3.3.90 \(\int x^5 (a+b x^3)^8 \, dx\) [290]

Optimal. Leaf size=34 \[ -\frac {a \left (a+b x^3\right )^9}{27 b^2}+\frac {\left (a+b x^3\right )^{10}}{30 b^2} \]

[Out]

-1/27*a*(b*x^3+a)^9/b^2+1/30*(b*x^3+a)^10/b^2

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Rubi [A]
time = 0.03, antiderivative size = 34, 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 {\left (a+b x^3\right )^{10}}{30 b^2}-\frac {a \left (a+b x^3\right )^9}{27 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^3)^8,x]

[Out]

-1/27*(a*(a + b*x^3)^9)/b^2 + (a + b*x^3)^10/(30*b^2)

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^5 \left (a+b x^3\right )^8 \, dx &=\frac {1}{3} \text {Subst}\left (\int x (a+b x)^8 \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-\frac {a (a+b x)^8}{b}+\frac {(a+b x)^9}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac {a \left (a+b x^3\right )^9}{27 b^2}+\frac {\left (a+b x^3\right )^{10}}{30 b^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(34)=68\).
time = 0.00, size = 108, normalized size = 3.18 \begin {gather*} \frac {a^8 x^6}{6}+\frac {8}{9} a^7 b x^9+\frac {7}{3} a^6 b^2 x^{12}+\frac {56}{15} a^5 b^3 x^{15}+\frac {35}{9} a^4 b^4 x^{18}+\frac {8}{3} a^3 b^5 x^{21}+\frac {7}{6} a^2 b^6 x^{24}+\frac {8}{27} a b^7 x^{27}+\frac {b^8 x^{30}}{30} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^3)^8,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(30)=60\).
time = 0.13, size = 91, normalized size = 2.68

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (30) = 60\).
time = 0.30, size = 90, normalized size = 2.65 \begin {gather*} \frac {1}{30} \, b^{8} x^{30} + \frac {8}{27} \, a b^{7} x^{27} + \frac {7}{6} \, a^{2} b^{6} x^{24} + \frac {8}{3} \, a^{3} b^{5} x^{21} + \frac {35}{9} \, a^{4} b^{4} x^{18} + \frac {56}{15} \, a^{5} b^{3} x^{15} + \frac {7}{3} \, a^{6} b^{2} x^{12} + \frac {8}{9} \, a^{7} b x^{9} + \frac {1}{6} \, a^{8} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (30) = 60\).
time = 0.35, size = 90, normalized size = 2.65 \begin {gather*} \frac {1}{30} \, b^{8} x^{30} + \frac {8}{27} \, a b^{7} x^{27} + \frac {7}{6} \, a^{2} b^{6} x^{24} + \frac {8}{3} \, a^{3} b^{5} x^{21} + \frac {35}{9} \, a^{4} b^{4} x^{18} + \frac {56}{15} \, a^{5} b^{3} x^{15} + \frac {7}{3} \, a^{6} b^{2} x^{12} + \frac {8}{9} \, a^{7} b x^{9} + \frac {1}{6} \, a^{8} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (27) = 54\).
time = 0.02, size = 107, normalized size = 3.15 \begin {gather*} \frac {a^{8} x^{6}}{6} + \frac {8 a^{7} b x^{9}}{9} + \frac {7 a^{6} b^{2} x^{12}}{3} + \frac {56 a^{5} b^{3} x^{15}}{15} + \frac {35 a^{4} b^{4} x^{18}}{9} + \frac {8 a^{3} b^{5} x^{21}}{3} + \frac {7 a^{2} b^{6} x^{24}}{6} + \frac {8 a b^{7} x^{27}}{27} + \frac {b^{8} x^{30}}{30} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (30) = 60\).
time = 1.18, size = 90, normalized size = 2.65 \begin {gather*} \frac {1}{30} \, b^{8} x^{30} + \frac {8}{27} \, a b^{7} x^{27} + \frac {7}{6} \, a^{2} b^{6} x^{24} + \frac {8}{3} \, a^{3} b^{5} x^{21} + \frac {35}{9} \, a^{4} b^{4} x^{18} + \frac {56}{15} \, a^{5} b^{3} x^{15} + \frac {7}{3} \, a^{6} b^{2} x^{12} + \frac {8}{9} \, a^{7} b x^{9} + \frac {1}{6} \, a^{8} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.09, size = 90, normalized size = 2.65 \begin {gather*} \frac {a^8\,x^6}{6}+\frac {8\,a^7\,b\,x^9}{9}+\frac {7\,a^6\,b^2\,x^{12}}{3}+\frac {56\,a^5\,b^3\,x^{15}}{15}+\frac {35\,a^4\,b^4\,x^{18}}{9}+\frac {8\,a^3\,b^5\,x^{21}}{3}+\frac {7\,a^2\,b^6\,x^{24}}{6}+\frac {8\,a\,b^7\,x^{27}}{27}+\frac {b^8\,x^{30}}{30} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*x^3)^8,x)

[Out]

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

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