3.2578 \(\int x^{-1-3 n} (a+b x^n)^8 \, dx\)

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

[Out]

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

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 3*n)*(a + b*x^n)^8,x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 114, normalized size = 0.86 \[ \frac {-\frac {1}{3} a^8 x^{-3 n}-4 a^7 b x^{-2 n}-28 a^6 b^2 x^{-n}+56 a^5 b^3 n \log (x)+70 a^4 b^4 x^n+28 a^3 b^5 x^{2 n}+\frac {28}{3} a^2 b^6 x^{3 n}+2 a b^7 x^{4 n}+\frac {1}{5} b^8 x^{5 n}}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.49, size = 116, normalized size = 0.87 \[ \frac {840 \, a^{5} b^{3} n x^{3 \, n} \log \relax (x) + 3 \, b^{8} x^{8 \, n} + 30 \, a b^{7} x^{7 \, n} + 140 \, a^{2} b^{6} x^{6 \, n} + 420 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} - 420 \, a^{6} b^{2} x^{2 \, n} - 60 \, a^{7} b x^{n} - 5 \, a^{8}}{15 \, n x^{3 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.29, size = 116, normalized size = 0.87 \[ \frac {840 \, a^{5} b^{3} n x^{3 \, n} \log \relax (x) + 3 \, b^{8} x^{8 \, n} + 30 \, a b^{7} x^{7 \, n} + 140 \, a^{2} b^{6} x^{6 \, n} + 420 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} - 420 \, a^{6} b^{2} x^{2 \, n} - 60 \, a^{7} b x^{n} - 5 \, a^{8}}{15 \, n x^{3 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.61, size = 131, normalized size = 0.98 \[ 56 \, a^{5} b^{3} \log \relax (x) + \frac {b^{8} x^{5 \, n}}{5 \, n} + \frac {2 \, a b^{7} x^{4 \, n}}{n} + \frac {28 \, a^{2} b^{6} x^{3 \, n}}{3 \, n} + \frac {28 \, a^{3} b^{5} x^{2 \, n}}{n} + \frac {70 \, a^{4} b^{4} x^{n}}{n} - \frac {a^{8}}{3 \, n x^{3 \, n}} - \frac {4 \, a^{7} b}{n x^{2 \, n}} - \frac {28 \, a^{6} b^{2}}{n x^{n}} \]

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]

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

________________________________________________________________________________________

mupad [B]  time = 1.60, size = 131, normalized size = 0.98 \[ \frac {b^8\,x^{5\,n}}{5\,n}-\frac {a^8}{3\,n\,x^{3\,n}}+56\,a^5\,b^3\,\ln \relax (x)-\frac {28\,a^6\,b^2}{n\,x^n}+\frac {28\,a^3\,b^5\,x^{2\,n}}{n}+\frac {28\,a^2\,b^6\,x^{3\,n}}{3\,n}-\frac {4\,a^7\,b}{n\,x^{2\,n}}+\frac {2\,a\,b^7\,x^{4\,n}}{n}+\frac {70\,a^4\,b^4\,x^n}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^8/x^(3*n + 1),x)

[Out]

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

________________________________________________________________________________________

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

________________________________________________________________________________________