3.287 \(\int x^{14} (a+b x^3)^8 \, dx\)

Optimal. Leaf size=91 \[ \frac {a^4 \left (a+b x^3\right )^9}{27 b^5}-\frac {2 a^3 \left (a+b x^3\right )^{10}}{15 b^5}+\frac {2 a^2 \left (a+b x^3\right )^{11}}{11 b^5}+\frac {\left (a+b x^3\right )^{13}}{39 b^5}-\frac {a \left (a+b x^3\right )^{12}}{9 b^5} \]

[Out]

1/27*a^4*(b*x^3+a)^9/b^5-2/15*a^3*(b*x^3+a)^10/b^5+2/11*a^2*(b*x^3+a)^11/b^5-1/9*a*(b*x^3+a)^12/b^5+1/39*(b*x^
3+a)^13/b^5

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Rubi [A]  time = 0.14, antiderivative size = 91, 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} \[ \frac {2 a^2 \left (a+b x^3\right )^{11}}{11 b^5}-\frac {2 a^3 \left (a+b x^3\right )^{10}}{15 b^5}+\frac {a^4 \left (a+b x^3\right )^9}{27 b^5}+\frac {\left (a+b x^3\right )^{13}}{39 b^5}-\frac {a \left (a+b x^3\right )^{12}}{9 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^14*(a + b*x^3)^8,x]

[Out]

(a^4*(a + b*x^3)^9)/(27*b^5) - (2*a^3*(a + b*x^3)^10)/(15*b^5) + (2*a^2*(a + b*x^3)^11)/(11*b^5) - (a*(a + b*x
^3)^12)/(9*b^5) + (a + b*x^3)^13/(39*b^5)

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^{14} \left (a+b x^3\right )^8 \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x^4 (a+b x)^8 \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {a^4 (a+b x)^8}{b^4}-\frac {4 a^3 (a+b x)^9}{b^4}+\frac {6 a^2 (a+b x)^{10}}{b^4}-\frac {4 a (a+b x)^{11}}{b^4}+\frac {(a+b x)^{12}}{b^4}\right ) \, dx,x,x^3\right )\\ &=\frac {a^4 \left (a+b x^3\right )^9}{27 b^5}-\frac {2 a^3 \left (a+b x^3\right )^{10}}{15 b^5}+\frac {2 a^2 \left (a+b x^3\right )^{11}}{11 b^5}-\frac {a \left (a+b x^3\right )^{12}}{9 b^5}+\frac {\left (a+b x^3\right )^{13}}{39 b^5}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 108, normalized size = 1.19 \[ \frac {a^8 x^{15}}{15}+\frac {4}{9} a^7 b x^{18}+\frac {4}{3} a^6 b^2 x^{21}+\frac {7}{3} a^5 b^3 x^{24}+\frac {70}{27} a^4 b^4 x^{27}+\frac {28}{15} a^3 b^5 x^{30}+\frac {28}{33} a^2 b^6 x^{33}+\frac {2}{9} a b^7 x^{36}+\frac {b^8 x^{39}}{39} \]

Antiderivative was successfully verified.

[In]

Integrate[x^14*(a + b*x^3)^8,x]

[Out]

(a^8*x^15)/15 + (4*a^7*b*x^18)/9 + (4*a^6*b^2*x^21)/3 + (7*a^5*b^3*x^24)/3 + (70*a^4*b^4*x^27)/27 + (28*a^3*b^
5*x^30)/15 + (28*a^2*b^6*x^33)/33 + (2*a*b^7*x^36)/9 + (b^8*x^39)/39

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fricas [A]  time = 0.78, size = 90, normalized size = 0.99 \[ \frac {1}{39} x^{39} b^{8} + \frac {2}{9} x^{36} b^{7} a + \frac {28}{33} x^{33} b^{6} a^{2} + \frac {28}{15} x^{30} b^{5} a^{3} + \frac {70}{27} x^{27} b^{4} a^{4} + \frac {7}{3} x^{24} b^{3} a^{5} + \frac {4}{3} x^{21} b^{2} a^{6} + \frac {4}{9} x^{18} b a^{7} + \frac {1}{15} x^{15} a^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14*(b*x^3+a)^8,x, algorithm="fricas")

[Out]

1/39*x^39*b^8 + 2/9*x^36*b^7*a + 28/33*x^33*b^6*a^2 + 28/15*x^30*b^5*a^3 + 70/27*x^27*b^4*a^4 + 7/3*x^24*b^3*a
^5 + 4/3*x^21*b^2*a^6 + 4/9*x^18*b*a^7 + 1/15*x^15*a^8

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giac [A]  time = 0.21, size = 90, normalized size = 0.99 \[ \frac {1}{39} \, b^{8} x^{39} + \frac {2}{9} \, a b^{7} x^{36} + \frac {28}{33} \, a^{2} b^{6} x^{33} + \frac {28}{15} \, a^{3} b^{5} x^{30} + \frac {70}{27} \, a^{4} b^{4} x^{27} + \frac {7}{3} \, a^{5} b^{3} x^{24} + \frac {4}{3} \, a^{6} b^{2} x^{21} + \frac {4}{9} \, a^{7} b x^{18} + \frac {1}{15} \, a^{8} x^{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14*(b*x^3+a)^8,x, algorithm="giac")

[Out]

1/39*b^8*x^39 + 2/9*a*b^7*x^36 + 28/33*a^2*b^6*x^33 + 28/15*a^3*b^5*x^30 + 70/27*a^4*b^4*x^27 + 7/3*a^5*b^3*x^
24 + 4/3*a^6*b^2*x^21 + 4/9*a^7*b*x^18 + 1/15*a^8*x^15

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maple [A]  time = 0.00, size = 91, normalized size = 1.00 \[ \frac {1}{39} b^{8} x^{39}+\frac {2}{9} a \,b^{7} x^{36}+\frac {28}{33} a^{2} b^{6} x^{33}+\frac {28}{15} a^{3} b^{5} x^{30}+\frac {70}{27} a^{4} b^{4} x^{27}+\frac {7}{3} a^{5} b^{3} x^{24}+\frac {4}{3} a^{6} b^{2} x^{21}+\frac {4}{9} a^{7} b \,x^{18}+\frac {1}{15} a^{8} x^{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14*(b*x^3+a)^8,x)

[Out]

1/39*b^8*x^39+2/9*a*b^7*x^36+28/33*a^2*b^6*x^33+28/15*a^3*b^5*x^30+70/27*a^4*b^4*x^27+7/3*a^5*b^3*x^24+4/3*a^6
*b^2*x^21+4/9*a^7*b*x^18+1/15*a^8*x^15

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maxima [A]  time = 1.34, size = 90, normalized size = 0.99 \[ \frac {1}{39} \, b^{8} x^{39} + \frac {2}{9} \, a b^{7} x^{36} + \frac {28}{33} \, a^{2} b^{6} x^{33} + \frac {28}{15} \, a^{3} b^{5} x^{30} + \frac {70}{27} \, a^{4} b^{4} x^{27} + \frac {7}{3} \, a^{5} b^{3} x^{24} + \frac {4}{3} \, a^{6} b^{2} x^{21} + \frac {4}{9} \, a^{7} b x^{18} + \frac {1}{15} \, a^{8} x^{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14*(b*x^3+a)^8,x, algorithm="maxima")

[Out]

1/39*b^8*x^39 + 2/9*a*b^7*x^36 + 28/33*a^2*b^6*x^33 + 28/15*a^3*b^5*x^30 + 70/27*a^4*b^4*x^27 + 7/3*a^5*b^3*x^
24 + 4/3*a^6*b^2*x^21 + 4/9*a^7*b*x^18 + 1/15*a^8*x^15

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mupad [B]  time = 0.09, size = 90, normalized size = 0.99 \[ \frac {a^8\,x^{15}}{15}+\frac {4\,a^7\,b\,x^{18}}{9}+\frac {4\,a^6\,b^2\,x^{21}}{3}+\frac {7\,a^5\,b^3\,x^{24}}{3}+\frac {70\,a^4\,b^4\,x^{27}}{27}+\frac {28\,a^3\,b^5\,x^{30}}{15}+\frac {28\,a^2\,b^6\,x^{33}}{33}+\frac {2\,a\,b^7\,x^{36}}{9}+\frac {b^8\,x^{39}}{39} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14*(a + b*x^3)^8,x)

[Out]

(a^8*x^15)/15 + (b^8*x^39)/39 + (4*a^7*b*x^18)/9 + (2*a*b^7*x^36)/9 + (4*a^6*b^2*x^21)/3 + (7*a^5*b^3*x^24)/3
+ (70*a^4*b^4*x^27)/27 + (28*a^3*b^5*x^30)/15 + (28*a^2*b^6*x^33)/33

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sympy [A]  time = 0.11, size = 107, normalized size = 1.18 \[ \frac {a^{8} x^{15}}{15} + \frac {4 a^{7} b x^{18}}{9} + \frac {4 a^{6} b^{2} x^{21}}{3} + \frac {7 a^{5} b^{3} x^{24}}{3} + \frac {70 a^{4} b^{4} x^{27}}{27} + \frac {28 a^{3} b^{5} x^{30}}{15} + \frac {28 a^{2} b^{6} x^{33}}{33} + \frac {2 a b^{7} x^{36}}{9} + \frac {b^{8} x^{39}}{39} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**14*(b*x**3+a)**8,x)

[Out]

a**8*x**15/15 + 4*a**7*b*x**18/9 + 4*a**6*b**2*x**21/3 + 7*a**5*b**3*x**24/3 + 70*a**4*b**4*x**27/27 + 28*a**3
*b**5*x**30/15 + 28*a**2*b**6*x**33/33 + 2*a*b**7*x**36/9 + b**8*x**39/39

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