\(\int x^{-1-8 n} (a+b x^n)^8 \, dx\) [2583]

Optimal result

Integrand size = 17, antiderivative size = 140 \[ \int x^{-1-8 n} \left (a+b x^n\right )^8 \, dx=-\frac {a^8 x^{-8 n}}{8 n}-\frac {8 a^7 b x^{-7 n}}{7 n}-\frac {14 a^6 b^2 x^{-6 n}}{3 n}-\frac {56 a^5 b^3 x^{-5 n}}{5 n}-\frac {35 a^4 b^4 x^{-4 n}}{2 n}-\frac {56 a^3 b^5 x^{-3 n}}{3 n}-\frac {14 a^2 b^6 x^{-2 n}}{n}-\frac {8 a b^7 x^{-n}}{n}+b^8 \log (x) \]

[Out]

-1/8*a^8/n/(x^(8*n))-8/7*a^7*b/n/(x^(7*n))-14/3*a^6*b^2/n/(x^(6*n))-56/5*a^5*b^3/n/(x^(5*n))-35/2*a^4*b^4/n/(x
^(4*n))-56/3*a^3*b^5/n/(x^(3*n))-14*a^2*b^6/n/(x^(2*n))-8*a*b^7/n/(x^n)+b^8*ln(x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 140, 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} \[ \int x^{-1-8 n} \left (a+b x^n\right )^8 \, dx=-\frac {a^8 x^{-8 n}}{8 n}-\frac {8 a^7 b x^{-7 n}}{7 n}-\frac {14 a^6 b^2 x^{-6 n}}{3 n}-\frac {56 a^5 b^3 x^{-5 n}}{5 n}-\frac {35 a^4 b^4 x^{-4 n}}{2 n}-\frac {56 a^3 b^5 x^{-3 n}}{3 n}-\frac {14 a^2 b^6 x^{-2 n}}{n}-\frac {8 a b^7 x^{-n}}{n}+b^8 \log (x) \]

[In]

Int[x^(-1 - 8*n)*(a + b*x^n)^8,x]

[Out]

-1/8*a^8/(n*x^(8*n)) - (8*a^7*b)/(7*n*x^(7*n)) - (14*a^6*b^2)/(3*n*x^(6*n)) - (56*a^5*b^3)/(5*n*x^(5*n)) - (35
*a^4*b^4)/(2*n*x^(4*n)) - (56*a^3*b^5)/(3*n*x^(3*n)) - (14*a^2*b^6)/(n*x^(2*n)) - (8*a*b^7)/(n*x^n) + b^8*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^9} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^8}{x^9}+\frac {8 a^7 b}{x^8}+\frac {28 a^6 b^2}{x^7}+\frac {56 a^5 b^3}{x^6}+\frac {70 a^4 b^4}{x^5}+\frac {56 a^3 b^5}{x^4}+\frac {28 a^2 b^6}{x^3}+\frac {8 a b^7}{x^2}+\frac {b^8}{x}\right ) \, dx,x,x^n\right )}{n} \\ & = -\frac {a^8 x^{-8 n}}{8 n}-\frac {8 a^7 b x^{-7 n}}{7 n}-\frac {14 a^6 b^2 x^{-6 n}}{3 n}-\frac {56 a^5 b^3 x^{-5 n}}{5 n}-\frac {35 a^4 b^4 x^{-4 n}}{2 n}-\frac {56 a^3 b^5 x^{-3 n}}{3 n}-\frac {14 a^2 b^6 x^{-2 n}}{n}-\frac {8 a b^7 x^{-n}}{n}+b^8 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.81 \[ \int x^{-1-8 n} \left (a+b x^n\right )^8 \, dx=-\frac {a x^{-8 n} \left (105 a^7+960 a^6 b x^n+3920 a^5 b^2 x^{2 n}+9408 a^4 b^3 x^{3 n}+14700 a^3 b^4 x^{4 n}+15680 a^2 b^5 x^{5 n}+11760 a b^6 x^{6 n}+6720 b^7 x^{7 n}\right )}{840 n}+\frac {b^8 \log \left (x^n\right )}{n} \]

[In]

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

[Out]

-1/840*(a*(105*a^7 + 960*a^6*b*x^n + 3920*a^5*b^2*x^(2*n) + 9408*a^4*b^3*x^(3*n) + 14700*a^3*b^4*x^(4*n) + 156
80*a^2*b^5*x^(5*n) + 11760*a*b^6*x^(6*n) + 6720*b^7*x^(7*n)))/(n*x^(8*n)) + (b^8*Log[x^n])/n

Maple [A] (verified)

Time = 4.37 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.92

method	result	size
risch	\(b^{8} \ln \left (x \right )-\frac {8 a \,b^{7} x^{-n}}{n}-\frac {14 a^{2} b^{6} x^{-2 n}}{n}-\frac {56 a^{3} b^{5} x^{-3 n}}{3 n}-\frac {35 a^{4} b^{4} x^{-4 n}}{2 n}-\frac {56 a^{5} b^{3} x^{-5 n}}{5 n}-\frac {14 a^{6} b^{2} x^{-6 n}}{3 n}-\frac {8 a^{7} b \,x^{-7 n}}{7 n}-\frac {a^{8} x^{-8 n}}{8 n}\)	\(129\)

[In]

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

[Out]

b^8*ln(x)-8*a*b^7/n/(x^n)-14*a^2*b^6/n/(x^n)^2-56/3*a^3*b^5/n/(x^n)^3-35/2*a^4*b^4/n/(x^n)^4-56/5*a^5*b^3/n/(x
^n)^5-14/3*a^6*b^2/n/(x^n)^6-8/7*a^7*b/n/(x^n)^7-1/8*a^8/n/(x^n)^8

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.83 \[ \int x^{-1-8 n} \left (a+b x^n\right )^8 \, dx=\frac {840 \, b^{8} n x^{8 \, n} \log \left (x\right ) - 6720 \, a b^{7} x^{7 \, n} - 11760 \, a^{2} b^{6} x^{6 \, n} - 15680 \, a^{3} b^{5} x^{5 \, n} - 14700 \, a^{4} b^{4} x^{4 \, n} - 9408 \, a^{5} b^{3} x^{3 \, n} - 3920 \, a^{6} b^{2} x^{2 \, n} - 960 \, a^{7} b x^{n} - 105 \, a^{8}}{840 \, n x^{8 \, n}} \]

[In]

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

[Out]

1/840*(840*b^8*n*x^(8*n)*log(x) - 6720*a*b^7*x^(7*n) - 11760*a^2*b^6*x^(6*n) - 15680*a^3*b^5*x^(5*n) - 14700*a
^4*b^4*x^(4*n) - 9408*a^5*b^3*x^(3*n) - 3920*a^6*b^2*x^(2*n) - 960*a^7*b*x^n - 105*a^8)/(n*x^(8*n))

Sympy [A] (verification not implemented)

Time = 29.88 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.97 \[ \int x^{-1-8 n} \left (a+b x^n\right )^8 \, dx=\begin {cases} - \frac {a^{8} x^{- 8 n}}{8 n} - \frac {8 a^{7} b x^{- 7 n}}{7 n} - \frac {14 a^{6} b^{2} x^{- 6 n}}{3 n} - \frac {56 a^{5} b^{3} x^{- 5 n}}{5 n} - \frac {35 a^{4} b^{4} x^{- 4 n}}{2 n} - \frac {56 a^{3} b^{5} x^{- 3 n}}{3 n} - \frac {14 a^{2} b^{6} x^{- 2 n}}{n} - \frac {8 a b^{7} x^{- n}}{n} + b^{8} \log {\left (x \right )} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-a**8/(8*n*x**(8*n)) - 8*a**7*b/(7*n*x**(7*n)) - 14*a**6*b**2/(3*n*x**(6*n)) - 56*a**5*b**3/(5*n*x*
*(5*n)) - 35*a**4*b**4/(2*n*x**(4*n)) - 56*a**3*b**5/(3*n*x**(3*n)) - 14*a**2*b**6/(n*x**(2*n)) - 8*a*b**7/(n*
x**n) + b**8*log(x), Ne(n, 0)), ((a + b)**8*log(x), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.01 \[ \int x^{-1-8 n} \left (a+b x^n\right )^8 \, dx=b^{8} \log \left (x\right ) - \frac {a^{8}}{8 \, n x^{8 \, n}} - \frac {8 \, a^{7} b}{7 \, n x^{7 \, n}} - \frac {14 \, a^{6} b^{2}}{3 \, n x^{6 \, n}} - \frac {56 \, a^{5} b^{3}}{5 \, n x^{5 \, n}} - \frac {35 \, a^{4} b^{4}}{2 \, n x^{4 \, n}} - \frac {56 \, a^{3} b^{5}}{3 \, n x^{3 \, n}} - \frac {14 \, a^{2} b^{6}}{n x^{2 \, n}} - \frac {8 \, a b^{7}}{n x^{n}} \]

[In]

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

[Out]

b^8*log(x) - 1/8*a^8/(n*x^(8*n)) - 8/7*a^7*b/(n*x^(7*n)) - 14/3*a^6*b^2/(n*x^(6*n)) - 56/5*a^5*b^3/(n*x^(5*n))
- 35/2*a^4*b^4/(n*x^(4*n)) - 56/3*a^3*b^5/(n*x^(3*n)) - 14*a^2*b^6/(n*x^(2*n)) - 8*a*b^7/(n*x^n)

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.83 \[ \int x^{-1-8 n} \left (a+b x^n\right )^8 \, dx=\frac {840 \, b^{8} n x^{8 \, n} \log \left (x\right ) - 6720 \, a b^{7} x^{7 \, n} - 11760 \, a^{2} b^{6} x^{6 \, n} - 15680 \, a^{3} b^{5} x^{5 \, n} - 14700 \, a^{4} b^{4} x^{4 \, n} - 9408 \, a^{5} b^{3} x^{3 \, n} - 3920 \, a^{6} b^{2} x^{2 \, n} - 960 \, a^{7} b x^{n} - 105 \, a^{8}}{840 \, n x^{8 \, n}} \]

[In]

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

[Out]

1/840*(840*b^8*n*x^(8*n)*log(x) - 6720*a*b^7*x^(7*n) - 11760*a^2*b^6*x^(6*n) - 15680*a^3*b^5*x^(5*n) - 14700*a
^4*b^4*x^(4*n) - 9408*a^5*b^3*x^(3*n) - 3920*a^6*b^2*x^(2*n) - 960*a^7*b*x^n - 105*a^8)/(n*x^(8*n))

Mupad [B] (verification not implemented)

Time = 5.94 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.01 \[ \int x^{-1-8 n} \left (a+b x^n\right )^8 \, dx=b^8\,\ln \left (x\right )-\frac {a^8}{8\,n\,x^{8\,n}}-\frac {14\,a^2\,b^6}{n\,x^{2\,n}}-\frac {56\,a^3\,b^5}{3\,n\,x^{3\,n}}-\frac {35\,a^4\,b^4}{2\,n\,x^{4\,n}}-\frac {56\,a^5\,b^3}{5\,n\,x^{5\,n}}-\frac {14\,a^6\,b^2}{3\,n\,x^{6\,n}}-\frac {8\,a\,b^7}{n\,x^n}-\frac {8\,a^7\,b}{7\,n\,x^{7\,n}} \]

[In]

int((a + b*x^n)^8/x^(8*n + 1),x)

[Out]

b^8*log(x) - a^8/(8*n*x^(8*n)) - (14*a^2*b^6)/(n*x^(2*n)) - (56*a^3*b^5)/(3*n*x^(3*n)) - (35*a^4*b^4)/(2*n*x^(
4*n)) - (56*a^5*b^3)/(5*n*x^(5*n)) - (14*a^6*b^2)/(3*n*x^(6*n)) - (8*a*b^7)/(n*x^n) - (8*a^7*b)/(7*n*x^(7*n))