3.2572 \(\int x^{-1+3 n} (a+b x^n)^8 \, dx\)

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

[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 = 0.04, 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 = {266, 43} \[ \frac {a^2 \left (a+b x^n\right )^9}{9 b^3 n}+\frac {\left (a+b x^n\right )^{11}}{11 b^3 n}-\frac {a \left (a+b x^n\right )^{10}}{5 b^3 n} \]

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 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^{-1+3 n} \left (a+b x^n\right )^8 \, dx &=\frac {\operatorname {Subst}\left (\int x^2 (a+b x)^8 \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {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 = 40, normalized size = 0.65 \[ \frac {\left (a+b x^n\right )^9 \left (a^2-9 a b x^n+45 b^2 x^{2 n}\right )}{495 b^3 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*x^n)^9*(a^2 - 9*a*b*x^n + 45*b^2*x^(2*n)))/(495*b^3*n)

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fricas [B]  time = 0.84, size = 113, normalized size = 1.82 \[ \frac {45 \, b^{8} x^{11 \, n} + 396 \, a b^{7} x^{10 \, n} + 1540 \, a^{2} b^{6} x^{9 \, n} + 3465 \, a^{3} b^{5} x^{8 \, n} + 4950 \, a^{4} b^{4} x^{7 \, n} + 4620 \, a^{5} b^{3} x^{6 \, n} + 2772 \, a^{6} b^{2} x^{5 \, n} + 990 \, a^{7} b x^{4 \, n} + 165 \, a^{8} x^{3 \, n}}{495 \, n} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{8} x^{3 \, n - 1}\,{d x} \]

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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maple [B]  time = 0.03, size = 136, normalized size = 2.19 \[ \frac {a^{8} x^{3 n}}{3 n}+\frac {2 a^{7} b \,x^{4 n}}{n}+\frac {28 a^{6} b^{2} x^{5 n}}{5 n}+\frac {28 a^{5} b^{3} x^{6 n}}{3 n}+\frac {10 a^{4} b^{4} x^{7 n}}{n}+\frac {7 a^{3} b^{5} x^{8 n}}{n}+\frac {28 a^{2} b^{6} x^{9 n}}{9 n}+\frac {4 a \,b^{7} x^{10 n}}{5 n}+\frac {b^{8} x^{11 n}}{11 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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]  time = 0.46, size = 135, normalized size = 2.18 \[ \frac {b^{8} x^{11 \, n}}{11 \, n} + \frac {4 \, a b^{7} x^{10 \, n}}{5 \, n} + \frac {28 \, a^{2} b^{6} x^{9 \, n}}{9 \, n} + \frac {7 \, a^{3} b^{5} x^{8 \, n}}{n} + \frac {10 \, a^{4} b^{4} x^{7 \, n}}{n} + \frac {28 \, a^{5} b^{3} x^{6 \, n}}{3 \, n} + \frac {28 \, a^{6} b^{2} x^{5 \, n}}{5 \, n} + \frac {2 \, a^{7} b x^{4 \, n}}{n} + \frac {a^{8} x^{3 \, n}}{3 \, n} \]

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

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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