3.290 \(\int x^5 (a+b x^3)^8 \, dx\)

Optimal. Leaf size=34 \[ \frac{\left (a+b x^3\right )^{10}}{30 b^2}-\frac{a \left (a+b x^3\right )^9}{27 b^2} \]

[Out]

-(a*(a + b*x^3)^9)/(27*b^2) + (a + b*x^3)^10/(30*b^2)

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Rubi [A]  time = 0.0429557, antiderivative size = 34, 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} \[ \frac{\left (a+b x^3\right )^{10}}{30 b^2}-\frac{a \left (a+b x^3\right )^9}{27 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^3)^8,x]

[Out]

-(a*(a + b*x^3)^9)/(27*b^2) + (a + b*x^3)^10/(30*b^2)

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^5 \left (a+b x^3\right )^8 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x (a+b x)^8 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^8}{b}+\frac{(a+b x)^9}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac{a \left (a+b x^3\right )^9}{27 b^2}+\frac{\left (a+b x^3\right )^{10}}{30 b^2}\\ \end{align*}

Mathematica [B]  time = 0.0026569, size = 108, normalized size = 3.18 \[ \frac{7}{6} a^2 b^6 x^{24}+\frac{8}{3} a^3 b^5 x^{21}+\frac{35}{9} a^4 b^4 x^{18}+\frac{56}{15} a^5 b^3 x^{15}+\frac{7}{3} a^6 b^2 x^{12}+\frac{8}{9} a^7 b x^9+\frac{a^8 x^6}{6}+\frac{8}{27} a b^7 x^{27}+\frac{b^8 x^{30}}{30} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^3)^8,x]

[Out]

(a^8*x^6)/6 + (8*a^7*b*x^9)/9 + (7*a^6*b^2*x^12)/3 + (56*a^5*b^3*x^15)/15 + (35*a^4*b^4*x^18)/9 + (8*a^3*b^5*x
^21)/3 + (7*a^2*b^6*x^24)/6 + (8*a*b^7*x^27)/27 + (b^8*x^30)/30

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Maple [B]  time = 0.001, size = 91, normalized size = 2.7 \begin{align*}{\frac{{b}^{8}{x}^{30}}{30}}+{\frac{8\,a{b}^{7}{x}^{27}}{27}}+{\frac{7\,{a}^{2}{b}^{6}{x}^{24}}{6}}+{\frac{8\,{a}^{3}{b}^{5}{x}^{21}}{3}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{18}}{9}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{15}}{15}}+{\frac{7\,{a}^{6}{b}^{2}{x}^{12}}{3}}+{\frac{8\,{a}^{7}b{x}^{9}}{9}}+{\frac{{a}^{8}{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x^3+a)^8,x)

[Out]

1/30*b^8*x^30+8/27*a*b^7*x^27+7/6*a^2*b^6*x^24+8/3*a^3*b^5*x^21+35/9*a^4*b^4*x^18+56/15*a^5*b^3*x^15+7/3*a^6*b
^2*x^12+8/9*a^7*b*x^9+1/6*a^8*x^6

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Maxima [B]  time = 0.95425, size = 122, normalized size = 3.59 \begin{align*} \frac{1}{30} \, b^{8} x^{30} + \frac{8}{27} \, a b^{7} x^{27} + \frac{7}{6} \, a^{2} b^{6} x^{24} + \frac{8}{3} \, a^{3} b^{5} x^{21} + \frac{35}{9} \, a^{4} b^{4} x^{18} + \frac{56}{15} \, a^{5} b^{3} x^{15} + \frac{7}{3} \, a^{6} b^{2} x^{12} + \frac{8}{9} \, a^{7} b x^{9} + \frac{1}{6} \, a^{8} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^3+a)^8,x, algorithm="maxima")

[Out]

1/30*b^8*x^30 + 8/27*a*b^7*x^27 + 7/6*a^2*b^6*x^24 + 8/3*a^3*b^5*x^21 + 35/9*a^4*b^4*x^18 + 56/15*a^5*b^3*x^15
+ 7/3*a^6*b^2*x^12 + 8/9*a^7*b*x^9 + 1/6*a^8*x^6

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Fricas [B]  time = 1.47272, size = 217, normalized size = 6.38 \begin{align*} \frac{1}{30} x^{30} b^{8} + \frac{8}{27} x^{27} b^{7} a + \frac{7}{6} x^{24} b^{6} a^{2} + \frac{8}{3} x^{21} b^{5} a^{3} + \frac{35}{9} x^{18} b^{4} a^{4} + \frac{56}{15} x^{15} b^{3} a^{5} + \frac{7}{3} x^{12} b^{2} a^{6} + \frac{8}{9} x^{9} b a^{7} + \frac{1}{6} x^{6} a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^3+a)^8,x, algorithm="fricas")

[Out]

1/30*x^30*b^8 + 8/27*x^27*b^7*a + 7/6*x^24*b^6*a^2 + 8/3*x^21*b^5*a^3 + 35/9*x^18*b^4*a^4 + 56/15*x^15*b^3*a^5
+ 7/3*x^12*b^2*a^6 + 8/9*x^9*b*a^7 + 1/6*x^6*a^8

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Sympy [B]  time = 0.086991, size = 107, normalized size = 3.15 \begin{align*} \frac{a^{8} x^{6}}{6} + \frac{8 a^{7} b x^{9}}{9} + \frac{7 a^{6} b^{2} x^{12}}{3} + \frac{56 a^{5} b^{3} x^{15}}{15} + \frac{35 a^{4} b^{4} x^{18}}{9} + \frac{8 a^{3} b^{5} x^{21}}{3} + \frac{7 a^{2} b^{6} x^{24}}{6} + \frac{8 a b^{7} x^{27}}{27} + \frac{b^{8} x^{30}}{30} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x**3+a)**8,x)

[Out]

a**8*x**6/6 + 8*a**7*b*x**9/9 + 7*a**6*b**2*x**12/3 + 56*a**5*b**3*x**15/15 + 35*a**4*b**4*x**18/9 + 8*a**3*b*
*5*x**21/3 + 7*a**2*b**6*x**24/6 + 8*a*b**7*x**27/27 + b**8*x**30/30

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Giac [B]  time = 1.08103, size = 122, normalized size = 3.59 \begin{align*} \frac{1}{30} \, b^{8} x^{30} + \frac{8}{27} \, a b^{7} x^{27} + \frac{7}{6} \, a^{2} b^{6} x^{24} + \frac{8}{3} \, a^{3} b^{5} x^{21} + \frac{35}{9} \, a^{4} b^{4} x^{18} + \frac{56}{15} \, a^{5} b^{3} x^{15} + \frac{7}{3} \, a^{6} b^{2} x^{12} + \frac{8}{9} \, a^{7} b x^{9} + \frac{1}{6} \, a^{8} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^3+a)^8,x, algorithm="giac")

[Out]

1/30*b^8*x^30 + 8/27*a*b^7*x^27 + 7/6*a^2*b^6*x^24 + 8/3*a^3*b^5*x^21 + 35/9*a^4*b^4*x^18 + 56/15*a^5*b^3*x^15
+ 7/3*a^6*b^2*x^12 + 8/9*a^7*b*x^9 + 1/6*a^8*x^6