3.288 \(\int x^{11} (a+b x^3)^8 \, dx\)

Optimal. Leaf size=72 \[ \frac{a^2 \left (a+b x^3\right )^{10}}{10 b^4}-\frac{a^3 \left (a+b x^3\right )^9}{27 b^4}+\frac{\left (a+b x^3\right )^{12}}{36 b^4}-\frac{a \left (a+b x^3\right )^{11}}{11 b^4} \]

[Out]

-(a^3*(a + b*x^3)^9)/(27*b^4) + (a^2*(a + b*x^3)^10)/(10*b^4) - (a*(a + b*x^3)^11)/(11*b^4) + (a + b*x^3)^12/(
36*b^4)

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Rubi [A]  time = 0.109987, antiderivative size = 72, 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} \[ \frac{a^2 \left (a+b x^3\right )^{10}}{10 b^4}-\frac{a^3 \left (a+b x^3\right )^9}{27 b^4}+\frac{\left (a+b x^3\right )^{12}}{36 b^4}-\frac{a \left (a+b x^3\right )^{11}}{11 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^11*(a + b*x^3)^8,x]

[Out]

-(a^3*(a + b*x^3)^9)/(27*b^4) + (a^2*(a + b*x^3)^10)/(10*b^4) - (a*(a + b*x^3)^11)/(11*b^4) + (a + b*x^3)^12/(
36*b^4)

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 )^8 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^3 (a+b x)^8 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^8}{b^3}+\frac{3 a^2 (a+b x)^9}{b^3}-\frac{3 a (a+b x)^{10}}{b^3}+\frac{(a+b x)^{11}}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^3 \left (a+b x^3\right )^9}{27 b^4}+\frac{a^2 \left (a+b x^3\right )^{10}}{10 b^4}-\frac{a \left (a+b x^3\right )^{11}}{11 b^4}+\frac{\left (a+b x^3\right )^{12}}{36 b^4}\\ \end{align*}

Mathematica [A]  time = 0.0028274, size = 108, normalized size = 1.5 \[ \frac{14}{15} a^2 b^6 x^{30}+\frac{56}{27} a^3 b^5 x^{27}+\frac{35}{12} a^4 b^4 x^{24}+\frac{8}{3} a^5 b^3 x^{21}+\frac{14}{9} a^6 b^2 x^{18}+\frac{8}{15} a^7 b x^{15}+\frac{a^8 x^{12}}{12}+\frac{8}{33} a b^7 x^{33}+\frac{b^8 x^{36}}{36} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11*(a + b*x^3)^8,x]

[Out]

(a^8*x^12)/12 + (8*a^7*b*x^15)/15 + (14*a^6*b^2*x^18)/9 + (8*a^5*b^3*x^21)/3 + (35*a^4*b^4*x^24)/12 + (56*a^3*
b^5*x^27)/27 + (14*a^2*b^6*x^30)/15 + (8*a*b^7*x^33)/33 + (b^8*x^36)/36

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Maple [A]  time = 0.001, size = 91, normalized size = 1.3 \begin{align*}{\frac{{b}^{8}{x}^{36}}{36}}+{\frac{8\,a{b}^{7}{x}^{33}}{33}}+{\frac{14\,{b}^{6}{a}^{2}{x}^{30}}{15}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{27}}{27}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{24}}{12}}+{\frac{8\,{a}^{5}{b}^{3}{x}^{21}}{3}}+{\frac{14\,{a}^{6}{b}^{2}{x}^{18}}{9}}+{\frac{8\,{a}^{7}b{x}^{15}}{15}}+{\frac{{a}^{8}{x}^{12}}{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(b*x^3+a)^8,x)

[Out]

1/36*b^8*x^36+8/33*a*b^7*x^33+14/15*b^6*a^2*x^30+56/27*a^3*b^5*x^27+35/12*a^4*b^4*x^24+8/3*a^5*b^3*x^21+14/9*a
^6*b^2*x^18+8/15*a^7*b*x^15+1/12*a^8*x^12

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Maxima [A]  time = 0.962593, size = 122, normalized size = 1.69 \begin{align*} \frac{1}{36} \, b^{8} x^{36} + \frac{8}{33} \, a b^{7} x^{33} + \frac{14}{15} \, a^{2} b^{6} x^{30} + \frac{56}{27} \, a^{3} b^{5} x^{27} + \frac{35}{12} \, a^{4} b^{4} x^{24} + \frac{8}{3} \, a^{5} b^{3} x^{21} + \frac{14}{9} \, a^{6} b^{2} x^{18} + \frac{8}{15} \, a^{7} b x^{15} + \frac{1}{12} \, a^{8} x^{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^8,x, algorithm="maxima")

[Out]

1/36*b^8*x^36 + 8/33*a*b^7*x^33 + 14/15*a^2*b^6*x^30 + 56/27*a^3*b^5*x^27 + 35/12*a^4*b^4*x^24 + 8/3*a^5*b^3*x
^21 + 14/9*a^6*b^2*x^18 + 8/15*a^7*b*x^15 + 1/12*a^8*x^12

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Fricas [A]  time = 1.43355, size = 228, normalized size = 3.17 \begin{align*} \frac{1}{36} x^{36} b^{8} + \frac{8}{33} x^{33} b^{7} a + \frac{14}{15} x^{30} b^{6} a^{2} + \frac{56}{27} x^{27} b^{5} a^{3} + \frac{35}{12} x^{24} b^{4} a^{4} + \frac{8}{3} x^{21} b^{3} a^{5} + \frac{14}{9} x^{18} b^{2} a^{6} + \frac{8}{15} x^{15} b a^{7} + \frac{1}{12} x^{12} a^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^8,x, algorithm="fricas")

[Out]

1/36*x^36*b^8 + 8/33*x^33*b^7*a + 14/15*x^30*b^6*a^2 + 56/27*x^27*b^5*a^3 + 35/12*x^24*b^4*a^4 + 8/3*x^21*b^3*
a^5 + 14/9*x^18*b^2*a^6 + 8/15*x^15*b*a^7 + 1/12*x^12*a^8

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Sympy [A]  time = 0.113184, size = 107, normalized size = 1.49 \begin{align*} \frac{a^{8} x^{12}}{12} + \frac{8 a^{7} b x^{15}}{15} + \frac{14 a^{6} b^{2} x^{18}}{9} + \frac{8 a^{5} b^{3} x^{21}}{3} + \frac{35 a^{4} b^{4} x^{24}}{12} + \frac{56 a^{3} b^{5} x^{27}}{27} + \frac{14 a^{2} b^{6} x^{30}}{15} + \frac{8 a b^{7} x^{33}}{33} + \frac{b^{8} x^{36}}{36} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(b*x**3+a)**8,x)

[Out]

a**8*x**12/12 + 8*a**7*b*x**15/15 + 14*a**6*b**2*x**18/9 + 8*a**5*b**3*x**21/3 + 35*a**4*b**4*x**24/12 + 56*a*
*3*b**5*x**27/27 + 14*a**2*b**6*x**30/15 + 8*a*b**7*x**33/33 + b**8*x**36/36

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Giac [A]  time = 1.09656, size = 122, normalized size = 1.69 \begin{align*} \frac{1}{36} \, b^{8} x^{36} + \frac{8}{33} \, a b^{7} x^{33} + \frac{14}{15} \, a^{2} b^{6} x^{30} + \frac{56}{27} \, a^{3} b^{5} x^{27} + \frac{35}{12} \, a^{4} b^{4} x^{24} + \frac{8}{3} \, a^{5} b^{3} x^{21} + \frac{14}{9} \, a^{6} b^{2} x^{18} + \frac{8}{15} \, a^{7} b x^{15} + \frac{1}{12} \, a^{8} x^{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^8,x, algorithm="giac")

[Out]

1/36*b^8*x^36 + 8/33*a*b^7*x^33 + 14/15*a^2*b^6*x^30 + 56/27*a^3*b^5*x^27 + 35/12*a^4*b^4*x^24 + 8/3*a^5*b^3*x
^21 + 14/9*a^6*b^2*x^18 + 8/15*a^7*b*x^15 + 1/12*a^8*x^12