\(\int x^{-1-14 n} (a+b x^n)^8 \, dx\) [2589]

Optimal result

Integrand size = 17, antiderivative size = 151 \[ \int x^{-1-14 n} \left (a+b x^n\right )^8 \, dx=-\frac {a^8 x^{-14 n}}{14 n}-\frac {8 a^7 b x^{-13 n}}{13 n}-\frac {7 a^6 b^2 x^{-12 n}}{3 n}-\frac {56 a^5 b^3 x^{-11 n}}{11 n}-\frac {7 a^4 b^4 x^{-10 n}}{n}-\frac {56 a^3 b^5 x^{-9 n}}{9 n}-\frac {7 a^2 b^6 x^{-8 n}}{2 n}-\frac {8 a b^7 x^{-7 n}}{7 n}-\frac {b^8 x^{-6 n}}{6 n} \]

[Out]

-1/14*a^8/n/(x^(14*n))-8/13*a^7*b/n/(x^(13*n))-7/3*a^6*b^2/n/(x^(12*n))-56/11*a^5*b^3/n/(x^(11*n))-7*a^4*b^4/n
/(x^(10*n))-56/9*a^3*b^5/n/(x^(9*n))-7/2*a^2*b^6/n/(x^(8*n))-8/7*a*b^7/n/(x^(7*n))-1/6*b^8/n/(x^(6*n))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 151, 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} \[ \int x^{-1-14 n} \left (a+b x^n\right )^8 \, dx=-\frac {a^8 x^{-14 n}}{14 n}-\frac {8 a^7 b x^{-13 n}}{13 n}-\frac {7 a^6 b^2 x^{-12 n}}{3 n}-\frac {56 a^5 b^3 x^{-11 n}}{11 n}-\frac {7 a^4 b^4 x^{-10 n}}{n}-\frac {56 a^3 b^5 x^{-9 n}}{9 n}-\frac {7 a^2 b^6 x^{-8 n}}{2 n}-\frac {8 a b^7 x^{-7 n}}{7 n}-\frac {b^8 x^{-6 n}}{6 n} \]

[In]

Int[x^(-1 - 14*n)*(a + b*x^n)^8,x]

[Out]

-1/14*a^8/(n*x^(14*n)) - (8*a^7*b)/(13*n*x^(13*n)) - (7*a^6*b^2)/(3*n*x^(12*n)) - (56*a^5*b^3)/(11*n*x^(11*n))
- (7*a^4*b^4)/(n*x^(10*n)) - (56*a^3*b^5)/(9*n*x^(9*n)) - (7*a^2*b^6)/(2*n*x^(8*n)) - (8*a*b^7)/(7*n*x^(7*n))
- b^8/(6*n*x^(6*n))

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^{15}} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {Subst}\left (\int \left (\frac {a^8}{x^{15}}+\frac {8 a^7 b}{x^{14}}+\frac {28 a^6 b^2}{x^{13}}+\frac {56 a^5 b^3}{x^{12}}+\frac {70 a^4 b^4}{x^{11}}+\frac {56 a^3 b^5}{x^{10}}+\frac {28 a^2 b^6}{x^9}+\frac {8 a b^7}{x^8}+\frac {b^8}{x^7}\right ) \, dx,x,x^n\right )}{n} \\ & = -\frac {a^8 x^{-14 n}}{14 n}-\frac {8 a^7 b x^{-13 n}}{13 n}-\frac {7 a^6 b^2 x^{-12 n}}{3 n}-\frac {56 a^5 b^3 x^{-11 n}}{11 n}-\frac {7 a^4 b^4 x^{-10 n}}{n}-\frac {56 a^3 b^5 x^{-9 n}}{9 n}-\frac {7 a^2 b^6 x^{-8 n}}{2 n}-\frac {8 a b^7 x^{-7 n}}{7 n}-\frac {b^8 x^{-6 n}}{6 n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75 \[ \int x^{-1-14 n} \left (a+b x^n\right )^8 \, dx=\frac {x^{-14 n} \left (-1287 a^8-11088 a^7 b x^n-42042 a^6 b^2 x^{2 n}-91728 a^5 b^3 x^{3 n}-126126 a^4 b^4 x^{4 n}-112112 a^3 b^5 x^{5 n}-63063 a^2 b^6 x^{6 n}-20592 a b^7 x^{7 n}-3003 b^8 x^{8 n}\right )}{18018 n} \]

[In]

Integrate[x^(-1 - 14*n)*(a + b*x^n)^8,x]

[Out]

(-1287*a^8 - 11088*a^7*b*x^n - 42042*a^6*b^2*x^(2*n) - 91728*a^5*b^3*x^(3*n) - 126126*a^4*b^4*x^(4*n) - 112112
*a^3*b^5*x^(5*n) - 63063*a^2*b^6*x^(6*n) - 20592*a*b^7*x^(7*n) - 3003*b^8*x^(8*n))/(18018*n*x^(14*n))

Maple [A] (verified)

Time = 8.52 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.90

method	result	size
risch	\(-\frac {b^{8} x^{-6 n}}{6 n}-\frac {8 a \,b^{7} x^{-7 n}}{7 n}-\frac {7 a^{2} b^{6} x^{-8 n}}{2 n}-\frac {56 a^{3} b^{5} x^{-9 n}}{9 n}-\frac {7 a^{4} b^{4} x^{-10 n}}{n}-\frac {56 a^{5} b^{3} x^{-11 n}}{11 n}-\frac {7 a^{6} b^{2} x^{-12 n}}{3 n}-\frac {8 a^{7} b \,x^{-13 n}}{13 n}-\frac {a^{8} x^{-14 n}}{14 n}\)	\(136\)
parallelrisch	\(\frac {-3003 b^{8} x^{-1-14 n} x^{8 n} x -20592 a \,b^{7} x^{-1-14 n} x^{7 n} x -63063 a^{2} b^{6} x^{-1-14 n} x^{6 n} x -112112 a^{3} b^{5} x^{-1-14 n} x^{5 n} x -126126 a^{4} b^{4} x^{-1-14 n} x^{4 n} x -91728 a^{5} b^{3} x^{-1-14 n} x^{3 n} x -42042 a^{6} b^{2} x^{-1-14 n} x^{2 n} x -11088 a^{7} b \,x^{-1-14 n} x^{n} x -1287 a^{8} x^{-1-14 n} x}{18018 n}\)	\(179\)

[In]

int(x^(-1-14*n)*(a+b*x^n)^8,x,method=_RETURNVERBOSE)

[Out]

-1/6*b^8/n/(x^n)^6-8/7*a*b^7/n/(x^n)^7-7/2*a^2*b^6/n/(x^n)^8-56/9*a^3*b^5/n/(x^n)^9-7*a^4*b^4/n/(x^n)^10-56/11
*a^5*b^3/n/(x^n)^11-7/3*a^6*b^2/n/(x^n)^12-8/13*a^7*b/n/(x^n)^13-1/14*a^8/n/(x^n)^14

Fricas [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75 \[ \int x^{-1-14 n} \left (a+b x^n\right )^8 \, dx=-\frac {3003 \, b^{8} x^{8 \, n} + 20592 \, a b^{7} x^{7 \, n} + 63063 \, a^{2} b^{6} x^{6 \, n} + 112112 \, a^{3} b^{5} x^{5 \, n} + 126126 \, a^{4} b^{4} x^{4 \, n} + 91728 \, a^{5} b^{3} x^{3 \, n} + 42042 \, a^{6} b^{2} x^{2 \, n} + 11088 \, a^{7} b x^{n} + 1287 \, a^{8}}{18018 \, n x^{14 \, n}} \]

[In]

integrate(x^(-1-14*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

-1/18018*(3003*b^8*x^(8*n) + 20592*a*b^7*x^(7*n) + 63063*a^2*b^6*x^(6*n) + 112112*a^3*b^5*x^(5*n) + 126126*a^4
*b^4*x^(4*n) + 91728*a^5*b^3*x^(3*n) + 42042*a^6*b^2*x^(2*n) + 11088*a^7*b*x^n + 1287*a^8)/(n*x^(14*n))

Sympy [A] (verification not implemented)

Time = 1.97 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.53 \[ \int x^{-1-14 n} \left (a+b x^n\right )^8 \, dx=\begin {cases} - \frac {a^{8} x x^{- 14 n - 1}}{14 n} - \frac {8 a^{7} b x x^{n} x^{- 14 n - 1}}{13 n} - \frac {7 a^{6} b^{2} x x^{2 n} x^{- 14 n - 1}}{3 n} - \frac {56 a^{5} b^{3} x x^{3 n} x^{- 14 n - 1}}{11 n} - \frac {7 a^{4} b^{4} x x^{4 n} x^{- 14 n - 1}}{n} - \frac {56 a^{3} b^{5} x x^{5 n} x^{- 14 n - 1}}{9 n} - \frac {7 a^{2} b^{6} x x^{6 n} x^{- 14 n - 1}}{2 n} - \frac {8 a b^{7} x x^{7 n} x^{- 14 n - 1}}{7 n} - \frac {b^{8} x x^{8 n} x^{- 14 n - 1}}{6 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(x**(-1-14*n)*(a+b*x**n)**8,x)

[Out]

Piecewise((-a**8*x*x**(-14*n - 1)/(14*n) - 8*a**7*b*x*x**n*x**(-14*n - 1)/(13*n) - 7*a**6*b**2*x*x**(2*n)*x**(
-14*n - 1)/(3*n) - 56*a**5*b**3*x*x**(3*n)*x**(-14*n - 1)/(11*n) - 7*a**4*b**4*x*x**(4*n)*x**(-14*n - 1)/n - 5
6*a**3*b**5*x*x**(5*n)*x**(-14*n - 1)/(9*n) - 7*a**2*b**6*x*x**(6*n)*x**(-14*n - 1)/(2*n) - 8*a*b**7*x*x**(7*n
)*x**(-14*n - 1)/(7*n) - b**8*x*x**(8*n)*x**(-14*n - 1)/(6*n), Ne(n, 0)), ((a + b)**8*log(x), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.01 \[ \int x^{-1-14 n} \left (a+b x^n\right )^8 \, dx=-\frac {a^{8}}{14 \, n x^{14 \, n}} - \frac {8 \, a^{7} b}{13 \, n x^{13 \, n}} - \frac {7 \, a^{6} b^{2}}{3 \, n x^{12 \, n}} - \frac {56 \, a^{5} b^{3}}{11 \, n x^{11 \, n}} - \frac {7 \, a^{4} b^{4}}{n x^{10 \, n}} - \frac {56 \, a^{3} b^{5}}{9 \, n x^{9 \, n}} - \frac {7 \, a^{2} b^{6}}{2 \, n x^{8 \, n}} - \frac {8 \, a b^{7}}{7 \, n x^{7 \, n}} - \frac {b^{8}}{6 \, n x^{6 \, n}} \]

[In]

integrate(x^(-1-14*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

-1/14*a^8/(n*x^(14*n)) - 8/13*a^7*b/(n*x^(13*n)) - 7/3*a^6*b^2/(n*x^(12*n)) - 56/11*a^5*b^3/(n*x^(11*n)) - 7*a
^4*b^4/(n*x^(10*n)) - 56/9*a^3*b^5/(n*x^(9*n)) - 7/2*a^2*b^6/(n*x^(8*n)) - 8/7*a*b^7/(n*x^(7*n)) - 1/6*b^8/(n*
x^(6*n))

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75 \[ \int x^{-1-14 n} \left (a+b x^n\right )^8 \, dx=-\frac {3003 \, b^{8} x^{8 \, n} + 20592 \, a b^{7} x^{7 \, n} + 63063 \, a^{2} b^{6} x^{6 \, n} + 112112 \, a^{3} b^{5} x^{5 \, n} + 126126 \, a^{4} b^{4} x^{4 \, n} + 91728 \, a^{5} b^{3} x^{3 \, n} + 42042 \, a^{6} b^{2} x^{2 \, n} + 11088 \, a^{7} b x^{n} + 1287 \, a^{8}}{18018 \, n x^{14 \, n}} \]

[In]

integrate(x^(-1-14*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

-1/18018*(3003*b^8*x^(8*n) + 20592*a*b^7*x^(7*n) + 63063*a^2*b^6*x^(6*n) + 112112*a^3*b^5*x^(5*n) + 126126*a^4
*b^4*x^(4*n) + 91728*a^5*b^3*x^(3*n) + 42042*a^6*b^2*x^(2*n) + 11088*a^7*b*x^n + 1287*a^8)/(n*x^(14*n))

Mupad [B] (verification not implemented)

Time = 5.89 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.01 \[ \int x^{-1-14 n} \left (a+b x^n\right )^8 \, dx=-\frac {a^8}{14\,n\,x^{14\,n}}-\frac {b^8}{6\,n\,x^{6\,n}}-\frac {7\,a^2\,b^6}{2\,n\,x^{8\,n}}-\frac {56\,a^3\,b^5}{9\,n\,x^{9\,n}}-\frac {7\,a^4\,b^4}{n\,x^{10\,n}}-\frac {56\,a^5\,b^3}{11\,n\,x^{11\,n}}-\frac {7\,a^6\,b^2}{3\,n\,x^{12\,n}}-\frac {8\,a\,b^7}{7\,n\,x^{7\,n}}-\frac {8\,a^7\,b}{13\,n\,x^{13\,n}} \]

[In]

int((a + b*x^n)^8/x^(14*n + 1),x)

[Out]

- a^8/(14*n*x^(14*n)) - b^8/(6*n*x^(6*n)) - (7*a^2*b^6)/(2*n*x^(8*n)) - (56*a^3*b^5)/(9*n*x^(9*n)) - (7*a^4*b^
4)/(n*x^(10*n)) - (56*a^5*b^3)/(11*n*x^(11*n)) - (7*a^6*b^2)/(3*n*x^(12*n)) - (8*a*b^7)/(7*n*x^(7*n)) - (8*a^7
*b)/(13*n*x^(13*n))