3.259 \(\int x^{14} (a+b x^3)^5 \, dx\)

Optimal. Leaf size=69 \[ \frac{5}{12} a^2 b^3 x^{24}+\frac{10}{21} a^3 b^2 x^{21}+\frac{5}{18} a^4 b x^{18}+\frac{a^5 x^{15}}{15}+\frac{5}{27} a b^4 x^{27}+\frac{b^5 x^{30}}{30} \]

[Out]

(a^5*x^15)/15 + (5*a^4*b*x^18)/18 + (10*a^3*b^2*x^21)/21 + (5*a^2*b^3*x^24)/12 + (5*a*b^4*x^27)/27 + (b^5*x^30
)/30

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Rubi [A]  time = 0.0420508, antiderivative size = 69, 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} \[ \frac{5}{12} a^2 b^3 x^{24}+\frac{10}{21} a^3 b^2 x^{21}+\frac{5}{18} a^4 b x^{18}+\frac{a^5 x^{15}}{15}+\frac{5}{27} a b^4 x^{27}+\frac{b^5 x^{30}}{30} \]

Antiderivative was successfully verified.

[In]

Int[x^14*(a + b*x^3)^5,x]

[Out]

(a^5*x^15)/15 + (5*a^4*b*x^18)/18 + (10*a^3*b^2*x^21)/21 + (5*a^2*b^3*x^24)/12 + (5*a*b^4*x^27)/27 + (b^5*x^30
)/30

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 )^5 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^4 (a+b x)^5 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (a^5 x^4+5 a^4 b x^5+10 a^3 b^2 x^6+10 a^2 b^3 x^7+5 a b^4 x^8+b^5 x^9\right ) \, dx,x,x^3\right )\\ &=\frac{a^5 x^{15}}{15}+\frac{5}{18} a^4 b x^{18}+\frac{10}{21} a^3 b^2 x^{21}+\frac{5}{12} a^2 b^3 x^{24}+\frac{5}{27} a b^4 x^{27}+\frac{b^5 x^{30}}{30}\\ \end{align*}

Mathematica [A]  time = 0.0021888, size = 69, normalized size = 1. \[ \frac{5}{12} a^2 b^3 x^{24}+\frac{10}{21} a^3 b^2 x^{21}+\frac{5}{18} a^4 b x^{18}+\frac{a^5 x^{15}}{15}+\frac{5}{27} a b^4 x^{27}+\frac{b^5 x^{30}}{30} \]

Antiderivative was successfully verified.

[In]

Integrate[x^14*(a + b*x^3)^5,x]

[Out]

(a^5*x^15)/15 + (5*a^4*b*x^18)/18 + (10*a^3*b^2*x^21)/21 + (5*a^2*b^3*x^24)/12 + (5*a*b^4*x^27)/27 + (b^5*x^30
)/30

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Maple [A]  time = 0.002, size = 58, normalized size = 0.8 \begin{align*}{\frac{{a}^{5}{x}^{15}}{15}}+{\frac{5\,{a}^{4}b{x}^{18}}{18}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{21}}{21}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{24}}{12}}+{\frac{5\,a{b}^{4}{x}^{27}}{27}}+{\frac{{b}^{5}{x}^{30}}{30}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14*(b*x^3+a)^5,x)

[Out]

1/15*a^5*x^15+5/18*a^4*b*x^18+10/21*a^3*b^2*x^21+5/12*a^2*b^3*x^24+5/27*a*b^4*x^27+1/30*b^5*x^30

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Maxima [A]  time = 0.970618, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{30} \, b^{5} x^{30} + \frac{5}{27} \, a b^{4} x^{27} + \frac{5}{12} \, a^{2} b^{3} x^{24} + \frac{10}{21} \, a^{3} b^{2} x^{21} + \frac{5}{18} \, a^{4} b x^{18} + \frac{1}{15} \, a^{5} x^{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14*(b*x^3+a)^5,x, algorithm="maxima")

[Out]

1/30*b^5*x^30 + 5/27*a*b^4*x^27 + 5/12*a^2*b^3*x^24 + 10/21*a^3*b^2*x^21 + 5/18*a^4*b*x^18 + 1/15*a^5*x^15

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Fricas [A]  time = 1.45251, size = 146, normalized size = 2.12 \begin{align*} \frac{1}{30} x^{30} b^{5} + \frac{5}{27} x^{27} b^{4} a + \frac{5}{12} x^{24} b^{3} a^{2} + \frac{10}{21} x^{21} b^{2} a^{3} + \frac{5}{18} x^{18} b a^{4} + \frac{1}{15} x^{15} a^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14*(b*x^3+a)^5,x, algorithm="fricas")

[Out]

1/30*x^30*b^5 + 5/27*x^27*b^4*a + 5/12*x^24*b^3*a^2 + 10/21*x^21*b^2*a^3 + 5/18*x^18*b*a^4 + 1/15*x^15*a^5

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Sympy [A]  time = 0.085392, size = 66, normalized size = 0.96 \begin{align*} \frac{a^{5} x^{15}}{15} + \frac{5 a^{4} b x^{18}}{18} + \frac{10 a^{3} b^{2} x^{21}}{21} + \frac{5 a^{2} b^{3} x^{24}}{12} + \frac{5 a b^{4} x^{27}}{27} + \frac{b^{5} x^{30}}{30} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**14*(b*x**3+a)**5,x)

[Out]

a**5*x**15/15 + 5*a**4*b*x**18/18 + 10*a**3*b**2*x**21/21 + 5*a**2*b**3*x**24/12 + 5*a*b**4*x**27/27 + b**5*x*
*30/30

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Giac [A]  time = 1.15984, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{30} \, b^{5} x^{30} + \frac{5}{27} \, a b^{4} x^{27} + \frac{5}{12} \, a^{2} b^{3} x^{24} + \frac{10}{21} \, a^{3} b^{2} x^{21} + \frac{5}{18} \, a^{4} b x^{18} + \frac{1}{15} \, a^{5} x^{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14*(b*x^3+a)^5,x, algorithm="giac")

[Out]

1/30*b^5*x^30 + 5/27*a*b^4*x^27 + 5/12*a^2*b^3*x^24 + 10/21*a^3*b^2*x^21 + 5/18*a^4*b*x^18 + 1/15*a^5*x^15