\(\int x^{-1-13 n} (a+b x^n)^8 \, dx\) [2588]

Optimal result

Integrand size = 17, antiderivative size = 131 \[ \int x^{-1-13 n} \left (a+b x^n\right )^8 \, dx=-\frac {x^{-13 n} \left (a+b x^n\right )^9}{13 a n}+\frac {b x^{-12 n} \left (a+b x^n\right )^9}{39 a^2 n}-\frac {b^2 x^{-11 n} \left (a+b x^n\right )^9}{143 a^3 n}+\frac {b^3 x^{-10 n} \left (a+b x^n\right )^9}{715 a^4 n}-\frac {b^4 x^{-9 n} \left (a+b x^n\right )^9}{6435 a^5 n} \]

[Out]

-1/13*(a+b*x^n)^9/a/n/(x^(13*n))+1/39*b*(a+b*x^n)^9/a^2/n/(x^(12*n))-1/143*b^2*(a+b*x^n)^9/a^3/n/(x^(11*n))+1/
715*b^3*(a+b*x^n)^9/a^4/n/(x^(10*n))-1/6435*b^4*(a+b*x^n)^9/a^5/n/(x^(9*n))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {272, 47, 37} \[ \int x^{-1-13 n} \left (a+b x^n\right )^8 \, dx=-\frac {b^4 x^{-9 n} \left (a+b x^n\right )^9}{6435 a^5 n}+\frac {b^3 x^{-10 n} \left (a+b x^n\right )^9}{715 a^4 n}-\frac {b^2 x^{-11 n} \left (a+b x^n\right )^9}{143 a^3 n}+\frac {b x^{-12 n} \left (a+b x^n\right )^9}{39 a^2 n}-\frac {x^{-13 n} \left (a+b x^n\right )^9}{13 a n} \]

[In]

Int[x^(-1 - 13*n)*(a + b*x^n)^8,x]

[Out]

-1/13*(a + b*x^n)^9/(a*n*x^(13*n)) + (b*(a + b*x^n)^9)/(39*a^2*n*x^(12*n)) - (b^2*(a + b*x^n)^9)/(143*a^3*n*x^
(11*n)) + (b^3*(a + b*x^n)^9)/(715*a^4*n*x^(10*n)) - (b^4*(a + b*x^n)^9)/(6435*a^5*n*x^(9*n))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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^{14}} \, dx,x,x^n\right )}{n} \\ & = -\frac {x^{-13 n} \left (a+b x^n\right )^9}{13 a n}-\frac {(4 b) \text {Subst}\left (\int \frac {(a+b x)^8}{x^{13}} \, dx,x,x^n\right )}{13 a n} \\ & = -\frac {x^{-13 n} \left (a+b x^n\right )^9}{13 a n}+\frac {b x^{-12 n} \left (a+b x^n\right )^9}{39 a^2 n}+\frac {b^2 \text {Subst}\left (\int \frac {(a+b x)^8}{x^{12}} \, dx,x,x^n\right )}{13 a^2 n} \\ & = -\frac {x^{-13 n} \left (a+b x^n\right )^9}{13 a n}+\frac {b x^{-12 n} \left (a+b x^n\right )^9}{39 a^2 n}-\frac {b^2 x^{-11 n} \left (a+b x^n\right )^9}{143 a^3 n}-\frac {\left (2 b^3\right ) \text {Subst}\left (\int \frac {(a+b x)^8}{x^{11}} \, dx,x,x^n\right )}{143 a^3 n} \\ & = -\frac {x^{-13 n} \left (a+b x^n\right )^9}{13 a n}+\frac {b x^{-12 n} \left (a+b x^n\right )^9}{39 a^2 n}-\frac {b^2 x^{-11 n} \left (a+b x^n\right )^9}{143 a^3 n}+\frac {b^3 x^{-10 n} \left (a+b x^n\right )^9}{715 a^4 n}+\frac {b^4 \text {Subst}\left (\int \frac {(a+b x)^8}{x^{10}} \, dx,x,x^n\right )}{715 a^4 n} \\ & = -\frac {x^{-13 n} \left (a+b x^n\right )^9}{13 a n}+\frac {b x^{-12 n} \left (a+b x^n\right )^9}{39 a^2 n}-\frac {b^2 x^{-11 n} \left (a+b x^n\right )^9}{143 a^3 n}+\frac {b^3 x^{-10 n} \left (a+b x^n\right )^9}{715 a^4 n}-\frac {b^4 x^{-9 n} \left (a+b x^n\right )^9}{6435 a^5 n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86 \[ \int x^{-1-13 n} \left (a+b x^n\right )^8 \, dx=\frac {x^{-13 n} \left (-495 a^8-4290 a^7 b x^n-16380 a^6 b^2 x^{2 n}-36036 a^5 b^3 x^{3 n}-50050 a^4 b^4 x^{4 n}-45045 a^3 b^5 x^{5 n}-25740 a^2 b^6 x^{6 n}-8580 a b^7 x^{7 n}-1287 b^8 x^{8 n}\right )}{6435 n} \]

[In]

Integrate[x^(-1 - 13*n)*(a + b*x^n)^8,x]

[Out]

(-495*a^8 - 4290*a^7*b*x^n - 16380*a^6*b^2*x^(2*n) - 36036*a^5*b^3*x^(3*n) - 50050*a^4*b^4*x^(4*n) - 45045*a^3
*b^5*x^(5*n) - 25740*a^2*b^6*x^(6*n) - 8580*a*b^7*x^(7*n) - 1287*b^8*x^(8*n))/(6435*n*x^(13*n))

Maple [A] (verified)

Time = 8.61 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.04

method	result	size
risch	\(-\frac {b^{8} x^{-5 n}}{5 n}-\frac {4 a \,b^{7} x^{-6 n}}{3 n}-\frac {4 a^{2} b^{6} x^{-7 n}}{n}-\frac {7 a^{3} b^{5} x^{-8 n}}{n}-\frac {70 a^{4} b^{4} x^{-9 n}}{9 n}-\frac {28 a^{5} b^{3} x^{-10 n}}{5 n}-\frac {28 a^{6} b^{2} x^{-11 n}}{11 n}-\frac {2 a^{7} b \,x^{-12 n}}{3 n}-\frac {a^{8} x^{-13 n}}{13 n}\)	\(136\)
parallelrisch	\(\frac {-1287 b^{8} x^{-1-13 n} x^{8 n} x -8580 a \,b^{7} x^{-1-13 n} x^{7 n} x -25740 a^{2} b^{6} x^{-1-13 n} x^{6 n} x -45045 a^{3} b^{5} x^{-1-13 n} x^{5 n} x -50050 a^{4} b^{4} x^{-1-13 n} x^{4 n} x -36036 a^{5} b^{3} x^{-1-13 n} x^{3 n} x -16380 a^{6} b^{2} x^{-1-13 n} x^{2 n} x -4290 a^{7} b \,x^{-1-13 n} x^{n} x -495 a^{8} x^{-1-13 n} x}{6435 n}\)	\(179\)

[In]

int(x^(-1-13*n)*(a+b*x^n)^8,x,method=_RETURNVERBOSE)

[Out]

-1/5*b^8/n/(x^n)^5-4/3*a*b^7/n/(x^n)^6-4*a^2*b^6/n/(x^n)^7-7*a^3*b^5/n/(x^n)^8-70/9*a^4*b^4/n/(x^n)^9-28/5*a^5
*b^3/n/(x^n)^10-28/11*a^6*b^2/n/(x^n)^11-2/3*a^7*b/n/(x^n)^12-1/13*a^8/n/(x^n)^13

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86 \[ \int x^{-1-13 n} \left (a+b x^n\right )^8 \, dx=-\frac {1287 \, b^{8} x^{8 \, n} + 8580 \, a b^{7} x^{7 \, n} + 25740 \, a^{2} b^{6} x^{6 \, n} + 45045 \, a^{3} b^{5} x^{5 \, n} + 50050 \, a^{4} b^{4} x^{4 \, n} + 36036 \, a^{5} b^{3} x^{3 \, n} + 16380 \, a^{6} b^{2} x^{2 \, n} + 4290 \, a^{7} b x^{n} + 495 \, a^{8}}{6435 \, n x^{13 \, n}} \]

[In]

integrate(x^(-1-13*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

-1/6435*(1287*b^8*x^(8*n) + 8580*a*b^7*x^(7*n) + 25740*a^2*b^6*x^(6*n) + 45045*a^3*b^5*x^(5*n) + 50050*a^4*b^4
*x^(4*n) + 36036*a^5*b^3*x^(3*n) + 16380*a^6*b^2*x^(2*n) + 4290*a^7*b*x^n + 495*a^8)/(n*x^(13*n))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (110) = 220\).

Time = 1.97 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.76 \[ \int x^{-1-13 n} \left (a+b x^n\right )^8 \, dx=\begin {cases} - \frac {a^{8} x x^{- 13 n - 1}}{13 n} - \frac {2 a^{7} b x x^{n} x^{- 13 n - 1}}{3 n} - \frac {28 a^{6} b^{2} x x^{2 n} x^{- 13 n - 1}}{11 n} - \frac {28 a^{5} b^{3} x x^{3 n} x^{- 13 n - 1}}{5 n} - \frac {70 a^{4} b^{4} x x^{4 n} x^{- 13 n - 1}}{9 n} - \frac {7 a^{3} b^{5} x x^{5 n} x^{- 13 n - 1}}{n} - \frac {4 a^{2} b^{6} x x^{6 n} x^{- 13 n - 1}}{n} - \frac {4 a b^{7} x x^{7 n} x^{- 13 n - 1}}{3 n} - \frac {b^{8} x x^{8 n} x^{- 13 n - 1}}{5 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(x**(-1-13*n)*(a+b*x**n)**8,x)

[Out]

Piecewise((-a**8*x*x**(-13*n - 1)/(13*n) - 2*a**7*b*x*x**n*x**(-13*n - 1)/(3*n) - 28*a**6*b**2*x*x**(2*n)*x**(
-13*n - 1)/(11*n) - 28*a**5*b**3*x*x**(3*n)*x**(-13*n - 1)/(5*n) - 70*a**4*b**4*x*x**(4*n)*x**(-13*n - 1)/(9*n
) - 7*a**3*b**5*x*x**(5*n)*x**(-13*n - 1)/n - 4*a**2*b**6*x*x**(6*n)*x**(-13*n - 1)/n - 4*a*b**7*x*x**(7*n)*x*
*(-13*n - 1)/(3*n) - b**8*x*x**(8*n)*x**(-13*n - 1)/(5*n), Ne(n, 0)), ((a + b)**8*log(x), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.17 \[ \int x^{-1-13 n} \left (a+b x^n\right )^8 \, dx=-\frac {a^{8}}{13 \, n x^{13 \, n}} - \frac {2 \, a^{7} b}{3 \, n x^{12 \, n}} - \frac {28 \, a^{6} b^{2}}{11 \, n x^{11 \, n}} - \frac {28 \, a^{5} b^{3}}{5 \, n x^{10 \, n}} - \frac {70 \, a^{4} b^{4}}{9 \, n x^{9 \, n}} - \frac {7 \, a^{3} b^{5}}{n x^{8 \, n}} - \frac {4 \, a^{2} b^{6}}{n x^{7 \, n}} - \frac {4 \, a b^{7}}{3 \, n x^{6 \, n}} - \frac {b^{8}}{5 \, n x^{5 \, n}} \]

[In]

integrate(x^(-1-13*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

-1/13*a^8/(n*x^(13*n)) - 2/3*a^7*b/(n*x^(12*n)) - 28/11*a^6*b^2/(n*x^(11*n)) - 28/5*a^5*b^3/(n*x^(10*n)) - 70/
9*a^4*b^4/(n*x^(9*n)) - 7*a^3*b^5/(n*x^(8*n)) - 4*a^2*b^6/(n*x^(7*n)) - 4/3*a*b^7/(n*x^(6*n)) - 1/5*b^8/(n*x^(
5*n))

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86 \[ \int x^{-1-13 n} \left (a+b x^n\right )^8 \, dx=-\frac {1287 \, b^{8} x^{8 \, n} + 8580 \, a b^{7} x^{7 \, n} + 25740 \, a^{2} b^{6} x^{6 \, n} + 45045 \, a^{3} b^{5} x^{5 \, n} + 50050 \, a^{4} b^{4} x^{4 \, n} + 36036 \, a^{5} b^{3} x^{3 \, n} + 16380 \, a^{6} b^{2} x^{2 \, n} + 4290 \, a^{7} b x^{n} + 495 \, a^{8}}{6435 \, n x^{13 \, n}} \]

[In]

integrate(x^(-1-13*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

-1/6435*(1287*b^8*x^(8*n) + 8580*a*b^7*x^(7*n) + 25740*a^2*b^6*x^(6*n) + 45045*a^3*b^5*x^(5*n) + 50050*a^4*b^4
*x^(4*n) + 36036*a^5*b^3*x^(3*n) + 16380*a^6*b^2*x^(2*n) + 4290*a^7*b*x^n + 495*a^8)/(n*x^(13*n))

Mupad [B] (verification not implemented)

Time = 5.78 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.17 \[ \int x^{-1-13 n} \left (a+b x^n\right )^8 \, dx=-\frac {a^8}{13\,n\,x^{13\,n}}-\frac {b^8}{5\,n\,x^{5\,n}}-\frac {4\,a^2\,b^6}{n\,x^{7\,n}}-\frac {7\,a^3\,b^5}{n\,x^{8\,n}}-\frac {70\,a^4\,b^4}{9\,n\,x^{9\,n}}-\frac {28\,a^5\,b^3}{5\,n\,x^{10\,n}}-\frac {28\,a^6\,b^2}{11\,n\,x^{11\,n}}-\frac {4\,a\,b^7}{3\,n\,x^{6\,n}}-\frac {2\,a^7\,b}{3\,n\,x^{12\,n}} \]

[In]

int((a + b*x^n)^8/x^(13*n + 1),x)

[Out]

- a^8/(13*n*x^(13*n)) - b^8/(5*n*x^(5*n)) - (4*a^2*b^6)/(n*x^(7*n)) - (7*a^3*b^5)/(n*x^(8*n)) - (70*a^4*b^4)/(
9*n*x^(9*n)) - (28*a^5*b^3)/(5*n*x^(10*n)) - (28*a^6*b^2)/(11*n*x^(11*n)) - (4*a*b^7)/(3*n*x^(6*n)) - (2*a^7*b
)/(3*n*x^(12*n))