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]

1/27*a^2*(b*x^3+a)^9/b^3-1/15*a*(b*x^3+a)^10/b^3+1/33*(b*x^3+a)^11/b^3

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Rubi [A]  time = 0.08, antiderivative size = 53, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [B]  time = 0.00, size = 108, normalized size = 2.04 \[ \frac {a^8 x^9}{9}+\frac {2}{3} a^7 b x^{12}+\frac {28}{15} a^6 b^2 x^{15}+\frac {28}{9} a^5 b^3 x^{18}+\frac {10}{3} a^4 b^4 x^{21}+\frac {7}{3} a^3 b^5 x^{24}+\frac {28}{27} a^2 b^6 x^{27}+\frac {4}{15} a b^7 x^{30}+\frac {b^8 x^{33}}{33} \]

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

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fricas [A]  time = 0.68, size = 90, normalized size = 1.70 \[ \frac {1}{33} x^{33} b^{8} + \frac {4}{15} x^{30} b^{7} a + \frac {28}{27} x^{27} b^{6} a^{2} + \frac {7}{3} x^{24} b^{5} a^{3} + \frac {10}{3} x^{21} b^{4} a^{4} + \frac {28}{9} x^{18} b^{3} a^{5} + \frac {28}{15} x^{15} b^{2} a^{6} + \frac {2}{3} x^{12} b a^{7} + \frac {1}{9} x^{9} a^{8} \]

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

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giac [A]  time = 0.16, size = 90, normalized size = 1.70 \[ \frac {1}{33} \, b^{8} x^{33} + \frac {4}{15} \, a b^{7} x^{30} + \frac {28}{27} \, a^{2} b^{6} x^{27} + \frac {7}{3} \, a^{3} b^{5} x^{24} + \frac {10}{3} \, a^{4} b^{4} x^{21} + \frac {28}{9} \, a^{5} b^{3} x^{18} + \frac {28}{15} \, a^{6} b^{2} x^{15} + \frac {2}{3} \, a^{7} b x^{12} + \frac {1}{9} \, a^{8} x^{9} \]

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

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maple [A]  time = 0.00, size = 91, normalized size = 1.72 \[ \frac {1}{33} b^{8} x^{33}+\frac {4}{15} a \,b^{7} x^{30}+\frac {28}{27} a^{2} b^{6} x^{27}+\frac {7}{3} a^{3} b^{5} x^{24}+\frac {10}{3} a^{4} b^{4} x^{21}+\frac {28}{9} a^{5} b^{3} x^{18}+\frac {28}{15} a^{6} b^{2} x^{15}+\frac {2}{3} a^{7} b \,x^{12}+\frac {1}{9} a^{8} x^{9} \]

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*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^18+28/15*a^
6*b^2*x^15+2/3*a^7*b*x^12+1/9*a^8*x^9

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maxima [A]  time = 1.32, size = 90, normalized size = 1.70 \[ \frac {1}{33} \, b^{8} x^{33} + \frac {4}{15} \, a b^{7} x^{30} + \frac {28}{27} \, a^{2} b^{6} x^{27} + \frac {7}{3} \, a^{3} b^{5} x^{24} + \frac {10}{3} \, a^{4} b^{4} x^{21} + \frac {28}{9} \, a^{5} b^{3} x^{18} + \frac {28}{15} \, a^{6} b^{2} x^{15} + \frac {2}{3} \, a^{7} b x^{12} + \frac {1}{9} \, a^{8} x^{9} \]

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

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mupad [B]  time = 0.09, size = 90, normalized size = 1.70 \[ \frac {a^8\,x^9}{9}+\frac {2\,a^7\,b\,x^{12}}{3}+\frac {28\,a^6\,b^2\,x^{15}}{15}+\frac {28\,a^5\,b^3\,x^{18}}{9}+\frac {10\,a^4\,b^4\,x^{21}}{3}+\frac {7\,a^3\,b^5\,x^{24}}{3}+\frac {28\,a^2\,b^6\,x^{27}}{27}+\frac {4\,a\,b^7\,x^{30}}{15}+\frac {b^8\,x^{33}}{33} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(a + b*x^3)^8,x)

[Out]

(a^8*x^9)/9 + (b^8*x^33)/33 + (2*a^7*b*x^12)/3 + (4*a*b^7*x^30)/15 + (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

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sympy [B]  time = 0.11, size = 107, normalized size = 2.02 \[ \frac {a^{8} x^{9}}{9} + \frac {2 a^{7} b x^{12}}{3} + \frac {28 a^{6} b^{2} x^{15}}{15} + \frac {28 a^{5} b^{3} x^{18}}{9} + \frac {10 a^{4} b^{4} x^{21}}{3} + \frac {7 a^{3} b^{5} x^{24}}{3} + \frac {28 a^{2} b^{6} x^{27}}{27} + \frac {4 a b^{7} x^{30}}{15} + \frac {b^{8} x^{33}}{33} \]

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

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