3.2580 \(\int x^{-1-5 n} (a+b x^n)^8 \, dx\)

Optimal. Leaf size=133 \[ -\frac {a^8 x^{-5 n}}{5 n}-\frac {2 a^7 b x^{-4 n}}{n}-\frac {28 a^6 b^2 x^{-3 n}}{3 n}-\frac {28 a^5 b^3 x^{-2 n}}{n}-\frac {70 a^4 b^4 x^{-n}}{n}+56 a^3 b^5 \log (x)+\frac {28 a^2 b^6 x^n}{n}+\frac {4 a b^7 x^{2 n}}{n}+\frac {b^8 x^{3 n}}{3 n} \]

[Out]

-1/5*a^8/n/(x^(5*n))-2*a^7*b/n/(x^(4*n))-28/3*a^6*b^2/n/(x^(3*n))-28*a^5*b^3/n/(x^(2*n))-70*a^4*b^4/n/(x^n)+28
*a^2*b^6*x^n/n+4*a*b^7*x^(2*n)/n+1/3*b^8*x^(3*n)/n+56*a^3*b^5*ln(x)

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Rubi [A]  time = 0.06, antiderivative size = 133, 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} \[ -\frac {28 a^6 b^2 x^{-3 n}}{3 n}-\frac {28 a^5 b^3 x^{-2 n}}{n}-\frac {70 a^4 b^4 x^{-n}}{n}+\frac {28 a^2 b^6 x^n}{n}+56 a^3 b^5 \log (x)-\frac {2 a^7 b x^{-4 n}}{n}-\frac {a^8 x^{-5 n}}{5 n}+\frac {4 a b^7 x^{2 n}}{n}+\frac {b^8 x^{3 n}}{3 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 5*n)*(a + b*x^n)^8,x]

[Out]

-a^8/(5*n*x^(5*n)) - (2*a^7*b)/(n*x^(4*n)) - (28*a^6*b^2)/(3*n*x^(3*n)) - (28*a^5*b^3)/(n*x^(2*n)) - (70*a^4*b
^4)/(n*x^n) + (28*a^2*b^6*x^n)/n + (4*a*b^7*x^(2*n))/n + (b^8*x^(3*n))/(3*n) + 56*a^3*b^5*Log[x]

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^{-1-5 n} \left (a+b x^n\right )^8 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^6} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (28 a^2 b^6+\frac {a^8}{x^6}+\frac {8 a^7 b}{x^5}+\frac {28 a^6 b^2}{x^4}+\frac {56 a^5 b^3}{x^3}+\frac {70 a^4 b^4}{x^2}+\frac {56 a^3 b^5}{x}+8 a b^7 x+b^8 x^2\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a^8 x^{-5 n}}{5 n}-\frac {2 a^7 b x^{-4 n}}{n}-\frac {28 a^6 b^2 x^{-3 n}}{3 n}-\frac {28 a^5 b^3 x^{-2 n}}{n}-\frac {70 a^4 b^4 x^{-n}}{n}+\frac {28 a^2 b^6 x^n}{n}+\frac {4 a b^7 x^{2 n}}{n}+\frac {b^8 x^{3 n}}{3 n}+56 a^3 b^5 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 114, normalized size = 0.86 \[ \frac {-\frac {1}{5} a^8 x^{-5 n}-2 a^7 b x^{-4 n}-\frac {28}{3} a^6 b^2 x^{-3 n}-28 a^5 b^3 x^{-2 n}-70 a^4 b^4 x^{-n}+56 a^3 b^5 n \log (x)+28 a^2 b^6 x^n+4 a b^7 x^{2 n}+\frac {1}{3} b^8 x^{3 n}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 5*n)*(a + b*x^n)^8,x]

[Out]

(-1/5*a^8/x^(5*n) - (2*a^7*b)/x^(4*n) - (28*a^6*b^2)/(3*x^(3*n)) - (28*a^5*b^3)/x^(2*n) - (70*a^4*b^4)/x^n + 2
8*a^2*b^6*x^n + 4*a*b^7*x^(2*n) + (b^8*x^(3*n))/3 + 56*a^3*b^5*n*Log[x])/n

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fricas [A]  time = 0.77, size = 116, normalized size = 0.87 \[ \frac {840 \, a^{3} b^{5} n x^{5 \, n} \log \relax (x) + 5 \, b^{8} x^{8 \, n} + 60 \, a b^{7} x^{7 \, n} + 420 \, a^{2} b^{6} x^{6 \, n} - 1050 \, a^{4} b^{4} x^{4 \, n} - 420 \, a^{5} b^{3} x^{3 \, n} - 140 \, a^{6} b^{2} x^{2 \, n} - 30 \, a^{7} b x^{n} - 3 \, a^{8}}{15 \, n x^{5 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-5*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

1/15*(840*a^3*b^5*n*x^(5*n)*log(x) + 5*b^8*x^(8*n) + 60*a*b^7*x^(7*n) + 420*a^2*b^6*x^(6*n) - 1050*a^4*b^4*x^(
4*n) - 420*a^5*b^3*x^(3*n) - 140*a^6*b^2*x^(2*n) - 30*a^7*b*x^n - 3*a^8)/(n*x^(5*n))

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giac [A]  time = 0.31, size = 116, normalized size = 0.87 \[ \frac {840 \, a^{3} b^{5} n x^{5 \, n} \log \relax (x) + 5 \, b^{8} x^{8 \, n} + 60 \, a b^{7} x^{7 \, n} + 420 \, a^{2} b^{6} x^{6 \, n} - 1050 \, a^{4} b^{4} x^{4 \, n} - 420 \, a^{5} b^{3} x^{3 \, n} - 140 \, a^{6} b^{2} x^{2 \, n} - 30 \, a^{7} b x^{n} - 3 \, a^{8}}{15 \, n x^{5 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-5*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

1/15*(840*a^3*b^5*n*x^(5*n)*log(x) + 5*b^8*x^(8*n) + 60*a*b^7*x^(7*n) + 420*a^2*b^6*x^(6*n) - 1050*a^4*b^4*x^(
4*n) - 420*a^5*b^3*x^(3*n) - 140*a^6*b^2*x^(2*n) - 30*a^7*b*x^n - 3*a^8)/(n*x^(5*n))

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maple [A]  time = 0.03, size = 128, normalized size = 0.96 \[ 56 a^{3} b^{5} \ln \relax (x )-\frac {a^{8} x^{-5 n}}{5 n}-\frac {2 a^{7} b \,x^{-4 n}}{n}-\frac {28 a^{6} b^{2} x^{-3 n}}{3 n}-\frac {28 a^{5} b^{3} x^{-2 n}}{n}-\frac {70 a^{4} b^{4} x^{-n}}{n}+\frac {28 a^{2} b^{6} x^{n}}{n}+\frac {4 a \,b^{7} x^{2 n}}{n}+\frac {b^{8} x^{3 n}}{3 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-5*n)*(b*x^n+a)^8,x)

[Out]

56*a^3*b^5*ln(x)+1/3*b^8/n*(x^n)^3+4*a*b^7/n*(x^n)^2+28*a^2*b^6*x^n/n-70*a^4*b^4/n/(x^n)-28*a^5*b^3/n/(x^n)^2-
28/3*a^6*b^2/n/(x^n)^3-2*a^7*b/n/(x^n)^4-1/5*a^8/n/(x^n)^5

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maxima [A]  time = 0.65, size = 135, normalized size = 1.02 \[ 56 \, a^{3} b^{5} \log \relax (x) + \frac {b^{8} x^{3 \, n}}{3 \, n} + \frac {4 \, a b^{7} x^{2 \, n}}{n} + \frac {28 \, a^{2} b^{6} x^{n}}{n} - \frac {a^{8}}{5 \, n x^{5 \, n}} - \frac {2 \, a^{7} b}{n x^{4 \, n}} - \frac {28 \, a^{6} b^{2}}{3 \, n x^{3 \, n}} - \frac {28 \, a^{5} b^{3}}{n x^{2 \, n}} - \frac {70 \, a^{4} b^{4}}{n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-5*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

56*a^3*b^5*log(x) + 1/3*b^8*x^(3*n)/n + 4*a*b^7*x^(2*n)/n + 28*a^2*b^6*x^n/n - 1/5*a^8/(n*x^(5*n)) - 2*a^7*b/(
n*x^(4*n)) - 28/3*a^6*b^2/(n*x^(3*n)) - 28*a^5*b^3/(n*x^(2*n)) - 70*a^4*b^4/(n*x^n)

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mupad [B]  time = 1.55, size = 135, normalized size = 1.02 \[ \frac {b^8\,x^{3\,n}}{3\,n}-\frac {a^8}{5\,n\,x^{5\,n}}+56\,a^3\,b^5\,\ln \relax (x)-\frac {70\,a^4\,b^4}{n\,x^n}-\frac {28\,a^5\,b^3}{n\,x^{2\,n}}-\frac {28\,a^6\,b^2}{3\,n\,x^{3\,n}}+\frac {4\,a\,b^7\,x^{2\,n}}{n}-\frac {2\,a^7\,b}{n\,x^{4\,n}}+\frac {28\,a^2\,b^6\,x^n}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^8/x^(5*n + 1),x)

[Out]

(b^8*x^(3*n))/(3*n) - a^8/(5*n*x^(5*n)) + 56*a^3*b^5*log(x) - (70*a^4*b^4)/(n*x^n) - (28*a^5*b^3)/(n*x^(2*n))
- (28*a^6*b^2)/(3*n*x^(3*n)) + (4*a*b^7*x^(2*n))/n - (2*a^7*b)/(n*x^(4*n)) + (28*a^2*b^6*x^n)/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**(-1-5*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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