3.3.88 \(\int x^{11} (a+b x^3)^8 \, dx\) [288]

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

[Out]

-1/27*a^3*(b*x^3+a)^9/b^4+1/10*a^2*(b*x^3+a)^10/b^4-1/11*a*(b*x^3+a)^11/b^4+1/36*(b*x^3+a)^12/b^4

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Rubi [A]
time = 0.07, antiderivative size = 72, 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^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 {\left (a+b x^3\right )^{12}}{36 b^4}-\frac {a \left (a+b x^3\right )^{11}}{11 b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^11*(a + b*x^3)^8,x]

[Out]

-1/27*(a^3*(a + b*x^3)^9)/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 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^{11} \left (a+b x^3\right )^8 \, dx &=\frac {1}{3} \text {Subst}\left (\int x^3 (a+b x)^8 \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {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*}

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Mathematica [A]
time = 0.00, size = 108, normalized size = 1.50 \begin {gather*} \frac {a^8 x^{12}}{12}+\frac {8}{15} a^7 b x^{15}+\frac {14}{9} a^6 b^2 x^{18}+\frac {8}{3} a^5 b^3 x^{21}+\frac {35}{12} a^4 b^4 x^{24}+\frac {56}{27} a^3 b^5 x^{27}+\frac {14}{15} a^2 b^6 x^{30}+\frac {8}{33} a b^7 x^{33}+\frac {b^8 x^{36}}{36} \end {gather*}

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 = 0.14, size = 91, normalized size = 1.26

method	result	size
gosper	\(\frac {1}{36} b^{8} x^{36}+\frac {56}{27} a^{3} b^{5} x^{27}+\frac {14}{15} a^{2} b^{6} x^{30}+\frac {8}{33} a \,b^{7} x^{33}+\frac {8}{15} a^{7} b \,x^{15}+\frac {14}{9} a^{6} b^{2} x^{18}+\frac {8}{3} a^{5} b^{3} x^{21}+\frac {35}{12} a^{4} b^{4} x^{24}+\frac {1}{12} a^{8} x^{12}\)	\(91\)
default	\(\frac {1}{36} b^{8} x^{36}+\frac {56}{27} a^{3} b^{5} x^{27}+\frac {14}{15} a^{2} b^{6} x^{30}+\frac {8}{33} a \,b^{7} x^{33}+\frac {8}{15} a^{7} b \,x^{15}+\frac {14}{9} a^{6} b^{2} x^{18}+\frac {8}{3} a^{5} b^{3} x^{21}+\frac {35}{12} a^{4} b^{4} x^{24}+\frac {1}{12} a^{8} x^{12}\)	\(91\)
norman	\(\frac {1}{36} b^{8} x^{36}+\frac {56}{27} a^{3} b^{5} x^{27}+\frac {14}{15} a^{2} b^{6} x^{30}+\frac {8}{33} a \,b^{7} x^{33}+\frac {8}{15} a^{7} b \,x^{15}+\frac {14}{9} a^{6} b^{2} x^{18}+\frac {8}{3} a^{5} b^{3} x^{21}+\frac {35}{12} a^{4} b^{4} x^{24}+\frac {1}{12} a^{8} x^{12}\)	\(91\)
risch	\(\frac {1}{36} b^{8} x^{36}+\frac {56}{27} a^{3} b^{5} x^{27}+\frac {14}{15} a^{2} b^{6} x^{30}+\frac {8}{33} a \,b^{7} x^{33}+\frac {8}{15} a^{7} b \,x^{15}+\frac {14}{9} a^{6} b^{2} x^{18}+\frac {8}{3} a^{5} b^{3} x^{21}+\frac {35}{12} a^{4} b^{4} x^{24}+\frac {1}{12} a^{8} x^{12}\)	\(91\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 90, normalized size = 1.25 \begin {gather*} \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 {gather*}

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 = 0.34, size = 90, normalized size = 1.25 \begin {gather*} \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 {gather*}

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*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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Sympy [A]
time = 0.02, size = 107, normalized size = 1.49 \begin {gather*} \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 {gather*}

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 = 2.23, size = 90, normalized size = 1.25 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 1.00, size = 90, normalized size = 1.25 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(a + b*x^3)^8,x)

[Out]

(a^8*x^12)/12 + (b^8*x^36)/36 + (8*a^7*b*x^15)/15 + (8*a*b^7*x^33)/33 + (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

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