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

Optimal. Leaf size=53 \[ \frac{a^2 \left (a+b x^3\right )^9}{27 b^3}+\frac{\left (a+b x^3\right )^{11}}{33 b^3}-\frac{a \left (a+b x^3\right )^{10}}{15 b^3} \]

[Out]

(a^2*(a + b*x^3)^9)/(27*b^3) - (a*(a + b*x^3)^10)/(15*b^3) + (a + b*x^3)^11/(33*b^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0794962, antiderivative size = 53, 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} \[ \frac{a^2 \left (a+b x^3\right )^9}{27 b^3}+\frac{\left (a+b x^3\right )^{11}}{33 b^3}-\frac{a \left (a+b x^3\right )^{10}}{15 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*x^3)^8,x]

[Out]

(a^2*(a + b*x^3)^9)/(27*b^3) - (a*(a + b*x^3)^10)/(15*b^3) + (a + b*x^3)^11/(33*b^3)

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^8 \left (a+b x^3\right )^8 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^2 (a+b x)^8 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^8}{b^2}-\frac{2 a (a+b x)^9}{b^2}+\frac{(a+b x)^{10}}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac{a^2 \left (a+b x^3\right )^9}{27 b^3}-\frac{a \left (a+b x^3\right )^{10}}{15 b^3}+\frac{\left (a+b x^3\right )^{11}}{33 b^3}\\ \end{align*}

Mathematica [B]  time = 0.0029469, size = 108, normalized size = 2.04 \[ \frac{28}{27} a^2 b^6 x^{27}+\frac{7}{3} a^3 b^5 x^{24}+\frac{10}{3} a^4 b^4 x^{21}+\frac{28}{9} a^5 b^3 x^{18}+\frac{28}{15} a^6 b^2 x^{15}+\frac{2}{3} a^7 b x^{12}+\frac{a^8 x^9}{9}+\frac{4}{15} a b^7 x^{30}+\frac{b^8 x^{33}}{33} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a + b*x^3)^8,x]

[Out]

(a^8*x^9)/9 + (2*a^7*b*x^12)/3 + (28*a^6*b^2*x^15)/15 + (28*a^5*b^3*x^18)/9 + (10*a^4*b^4*x^21)/3 + (7*a^3*b^5
*x^24)/3 + (28*a^2*b^6*x^27)/27 + (4*a*b^7*x^30)/15 + (b^8*x^33)/33

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 91, normalized size = 1.7 \begin{align*}{\frac{{b}^{8}{x}^{33}}{33}}+{\frac{4\,a{b}^{7}{x}^{30}}{15}}+{\frac{28\,{b}^{6}{a}^{2}{x}^{27}}{27}}+{\frac{7\,{a}^{3}{b}^{5}{x}^{24}}{3}}+{\frac{10\,{a}^{4}{b}^{4}{x}^{21}}{3}}+{\frac{28\,{a}^{5}{b}^{3}{x}^{18}}{9}}+{\frac{28\,{a}^{6}{b}^{2}{x}^{15}}{15}}+{\frac{2\,{a}^{7}b{x}^{12}}{3}}+{\frac{{a}^{8}{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/33*b^8*x^33+4/15*a*b^7*x^30+28/27*b^6*a^2*x^27+7/3*a^3*b^5*x^24+10/3*a^4*b^4*x^21+28/9*a^5*b^3*x^18+28/15*a^
6*b^2*x^15+2/3*a^7*b*x^12+1/9*a^8*x^9

________________________________________________________________________________________

Maxima [A]  time = 0.949077, size = 122, normalized size = 2.3 \begin{align*} \frac{1}{33} \, b^{8} x^{33} + \frac{4}{15} \, a b^{7} x^{30} + \frac{28}{27} \, a^{2} b^{6} x^{27} + \frac{7}{3} \, a^{3} b^{5} x^{24} + \frac{10}{3} \, a^{4} b^{4} x^{21} + \frac{28}{9} \, a^{5} b^{3} x^{18} + \frac{28}{15} \, a^{6} b^{2} x^{15} + \frac{2}{3} \, a^{7} b x^{12} + \frac{1}{9} \, a^{8} x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/33*b^8*x^33 + 4/15*a*b^7*x^30 + 28/27*a^2*b^6*x^27 + 7/3*a^3*b^5*x^24 + 10/3*a^4*b^4*x^21 + 28/9*a^5*b^3*x^1
8 + 28/15*a^6*b^2*x^15 + 2/3*a^7*b*x^12 + 1/9*a^8*x^9

________________________________________________________________________________________

Fricas [A]  time = 1.44819, size = 223, normalized size = 4.21 \begin{align*} \frac{1}{33} x^{33} b^{8} + \frac{4}{15} x^{30} b^{7} a + \frac{28}{27} x^{27} b^{6} a^{2} + \frac{7}{3} x^{24} b^{5} a^{3} + \frac{10}{3} x^{21} b^{4} a^{4} + \frac{28}{9} x^{18} b^{3} a^{5} + \frac{28}{15} x^{15} b^{2} a^{6} + \frac{2}{3} x^{12} b a^{7} + \frac{1}{9} x^{9} a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/33*x^33*b^8 + 4/15*x^30*b^7*a + 28/27*x^27*b^6*a^2 + 7/3*x^24*b^5*a^3 + 10/3*x^21*b^4*a^4 + 28/9*x^18*b^3*a^
5 + 28/15*x^15*b^2*a^6 + 2/3*x^12*b*a^7 + 1/9*x^9*a^8

________________________________________________________________________________________

Sympy [B]  time = 0.087228, size = 107, normalized size = 2.02 \begin{align*} \frac{a^{8} x^{9}}{9} + \frac{2 a^{7} b x^{12}}{3} + \frac{28 a^{6} b^{2} x^{15}}{15} + \frac{28 a^{5} b^{3} x^{18}}{9} + \frac{10 a^{4} b^{4} x^{21}}{3} + \frac{7 a^{3} b^{5} x^{24}}{3} + \frac{28 a^{2} b^{6} x^{27}}{27} + \frac{4 a b^{7} x^{30}}{15} + \frac{b^{8} x^{33}}{33} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**8*x**9/9 + 2*a**7*b*x**12/3 + 28*a**6*b**2*x**15/15 + 28*a**5*b**3*x**18/9 + 10*a**4*b**4*x**21/3 + 7*a**3*
b**5*x**24/3 + 28*a**2*b**6*x**27/27 + 4*a*b**7*x**30/15 + b**8*x**33/33

________________________________________________________________________________________

Giac [A]  time = 1.15574, size = 122, normalized size = 2.3 \begin{align*} \frac{1}{33} \, b^{8} x^{33} + \frac{4}{15} \, a b^{7} x^{30} + \frac{28}{27} \, a^{2} b^{6} x^{27} + \frac{7}{3} \, a^{3} b^{5} x^{24} + \frac{10}{3} \, a^{4} b^{4} x^{21} + \frac{28}{9} \, a^{5} b^{3} x^{18} + \frac{28}{15} \, a^{6} b^{2} x^{15} + \frac{2}{3} \, a^{7} b x^{12} + \frac{1}{9} \, a^{8} x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/33*b^8*x^33 + 4/15*a*b^7*x^30 + 28/27*a^2*b^6*x^27 + 7/3*a^3*b^5*x^24 + 10/3*a^4*b^4*x^21 + 28/9*a^5*b^3*x^1
8 + 28/15*a^6*b^2*x^15 + 2/3*a^7*b*x^12 + 1/9*a^8*x^9