3.260 \(\int x^{11} (a+b x^3)^5 \, dx\)

Optimal. Leaf size=72 \[ -\frac {a^3 \left (a+b x^3\right )^6}{18 b^4}+\frac {a^2 \left (a+b x^3\right )^7}{7 b^4}+\frac {\left (a+b x^3\right )^9}{27 b^4}-\frac {a \left (a+b x^3\right )^8}{8 b^4} \]

[Out]

-1/18*a^3*(b*x^3+a)^6/b^4+1/7*a^2*(b*x^3+a)^7/b^4-1/8*a*(b*x^3+a)^8/b^4+1/27*(b*x^3+a)^9/b^4

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Rubi [A]  time = 0.09, 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 = {266, 43} \[ \frac {a^2 \left (a+b x^3\right )^7}{7 b^4}-\frac {a^3 \left (a+b x^3\right )^6}{18 b^4}+\frac {\left (a+b x^3\right )^9}{27 b^4}-\frac {a \left (a+b x^3\right )^8}{8 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^11*(a + b*x^3)^5,x]

[Out]

-(a^3*(a + b*x^3)^6)/(18*b^4) + (a^2*(a + b*x^3)^7)/(7*b^4) - (a*(a + b*x^3)^8)/(8*b^4) + (a + b*x^3)^9/(27*b^
4)

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^{11} \left (a+b x^3\right )^5 \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x^3 (a+b x)^5 \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {a^3 (a+b x)^5}{b^3}+\frac {3 a^2 (a+b x)^6}{b^3}-\frac {3 a (a+b x)^7}{b^3}+\frac {(a+b x)^8}{b^3}\right ) \, dx,x,x^3\right )\\ &=-\frac {a^3 \left (a+b x^3\right )^6}{18 b^4}+\frac {a^2 \left (a+b x^3\right )^7}{7 b^4}-\frac {a \left (a+b x^3\right )^8}{8 b^4}+\frac {\left (a+b x^3\right )^9}{27 b^4}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 69, normalized size = 0.96 \[ \frac {a^5 x^{12}}{12}+\frac {1}{3} a^4 b x^{15}+\frac {5}{9} a^3 b^2 x^{18}+\frac {10}{21} a^2 b^3 x^{21}+\frac {5}{24} a b^4 x^{24}+\frac {b^5 x^{27}}{27} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11*(a + b*x^3)^5,x]

[Out]

(a^5*x^12)/12 + (a^4*b*x^15)/3 + (5*a^3*b^2*x^18)/9 + (10*a^2*b^3*x^21)/21 + (5*a*b^4*x^24)/24 + (b^5*x^27)/27

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fricas [A]  time = 0.71, size = 57, normalized size = 0.79 \[ \frac {1}{27} x^{27} b^{5} + \frac {5}{24} x^{24} b^{4} a + \frac {10}{21} x^{21} b^{3} a^{2} + \frac {5}{9} x^{18} b^{2} a^{3} + \frac {1}{3} x^{15} b a^{4} + \frac {1}{12} x^{12} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^5,x, algorithm="fricas")

[Out]

1/27*x^27*b^5 + 5/24*x^24*b^4*a + 10/21*x^21*b^3*a^2 + 5/9*x^18*b^2*a^3 + 1/3*x^15*b*a^4 + 1/12*x^12*a^5

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giac [A]  time = 0.15, size = 57, normalized size = 0.79 \[ \frac {1}{27} \, b^{5} x^{27} + \frac {5}{24} \, a b^{4} x^{24} + \frac {10}{21} \, a^{2} b^{3} x^{21} + \frac {5}{9} \, a^{3} b^{2} x^{18} + \frac {1}{3} \, a^{4} b x^{15} + \frac {1}{12} \, a^{5} x^{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^5,x, algorithm="giac")

[Out]

1/27*b^5*x^27 + 5/24*a*b^4*x^24 + 10/21*a^2*b^3*x^21 + 5/9*a^3*b^2*x^18 + 1/3*a^4*b*x^15 + 1/12*a^5*x^12

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maple [A]  time = 0.00, size = 58, normalized size = 0.81 \[ \frac {1}{27} b^{5} x^{27}+\frac {5}{24} a \,b^{4} x^{24}+\frac {10}{21} a^{2} b^{3} x^{21}+\frac {5}{9} a^{3} b^{2} x^{18}+\frac {1}{3} a^{4} b \,x^{15}+\frac {1}{12} a^{5} x^{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(b*x^3+a)^5,x)

[Out]

1/27*b^5*x^27+5/24*a*b^4*x^24+10/21*a^2*b^3*x^21+5/9*a^3*b^2*x^18+1/3*a^4*b*x^15+1/12*a^5*x^12

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maxima [A]  time = 1.33, size = 57, normalized size = 0.79 \[ \frac {1}{27} \, b^{5} x^{27} + \frac {5}{24} \, a b^{4} x^{24} + \frac {10}{21} \, a^{2} b^{3} x^{21} + \frac {5}{9} \, a^{3} b^{2} x^{18} + \frac {1}{3} \, a^{4} b x^{15} + \frac {1}{12} \, a^{5} x^{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11*(b*x^3+a)^5,x, algorithm="maxima")

[Out]

1/27*b^5*x^27 + 5/24*a*b^4*x^24 + 10/21*a^2*b^3*x^21 + 5/9*a^3*b^2*x^18 + 1/3*a^4*b*x^15 + 1/12*a^5*x^12

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mupad [B]  time = 0.02, size = 57, normalized size = 0.79 \[ \frac {a^5\,x^{12}}{12}+\frac {a^4\,b\,x^{15}}{3}+\frac {5\,a^3\,b^2\,x^{18}}{9}+\frac {10\,a^2\,b^3\,x^{21}}{21}+\frac {5\,a\,b^4\,x^{24}}{24}+\frac {b^5\,x^{27}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11*(a + b*x^3)^5,x)

[Out]

(a^5*x^12)/12 + (b^5*x^27)/27 + (a^4*b*x^15)/3 + (5*a*b^4*x^24)/24 + (5*a^3*b^2*x^18)/9 + (10*a^2*b^3*x^21)/21

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sympy [A]  time = 0.08, size = 65, normalized size = 0.90 \[ \frac {a^{5} x^{12}}{12} + \frac {a^{4} b x^{15}}{3} + \frac {5 a^{3} b^{2} x^{18}}{9} + \frac {10 a^{2} b^{3} x^{21}}{21} + \frac {5 a b^{4} x^{24}}{24} + \frac {b^{5} x^{27}}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11*(b*x**3+a)**5,x)

[Out]

a**5*x**12/12 + a**4*b*x**15/3 + 5*a**3*b**2*x**18/9 + 10*a**2*b**3*x**21/21 + 5*a*b**4*x**24/24 + b**5*x**27/
27

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