3.3.61 \(\int x^8 (a+b x^3)^5 \, dx\) [261]

Optimal. Leaf size=53 \[ \frac {a^2 \left (a+b x^3\right )^6}{18 b^3}-\frac {2 a \left (a+b x^3\right )^7}{21 b^3}+\frac {\left (a+b x^3\right )^8}{24 b^3} \]

[Out]

1/18*a^2*(b*x^3+a)^6/b^3-2/21*a*(b*x^3+a)^7/b^3+1/24*(b*x^3+a)^8/b^3

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Rubi [A]
time = 0.05, 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 = {272, 45} \begin {gather*} \frac {a^2 \left (a+b x^3\right )^6}{18 b^3}+\frac {\left (a+b x^3\right )^8}{24 b^3}-\frac {2 a \left (a+b x^3\right )^7}{21 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*x^3)^5,x]

[Out]

(a^2*(a + b*x^3)^6)/(18*b^3) - (2*a*(a + b*x^3)^7)/(21*b^3) + (a + b*x^3)^8/(24*b^3)

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^8 \left (a+b x^3\right )^5 \, dx &=\frac {1}{3} \text {Subst}\left (\int x^2 (a+b x)^5 \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a^2 (a+b x)^5}{b^2}-\frac {2 a (a+b x)^6}{b^2}+\frac {(a+b x)^7}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac {a^2 \left (a+b x^3\right )^6}{18 b^3}-\frac {2 a \left (a+b x^3\right )^7}{21 b^3}+\frac {\left (a+b x^3\right )^8}{24 b^3}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 69, normalized size = 1.30 \begin {gather*} \frac {a^5 x^9}{9}+\frac {5}{12} a^4 b x^{12}+\frac {2}{3} a^3 b^2 x^{15}+\frac {5}{9} a^2 b^3 x^{18}+\frac {5}{21} a b^4 x^{21}+\frac {b^5 x^{24}}{24} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a + b*x^3)^5,x]

[Out]

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

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Maple [A]
time = 0.13, size = 58, normalized size = 1.09

method	result	size
gosper	\(\frac {2}{3} a^{3} b^{2} x^{15}+\frac {5}{9} a^{2} b^{3} x^{18}+\frac {5}{21} a \,b^{4} x^{21}+\frac {1}{24} b^{5} x^{24}+\frac {1}{9} a^{5} x^{9}+\frac {5}{12} a^{4} b \,x^{12}\)	\(58\)
default	\(\frac {2}{3} a^{3} b^{2} x^{15}+\frac {5}{9} a^{2} b^{3} x^{18}+\frac {5}{21} a \,b^{4} x^{21}+\frac {1}{24} b^{5} x^{24}+\frac {1}{9} a^{5} x^{9}+\frac {5}{12} a^{4} b \,x^{12}\)	\(58\)
norman	\(\frac {2}{3} a^{3} b^{2} x^{15}+\frac {5}{9} a^{2} b^{3} x^{18}+\frac {5}{21} a \,b^{4} x^{21}+\frac {1}{24} b^{5} x^{24}+\frac {1}{9} a^{5} x^{9}+\frac {5}{12} a^{4} b \,x^{12}\)	\(58\)
risch	\(\frac {2}{3} a^{3} b^{2} x^{15}+\frac {5}{9} a^{2} b^{3} x^{18}+\frac {5}{21} a \,b^{4} x^{21}+\frac {1}{24} b^{5} x^{24}+\frac {1}{9} a^{5} x^{9}+\frac {5}{12} a^{4} b \,x^{12}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(b*x^3+a)^5,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.30, size = 57, normalized size = 1.08 \begin {gather*} \frac {1}{24} \, b^{5} x^{24} + \frac {5}{21} \, a b^{4} x^{21} + \frac {5}{9} \, a^{2} b^{3} x^{18} + \frac {2}{3} \, a^{3} b^{2} x^{15} + \frac {5}{12} \, a^{4} b x^{12} + \frac {1}{9} \, a^{5} x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^3+a)^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.34, size = 57, normalized size = 1.08 \begin {gather*} \frac {1}{24} \, b^{5} x^{24} + \frac {5}{21} \, a b^{4} x^{21} + \frac {5}{9} \, a^{2} b^{3} x^{18} + \frac {2}{3} \, a^{3} b^{2} x^{15} + \frac {5}{12} \, a^{4} b x^{12} + \frac {1}{9} \, a^{5} x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^3+a)^5,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.01, size = 66, normalized size = 1.25 \begin {gather*} \frac {a^{5} x^{9}}{9} + \frac {5 a^{4} b x^{12}}{12} + \frac {2 a^{3} b^{2} x^{15}}{3} + \frac {5 a^{2} b^{3} x^{18}}{9} + \frac {5 a b^{4} x^{21}}{21} + \frac {b^{5} x^{24}}{24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(b*x**3+a)**5,x)

[Out]

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

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Giac [A]
time = 1.08, size = 57, normalized size = 1.08 \begin {gather*} \frac {1}{24} \, b^{5} x^{24} + \frac {5}{21} \, a b^{4} x^{21} + \frac {5}{9} \, a^{2} b^{3} x^{18} + \frac {2}{3} \, a^{3} b^{2} x^{15} + \frac {5}{12} \, a^{4} b x^{12} + \frac {1}{9} \, a^{5} x^{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x^3+a)^5,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.03, size = 57, normalized size = 1.08 \begin {gather*} \frac {a^5\,x^9}{9}+\frac {5\,a^4\,b\,x^{12}}{12}+\frac {2\,a^3\,b^2\,x^{15}}{3}+\frac {5\,a^2\,b^3\,x^{18}}{9}+\frac {5\,a\,b^4\,x^{21}}{21}+\frac {b^5\,x^{24}}{24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(a + b*x^3)^5,x)

[Out]

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

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