\(\int x^{-1-4 n} (a+b x^n)^8 \, dx\) [2579]

Optimal result

Integrand size = 17, antiderivative size = 135 \[ \int x^{-1-4 n} \left (a+b x^n\right )^8 \, dx=-\frac {a^8 x^{-4 n}}{4 n}-\frac {8 a^7 b x^{-3 n}}{3 n}-\frac {14 a^6 b^2 x^{-2 n}}{n}-\frac {56 a^5 b^3 x^{-n}}{n}+\frac {56 a^3 b^5 x^n}{n}+\frac {14 a^2 b^6 x^{2 n}}{n}+\frac {8 a b^7 x^{3 n}}{3 n}+\frac {b^8 x^{4 n}}{4 n}+70 a^4 b^4 \log (x) \]

[Out]

-1/4*a^8/n/(x^(4*n))-8/3*a^7*b/n/(x^(3*n))-14*a^6*b^2/n/(x^(2*n))-56*a^5*b^3/n/(x^n)+56*a^3*b^5*x^n/n+14*a^2*b
^6*x^(2*n)/n+8/3*a*b^7*x^(3*n)/n+1/4*b^8*x^(4*n)/n+70*a^4*b^4*ln(x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {272, 45} \[ \int x^{-1-4 n} \left (a+b x^n\right )^8 \, dx=-\frac {a^8 x^{-4 n}}{4 n}-\frac {8 a^7 b x^{-3 n}}{3 n}-\frac {14 a^6 b^2 x^{-2 n}}{n}-\frac {56 a^5 b^3 x^{-n}}{n}+70 a^4 b^4 \log (x)+\frac {56 a^3 b^5 x^n}{n}+\frac {14 a^2 b^6 x^{2 n}}{n}+\frac {8 a b^7 x^{3 n}}{3 n}+\frac {b^8 x^{4 n}}{4 n} \]

[In]

Int[x^(-1 - 4*n)*(a + b*x^n)^8,x]

[Out]

-1/4*a^8/(n*x^(4*n)) - (8*a^7*b)/(3*n*x^(3*n)) - (14*a^6*b^2)/(n*x^(2*n)) - (56*a^5*b^3)/(n*x^n) + (56*a^3*b^5
*x^n)/n + (14*a^2*b^6*x^(2*n))/n + (8*a*b^7*x^(3*n))/(3*n) + (b^8*x^(4*n))/(4*n) + 70*a^4*b^4*Log[x]

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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b x)^8}{x^5} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (56 a^3 b^5+\frac {a^8}{x^5}+\frac {8 a^7 b}{x^4}+\frac {28 a^6 b^2}{x^3}+\frac {56 a^5 b^3}{x^2}+\frac {70 a^4 b^4}{x}+28 a^2 b^6 x+8 a b^7 x^2+b^8 x^3\right ) \, dx,x,x^n\right )}{n} \\ & = -\frac {a^8 x^{-4 n}}{4 n}-\frac {8 a^7 b x^{-3 n}}{3 n}-\frac {14 a^6 b^2 x^{-2 n}}{n}-\frac {56 a^5 b^3 x^{-n}}{n}+\frac {56 a^3 b^5 x^n}{n}+\frac {14 a^2 b^6 x^{2 n}}{n}+\frac {8 a b^7 x^{3 n}}{3 n}+\frac {b^8 x^{4 n}}{4 n}+70 a^4 b^4 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.86 \[ \int x^{-1-4 n} \left (a+b x^n\right )^8 \, dx=\frac {x^{-4 n} \left (-3 a^8-32 a^7 b x^n-168 a^6 b^2 x^{2 n}-672 a^5 b^3 x^{3 n}+672 a^3 b^5 x^{5 n}+168 a^2 b^6 x^{6 n}+32 a b^7 x^{7 n}+3 b^8 x^{8 n}\right )}{12 n}+\frac {70 a^4 b^4 \log \left (x^n\right )}{n} \]

[In]

Integrate[x^(-1 - 4*n)*(a + b*x^n)^8,x]

[Out]

(-3*a^8 - 32*a^7*b*x^n - 168*a^6*b^2*x^(2*n) - 672*a^5*b^3*x^(3*n) + 672*a^3*b^5*x^(5*n) + 168*a^2*b^6*x^(6*n)
+ 32*a*b^7*x^(7*n) + 3*b^8*x^(8*n))/(12*n*x^(4*n)) + (70*a^4*b^4*Log[x^n])/n

Maple [A] (verified)

Time = 4.17 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.95

method	result	size
risch	\(70 a^{4} b^{4} \ln \left (x \right )+\frac {b^{8} x^{4 n}}{4 n}+\frac {8 a \,b^{7} x^{3 n}}{3 n}+\frac {14 a^{2} b^{6} x^{2 n}}{n}+\frac {56 a^{3} b^{5} x^{n}}{n}-\frac {56 a^{5} b^{3} x^{-n}}{n}-\frac {14 a^{6} b^{2} x^{-2 n}}{n}-\frac {8 a^{7} b \,x^{-3 n}}{3 n}-\frac {a^{8} x^{-4 n}}{4 n}\)	\(128\)

[In]

int(x^(-1-4*n)*(a+b*x^n)^8,x,method=_RETURNVERBOSE)

[Out]

70*a^4*b^4*ln(x)+1/4*b^8/n*(x^n)^4+8/3*a*b^7/n*(x^n)^3+14*a^2*b^6/n*(x^n)^2+56*a^3*b^5*x^n/n-56*a^5*b^3/n/(x^n
)-14*a^6*b^2/n/(x^n)^2-8/3*a^7*b/n/(x^n)^3-1/4*a^8/n/(x^n)^4

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.86 \[ \int x^{-1-4 n} \left (a+b x^n\right )^8 \, dx=\frac {840 \, a^{4} b^{4} n x^{4 \, n} \log \left (x\right ) + 3 \, b^{8} x^{8 \, n} + 32 \, a b^{7} x^{7 \, n} + 168 \, a^{2} b^{6} x^{6 \, n} + 672 \, a^{3} b^{5} x^{5 \, n} - 672 \, a^{5} b^{3} x^{3 \, n} - 168 \, a^{6} b^{2} x^{2 \, n} - 32 \, a^{7} b x^{n} - 3 \, a^{8}}{12 \, n x^{4 \, n}} \]

[In]

integrate(x^(-1-4*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

1/12*(840*a^4*b^4*n*x^(4*n)*log(x) + 3*b^8*x^(8*n) + 32*a*b^7*x^(7*n) + 168*a^2*b^6*x^(6*n) + 672*a^3*b^5*x^(5
*n) - 672*a^5*b^3*x^(3*n) - 168*a^6*b^2*x^(2*n) - 32*a^7*b*x^n - 3*a^8)/(n*x^(4*n))

Sympy [A] (verification not implemented)

Time = 9.14 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.97 \[ \int x^{-1-4 n} \left (a+b x^n\right )^8 \, dx=\begin {cases} - \frac {a^{8} x^{- 4 n}}{4 n} - \frac {8 a^{7} b x^{- 3 n}}{3 n} - \frac {14 a^{6} b^{2} x^{- 2 n}}{n} - \frac {56 a^{5} b^{3} x^{- n}}{n} + 70 a^{4} b^{4} \log {\left (x \right )} + \frac {56 a^{3} b^{5} x^{n}}{n} + \frac {14 a^{2} b^{6} x^{2 n}}{n} + \frac {8 a b^{7} x^{3 n}}{3 n} + \frac {b^{8} x^{4 n}}{4 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(x**(-1-4*n)*(a+b*x**n)**8,x)

[Out]

Piecewise((-a**8/(4*n*x**(4*n)) - 8*a**7*b/(3*n*x**(3*n)) - 14*a**6*b**2/(n*x**(2*n)) - 56*a**5*b**3/(n*x**n)
+ 70*a**4*b**4*log(x) + 56*a**3*b**5*x**n/n + 14*a**2*b**6*x**(2*n)/n + 8*a*b**7*x**(3*n)/(3*n) + b**8*x**(4*n
)/(4*n), Ne(n, 0)), ((a + b)**8*log(x), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.99 \[ \int x^{-1-4 n} \left (a+b x^n\right )^8 \, dx=70 \, a^{4} b^{4} \log \left (x\right ) + \frac {b^{8} x^{4 \, n}}{4 \, n} + \frac {8 \, a b^{7} x^{3 \, n}}{3 \, n} + \frac {14 \, a^{2} b^{6} x^{2 \, n}}{n} + \frac {56 \, a^{3} b^{5} x^{n}}{n} - \frac {a^{8}}{4 \, n x^{4 \, n}} - \frac {8 \, a^{7} b}{3 \, n x^{3 \, n}} - \frac {14 \, a^{6} b^{2}}{n x^{2 \, n}} - \frac {56 \, a^{5} b^{3}}{n x^{n}} \]

[In]

integrate(x^(-1-4*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

70*a^4*b^4*log(x) + 1/4*b^8*x^(4*n)/n + 8/3*a*b^7*x^(3*n)/n + 14*a^2*b^6*x^(2*n)/n + 56*a^3*b^5*x^n/n - 1/4*a^
8/(n*x^(4*n)) - 8/3*a^7*b/(n*x^(3*n)) - 14*a^6*b^2/(n*x^(2*n)) - 56*a^5*b^3/(n*x^n)

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.86 \[ \int x^{-1-4 n} \left (a+b x^n\right )^8 \, dx=\frac {840 \, a^{4} b^{4} n x^{4 \, n} \log \left (x\right ) + 3 \, b^{8} x^{8 \, n} + 32 \, a b^{7} x^{7 \, n} + 168 \, a^{2} b^{6} x^{6 \, n} + 672 \, a^{3} b^{5} x^{5 \, n} - 672 \, a^{5} b^{3} x^{3 \, n} - 168 \, a^{6} b^{2} x^{2 \, n} - 32 \, a^{7} b x^{n} - 3 \, a^{8}}{12 \, n x^{4 \, n}} \]

[In]

integrate(x^(-1-4*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

1/12*(840*a^4*b^4*n*x^(4*n)*log(x) + 3*b^8*x^(8*n) + 32*a*b^7*x^(7*n) + 168*a^2*b^6*x^(6*n) + 672*a^3*b^5*x^(5
*n) - 672*a^5*b^3*x^(3*n) - 168*a^6*b^2*x^(2*n) - 32*a^7*b*x^n - 3*a^8)/(n*x^(4*n))

Mupad [B] (verification not implemented)

Time = 6.15 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.99 \[ \int x^{-1-4 n} \left (a+b x^n\right )^8 \, dx=\frac {b^8\,x^{4\,n}}{4\,n}-\frac {a^8}{4\,n\,x^{4\,n}}+70\,a^4\,b^4\,\ln \left (x\right )-\frac {56\,a^5\,b^3}{n\,x^n}+\frac {14\,a^2\,b^6\,x^{2\,n}}{n}-\frac {14\,a^6\,b^2}{n\,x^{2\,n}}+\frac {8\,a\,b^7\,x^{3\,n}}{3\,n}-\frac {8\,a^7\,b}{3\,n\,x^{3\,n}}+\frac {56\,a^3\,b^5\,x^n}{n} \]

[In]

int((a + b*x^n)^8/x^(4*n + 1),x)

[Out]

(b^8*x^(4*n))/(4*n) - a^8/(4*n*x^(4*n)) + 70*a^4*b^4*log(x) - (56*a^5*b^3)/(n*x^n) + (14*a^2*b^6*x^(2*n))/n -
(14*a^6*b^2)/(n*x^(2*n)) + (8*a*b^7*x^(3*n))/(3*n) - (8*a^7*b)/(3*n*x^(3*n)) + (56*a^3*b^5*x^n)/n