3.3.85 \(\int x^{20} (a+b x^3)^8 \, dx\) [285]

Optimal. Leaf size=129 \[ \frac {a^6 \left (a+b x^3\right )^9}{27 b^7}-\frac {a^5 \left (a+b x^3\right )^{10}}{5 b^7}+\frac {5 a^4 \left (a+b x^3\right )^{11}}{11 b^7}-\frac {5 a^3 \left (a+b x^3\right )^{12}}{9 b^7}+\frac {5 a^2 \left (a+b x^3\right )^{13}}{13 b^7}-\frac {a \left (a+b x^3\right )^{14}}{7 b^7}+\frac {\left (a+b x^3\right )^{15}}{45 b^7} \]

[Out]

1/27*a^6*(b*x^3+a)^9/b^7-1/5*a^5*(b*x^3+a)^10/b^7+5/11*a^4*(b*x^3+a)^11/b^7-5/9*a^3*(b*x^3+a)^12/b^7+5/13*a^2*
(b*x^3+a)^13/b^7-1/7*a*(b*x^3+a)^14/b^7+1/45*(b*x^3+a)^15/b^7

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Rubi [A]
time = 0.15, antiderivative size = 129, 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 = {272, 45} \begin {gather*} \frac {a^6 \left (a+b x^3\right )^9}{27 b^7}-\frac {a^5 \left (a+b x^3\right )^{10}}{5 b^7}+\frac {5 a^4 \left (a+b x^3\right )^{11}}{11 b^7}-\frac {5 a^3 \left (a+b x^3\right )^{12}}{9 b^7}+\frac {5 a^2 \left (a+b x^3\right )^{13}}{13 b^7}+\frac {\left (a+b x^3\right )^{15}}{45 b^7}-\frac {a \left (a+b x^3\right )^{14}}{7 b^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^20*(a + b*x^3)^8,x]

[Out]

(a^6*(a + b*x^3)^9)/(27*b^7) - (a^5*(a + b*x^3)^10)/(5*b^7) + (5*a^4*(a + b*x^3)^11)/(11*b^7) - (5*a^3*(a + b*
x^3)^12)/(9*b^7) + (5*a^2*(a + b*x^3)^13)/(13*b^7) - (a*(a + b*x^3)^14)/(7*b^7) + (a + b*x^3)^15/(45*b^7)

Rule 45

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 272

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^{20} \left (a+b x^3\right )^8 \, dx &=\frac {1}{3} \text {Subst}\left (\int x^6 (a+b x)^8 \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a^6 (a+b x)^8}{b^6}-\frac {6 a^5 (a+b x)^9}{b^6}+\frac {15 a^4 (a+b x)^{10}}{b^6}-\frac {20 a^3 (a+b x)^{11}}{b^6}+\frac {15 a^2 (a+b x)^{12}}{b^6}-\frac {6 a (a+b x)^{13}}{b^6}+\frac {(a+b x)^{14}}{b^6}\right ) \, dx,x,x^3\right )\\ &=\frac {a^6 \left (a+b x^3\right )^9}{27 b^7}-\frac {a^5 \left (a+b x^3\right )^{10}}{5 b^7}+\frac {5 a^4 \left (a+b x^3\right )^{11}}{11 b^7}-\frac {5 a^3 \left (a+b x^3\right )^{12}}{9 b^7}+\frac {5 a^2 \left (a+b x^3\right )^{13}}{13 b^7}-\frac {a \left (a+b x^3\right )^{14}}{7 b^7}+\frac {\left (a+b x^3\right )^{15}}{45 b^7}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 108, normalized size = 0.84 \begin {gather*} \frac {a^8 x^{21}}{21}+\frac {1}{3} a^7 b x^{24}+\frac {28}{27} a^6 b^2 x^{27}+\frac {28}{15} a^5 b^3 x^{30}+\frac {70}{33} a^4 b^4 x^{33}+\frac {14}{9} a^3 b^5 x^{36}+\frac {28}{39} a^2 b^6 x^{39}+\frac {4}{21} a b^7 x^{42}+\frac {b^8 x^{45}}{45} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^20*(a + b*x^3)^8,x]

[Out]

(a^8*x^21)/21 + (a^7*b*x^24)/3 + (28*a^6*b^2*x^27)/27 + (28*a^5*b^3*x^30)/15 + (70*a^4*b^4*x^33)/33 + (14*a^3*
b^5*x^36)/9 + (28*a^2*b^6*x^39)/39 + (4*a*b^7*x^42)/21 + (b^8*x^45)/45

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Maple [A]
time = 0.13, size = 91, normalized size = 0.71

method	result	size
gosper	\(\frac {28}{39} a^{2} b^{6} x^{39}+\frac {4}{21} a \,b^{7} x^{42}+\frac {1}{45} b^{8} x^{45}+\frac {14}{9} a^{3} b^{5} x^{36}+\frac {70}{33} a^{4} b^{4} x^{33}+\frac {28}{15} a^{5} b^{3} x^{30}+\frac {1}{21} a^{8} x^{21}+\frac {1}{3} b \,a^{7} x^{24}+\frac {28}{27} a^{6} b^{2} x^{27}\)	\(91\)
default	\(\frac {28}{39} a^{2} b^{6} x^{39}+\frac {4}{21} a \,b^{7} x^{42}+\frac {1}{45} b^{8} x^{45}+\frac {14}{9} a^{3} b^{5} x^{36}+\frac {70}{33} a^{4} b^{4} x^{33}+\frac {28}{15} a^{5} b^{3} x^{30}+\frac {1}{21} a^{8} x^{21}+\frac {1}{3} b \,a^{7} x^{24}+\frac {28}{27} a^{6} b^{2} x^{27}\)	\(91\)
risch	\(\frac {28}{39} a^{2} b^{6} x^{39}+\frac {4}{21} a \,b^{7} x^{42}+\frac {1}{45} b^{8} x^{45}+\frac {14}{9} a^{3} b^{5} x^{36}+\frac {70}{33} a^{4} b^{4} x^{33}+\frac {28}{15} a^{5} b^{3} x^{30}+\frac {1}{21} a^{8} x^{21}+\frac {1}{3} b \,a^{7} x^{24}+\frac {28}{27} a^{6} b^{2} x^{27}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^20*(b*x^3+a)^8,x,method=_RETURNVERBOSE)

[Out]

28/39*a^2*b^6*x^39+4/21*a*b^7*x^42+1/45*b^8*x^45+14/9*a^3*b^5*x^36+70/33*a^4*b^4*x^33+28/15*a^5*b^3*x^30+1/21*
a^8*x^21+1/3*b*a^7*x^24+28/27*a^6*b^2*x^27

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Maxima [A]
time = 0.29, size = 90, normalized size = 0.70 \begin {gather*} \frac {1}{45} \, b^{8} x^{45} + \frac {4}{21} \, a b^{7} x^{42} + \frac {28}{39} \, a^{2} b^{6} x^{39} + \frac {14}{9} \, a^{3} b^{5} x^{36} + \frac {70}{33} \, a^{4} b^{4} x^{33} + \frac {28}{15} \, a^{5} b^{3} x^{30} + \frac {28}{27} \, a^{6} b^{2} x^{27} + \frac {1}{3} \, a^{7} b x^{24} + \frac {1}{21} \, a^{8} x^{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^20*(b*x^3+a)^8,x, algorithm="maxima")

[Out]

1/45*b^8*x^45 + 4/21*a*b^7*x^42 + 28/39*a^2*b^6*x^39 + 14/9*a^3*b^5*x^36 + 70/33*a^4*b^4*x^33 + 28/15*a^5*b^3*
x^30 + 28/27*a^6*b^2*x^27 + 1/3*a^7*b*x^24 + 1/21*a^8*x^21

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Fricas [A]
time = 0.34, size = 90, normalized size = 0.70 \begin {gather*} \frac {1}{45} \, b^{8} x^{45} + \frac {4}{21} \, a b^{7} x^{42} + \frac {28}{39} \, a^{2} b^{6} x^{39} + \frac {14}{9} \, a^{3} b^{5} x^{36} + \frac {70}{33} \, a^{4} b^{4} x^{33} + \frac {28}{15} \, a^{5} b^{3} x^{30} + \frac {28}{27} \, a^{6} b^{2} x^{27} + \frac {1}{3} \, a^{7} b x^{24} + \frac {1}{21} \, a^{8} x^{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^20*(b*x^3+a)^8,x, algorithm="fricas")

[Out]

1/45*b^8*x^45 + 4/21*a*b^7*x^42 + 28/39*a^2*b^6*x^39 + 14/9*a^3*b^5*x^36 + 70/33*a^4*b^4*x^33 + 28/15*a^5*b^3*
x^30 + 28/27*a^6*b^2*x^27 + 1/3*a^7*b*x^24 + 1/21*a^8*x^21

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Sympy [A]
time = 0.02, size = 105, normalized size = 0.81 \begin {gather*} \frac {a^{8} x^{21}}{21} + \frac {a^{7} b x^{24}}{3} + \frac {28 a^{6} b^{2} x^{27}}{27} + \frac {28 a^{5} b^{3} x^{30}}{15} + \frac {70 a^{4} b^{4} x^{33}}{33} + \frac {14 a^{3} b^{5} x^{36}}{9} + \frac {28 a^{2} b^{6} x^{39}}{39} + \frac {4 a b^{7} x^{42}}{21} + \frac {b^{8} x^{45}}{45} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**20*(b*x**3+a)**8,x)

[Out]

a**8*x**21/21 + a**7*b*x**24/3 + 28*a**6*b**2*x**27/27 + 28*a**5*b**3*x**30/15 + 70*a**4*b**4*x**33/33 + 14*a*
*3*b**5*x**36/9 + 28*a**2*b**6*x**39/39 + 4*a*b**7*x**42/21 + b**8*x**45/45

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Giac [A]
time = 1.27, size = 90, normalized size = 0.70 \begin {gather*} \frac {1}{45} \, b^{8} x^{45} + \frac {4}{21} \, a b^{7} x^{42} + \frac {28}{39} \, a^{2} b^{6} x^{39} + \frac {14}{9} \, a^{3} b^{5} x^{36} + \frac {70}{33} \, a^{4} b^{4} x^{33} + \frac {28}{15} \, a^{5} b^{3} x^{30} + \frac {28}{27} \, a^{6} b^{2} x^{27} + \frac {1}{3} \, a^{7} b x^{24} + \frac {1}{21} \, a^{8} x^{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^20*(b*x^3+a)^8,x, algorithm="giac")

[Out]

1/45*b^8*x^45 + 4/21*a*b^7*x^42 + 28/39*a^2*b^6*x^39 + 14/9*a^3*b^5*x^36 + 70/33*a^4*b^4*x^33 + 28/15*a^5*b^3*
x^30 + 28/27*a^6*b^2*x^27 + 1/3*a^7*b*x^24 + 1/21*a^8*x^21

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Mupad [B]
time = 1.03, size = 90, normalized size = 0.70 \begin {gather*} \frac {a^8\,x^{21}}{21}+\frac {a^7\,b\,x^{24}}{3}+\frac {28\,a^6\,b^2\,x^{27}}{27}+\frac {28\,a^5\,b^3\,x^{30}}{15}+\frac {70\,a^4\,b^4\,x^{33}}{33}+\frac {14\,a^3\,b^5\,x^{36}}{9}+\frac {28\,a^2\,b^6\,x^{39}}{39}+\frac {4\,a\,b^7\,x^{42}}{21}+\frac {b^8\,x^{45}}{45} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^20*(a + b*x^3)^8,x)

[Out]

(a^8*x^21)/21 + (b^8*x^45)/45 + (a^7*b*x^24)/3 + (4*a*b^7*x^42)/21 + (28*a^6*b^2*x^27)/27 + (28*a^5*b^3*x^30)/
15 + (70*a^4*b^4*x^33)/33 + (14*a^3*b^5*x^36)/9 + (28*a^2*b^6*x^39)/39

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