]

[Out]

(a^3*x^15)/15 + (a^2*b*x^18)/6 + (a*b^2*x^21)/7 + (b^3*x^24)/24

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Maple [A]  time = 0.001, size = 36, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}{x}^{15}}{15}}+{\frac{{a}^{2}b{x}^{18}}{6}}+{\frac{a{b}^{2}{x}^{21}}{7}}+{\frac{{b}^{3}{x}^{24}}{24}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14*(b*x^3+a)^3,x)

[Out]

1/15*a^3*x^15+1/6*a^2*b*x^18+1/7*a*b^2*x^21+1/24*b^3*x^24

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Maxima [A]  time = 0.956449, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{24} \, b^{3} x^{24} + \frac{1}{7} \, a b^{2} x^{21} + \frac{1}{6} \, a^{2} b x^{18} + \frac{1}{15} \, a^{3} x^{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14*(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/24*b^3*x^24 + 1/7*a*b^2*x^21 + 1/6*a^2*b*x^18 + 1/15*a^3*x^15

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Fricas [A]  time = 1.54668, size = 88, normalized size = 2.05 \begin{align*} \frac{1}{24} x^{24} b^{3} + \frac{1}{7} x^{21} b^{2} a + \frac{1}{6} x^{18} b a^{2} + \frac{1}{15} x^{15} a^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14*(b*x^3+a)^3,x, algorithm="fricas")

[Out]

1/24*x^24*b^3 + 1/7*x^21*b^2*a + 1/6*x^18*b*a^2 + 1/15*x^15*a^3

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Sympy [A]  time = 0.07106, size = 36, normalized size = 0.84 \begin{align*} \frac{a^{3} x^{15}}{15} + \frac{a^{2} b x^{18}}{6} + \frac{a b^{2} x^{21}}{7} + \frac{b^{3} x^{24}}{24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**14*(b*x**3+a)**3,x)

[Out]

a**3*x**15/15 + a**2*b*x**18/6 + a*b**2*x**21/7 + b**3*x**24/24

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Giac [A]  time = 1.17303, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{24} \, b^{3} x^{24} + \frac{1}{7} \, a b^{2} x^{21} + \frac{1}{6} \, a^{2} b x^{18} + \frac{1}{15} \, a^{3} x^{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14*(b*x^3+a)^3,x, algorithm="giac")

[Out]

1/24*b^3*x^24 + 1/7*a*b^2*x^21 + 1/6*a^2*b*x^18 + 1/15*a^3*x^15