3.26.72 \(\int x^{-1+3 n} (a+b x^n)^8 \, dx\) [2572]

Optimal. Leaf size=62 \[ \frac {a^2 \left (a+b x^n\right )^9}{9 b^3 n}-\frac {a \left (a+b x^n\right )^{10}}{5 b^3 n}+\frac {\left (a+b x^n\right )^{11}}{11 b^3 n} \]

[Out]

1/9*a^2*(a+b*x^n)^9/b^3/n-1/5*a*(a+b*x^n)^10/b^3/n+1/11*(a+b*x^n)^11/b^3/n

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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {272, 45} \begin {gather*} \frac {a^2 \left (a+b x^n\right )^9}{9 b^3 n}+\frac {\left (a+b x^n\right )^{11}}{11 b^3 n}-\frac {a \left (a+b x^n\right )^{10}}{5 b^3 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 3*n)*(a + b*x^n)^8,x]

[Out]

(a^2*(a + b*x^n)^9)/(9*b^3*n) - (a*(a + b*x^n)^10)/(5*b^3*n) + (a + b*x^n)^11/(11*b^3*n)

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^{-1+3 n} \left (a+b x^n\right )^8 \, dx &=\frac {\text {Subst}\left (\int x^2 (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^2 (a+b x)^8}{b^2}-\frac {2 a (a+b x)^9}{b^2}+\frac {(a+b x)^{10}}{b^2}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {a^2 \left (a+b x^n\right )^9}{9 b^3 n}-\frac {a \left (a+b x^n\right )^{10}}{5 b^3 n}+\frac {\left (a+b x^n\right )^{11}}{11 b^3 n}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 113, normalized size = 1.82 \begin {gather*} \frac {x^{3 n} \left (165 a^8+990 a^7 b x^n+2772 a^6 b^2 x^{2 n}+4620 a^5 b^3 x^{3 n}+4950 a^4 b^4 x^{4 n}+3465 a^3 b^5 x^{5 n}+1540 a^2 b^6 x^{6 n}+396 a b^7 x^{7 n}+45 b^8 x^{8 n}\right )}{495 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + 3*n)*(a + b*x^n)^8,x]

[Out]

(x^(3*n)*(165*a^8 + 990*a^7*b*x^n + 2772*a^6*b^2*x^(2*n) + 4620*a^5*b^3*x^(3*n) + 4950*a^4*b^4*x^(4*n) + 3465*
a^3*b^5*x^(5*n) + 1540*a^2*b^6*x^(6*n) + 396*a*b^7*x^(7*n) + 45*b^8*x^(8*n)))/(495*n)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(135\) vs. \(2(56)=112\).
time = 0.23, size = 136, normalized size = 2.19

method	result	size
risch	\(\frac {b^{8} x^{11 n}}{11 n}+\frac {4 a \,b^{7} x^{10 n}}{5 n}+\frac {28 a^{2} b^{6} x^{9 n}}{9 n}+\frac {7 a^{3} b^{5} x^{8 n}}{n}+\frac {10 a^{4} b^{4} x^{7 n}}{n}+\frac {28 a^{5} b^{3} x^{6 n}}{3 n}+\frac {28 a^{6} b^{2} x^{5 n}}{5 n}+\frac {2 a^{7} b \,x^{4 n}}{n}+\frac {a^{8} x^{3 n}}{3 n}\)	\(136\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+3*n)*(a+b*x^n)^8,x,method=_RETURNVERBOSE)

[Out]

1/11*b^8/n*(x^n)^11+4/5*a*b^7/n*(x^n)^10+28/9*a^2*b^6/n*(x^n)^9+7*a^3*b^5/n*(x^n)^8+10*a^4*b^4/n*(x^n)^7+28/3*
a^5*b^3/n*(x^n)^6+28/5*a^6*b^2/n*(x^n)^5+2*a^7*b/n*(x^n)^4+1/3*a^8/n*(x^n)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (56) = 112\).
time = 0.29, size = 135, normalized size = 2.18 \begin {gather*} \frac {b^{8} x^{11 \, n}}{11 \, n} + \frac {4 \, a b^{7} x^{10 \, n}}{5 \, n} + \frac {28 \, a^{2} b^{6} x^{9 \, n}}{9 \, n} + \frac {7 \, a^{3} b^{5} x^{8 \, n}}{n} + \frac {10 \, a^{4} b^{4} x^{7 \, n}}{n} + \frac {28 \, a^{5} b^{3} x^{6 \, n}}{3 \, n} + \frac {28 \, a^{6} b^{2} x^{5 \, n}}{5 \, n} + \frac {2 \, a^{7} b x^{4 \, n}}{n} + \frac {a^{8} x^{3 \, n}}{3 \, n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+3*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

1/11*b^8*x^(11*n)/n + 4/5*a*b^7*x^(10*n)/n + 28/9*a^2*b^6*x^(9*n)/n + 7*a^3*b^5*x^(8*n)/n + 10*a^4*b^4*x^(7*n)
/n + 28/3*a^5*b^3*x^(6*n)/n + 28/5*a^6*b^2*x^(5*n)/n + 2*a^7*b*x^(4*n)/n + 1/3*a^8*x^(3*n)/n

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (56) = 112\).
time = 0.38, size = 113, normalized size = 1.82 \begin {gather*} \frac {45 \, b^{8} x^{11 \, n} + 396 \, a b^{7} x^{10 \, n} + 1540 \, a^{2} b^{6} x^{9 \, n} + 3465 \, a^{3} b^{5} x^{8 \, n} + 4950 \, a^{4} b^{4} x^{7 \, n} + 4620 \, a^{5} b^{3} x^{6 \, n} + 2772 \, a^{6} b^{2} x^{5 \, n} + 990 \, a^{7} b x^{4 \, n} + 165 \, a^{8} x^{3 \, n}}{495 \, n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+3*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

1/495*(45*b^8*x^(11*n) + 396*a*b^7*x^(10*n) + 1540*a^2*b^6*x^(9*n) + 3465*a^3*b^5*x^(8*n) + 4950*a^4*b^4*x^(7*
n) + 4620*a^5*b^3*x^(6*n) + 2772*a^6*b^2*x^(5*n) + 990*a^7*b*x^(4*n) + 165*a^8*x^(3*n))/n

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (49) = 98\).
time = 30.89, size = 141, normalized size = 2.27 \begin {gather*} \begin {cases} \frac {a^{8} x^{3 n}}{3 n} + \frac {2 a^{7} b x^{4 n}}{n} + \frac {28 a^{6} b^{2} x^{5 n}}{5 n} + \frac {28 a^{5} b^{3} x^{6 n}}{3 n} + \frac {10 a^{4} b^{4} x^{7 n}}{n} + \frac {7 a^{3} b^{5} x^{8 n}}{n} + \frac {28 a^{2} b^{6} x^{9 n}}{9 n} + \frac {4 a b^{7} x^{10 n}}{5 n} + \frac {b^{8} x^{11 n}}{11 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+3*n)*(a+b*x**n)**8,x)

[Out]

Piecewise((a**8*x**(3*n)/(3*n) + 2*a**7*b*x**(4*n)/n + 28*a**6*b**2*x**(5*n)/(5*n) + 28*a**5*b**3*x**(6*n)/(3*
n) + 10*a**4*b**4*x**(7*n)/n + 7*a**3*b**5*x**(8*n)/n + 28*a**2*b**6*x**(9*n)/(9*n) + 4*a*b**7*x**(10*n)/(5*n)
+ b**8*x**(11*n)/(11*n), Ne(n, 0)), ((a + b)**8*log(x), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+3*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^8*x^(3*n - 1), x)

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Mupad [B]
time = 1.45, size = 135, normalized size = 2.18 \begin {gather*} \frac {a^8\,x^{3\,n}}{3\,n}+\frac {b^8\,x^{11\,n}}{11\,n}+\frac {28\,a^6\,b^2\,x^{5\,n}}{5\,n}+\frac {28\,a^5\,b^3\,x^{6\,n}}{3\,n}+\frac {10\,a^4\,b^4\,x^{7\,n}}{n}+\frac {7\,a^3\,b^5\,x^{8\,n}}{n}+\frac {28\,a^2\,b^6\,x^{9\,n}}{9\,n}+\frac {2\,a^7\,b\,x^{4\,n}}{n}+\frac {4\,a\,b^7\,x^{10\,n}}{5\,n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3*n - 1)*(a + b*x^n)^8,x)

[Out]

(a^8*x^(3*n))/(3*n) + (b^8*x^(11*n))/(11*n) + (28*a^6*b^2*x^(5*n))/(5*n) + (28*a^5*b^3*x^(6*n))/(3*n) + (10*a^
4*b^4*x^(7*n))/n + (7*a^3*b^5*x^(8*n))/n + (28*a^2*b^6*x^(9*n))/(9*n) + (2*a^7*b*x^(4*n))/n + (4*a*b^7*x^(10*n
))/(5*n)

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