\(\int x^{-1-7 n} (a+b x^n)^8 \, dx\) [2582]

Optimal result

Integrand size = 17, antiderivative size = 134 \[ \int x^{-1-7 n} \left (a+b x^n\right )^8 \, dx=-\frac {a^8 x^{-7 n}}{7 n}-\frac {4 a^7 b x^{-6 n}}{3 n}-\frac {28 a^6 b^2 x^{-5 n}}{5 n}-\frac {14 a^5 b^3 x^{-4 n}}{n}-\frac {70 a^4 b^4 x^{-3 n}}{3 n}-\frac {28 a^3 b^5 x^{-2 n}}{n}-\frac {28 a^2 b^6 x^{-n}}{n}+\frac {b^8 x^n}{n}+8 a b^7 \log (x) \]

[Out]

-1/7*a^8/n/(x^(7*n))-4/3*a^7*b/n/(x^(6*n))-28/5*a^6*b^2/n/(x^(5*n))-14*a^5*b^3/n/(x^(4*n))-70/3*a^4*b^4/n/(x^(
3*n))-28*a^3*b^5/n/(x^(2*n))-28*a^2*b^6/n/(x^n)+b^8*x^n/n+8*a*b^7*ln(x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 134, 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} \[ \int x^{-1-7 n} \left (a+b x^n\right )^8 \, dx=-\frac {a^8 x^{-7 n}}{7 n}-\frac {4 a^7 b x^{-6 n}}{3 n}-\frac {28 a^6 b^2 x^{-5 n}}{5 n}-\frac {14 a^5 b^3 x^{-4 n}}{n}-\frac {70 a^4 b^4 x^{-3 n}}{3 n}-\frac {28 a^3 b^5 x^{-2 n}}{n}-\frac {28 a^2 b^6 x^{-n}}{n}+8 a b^7 \log (x)+\frac {b^8 x^n}{n} \]

[In]

Int[x^(-1 - 7*n)*(a + b*x^n)^8,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87 \[ \int x^{-1-7 n} \left (a+b x^n\right )^8 \, dx=\frac {x^{-7 n} \left (-15 a^8-140 a^7 b x^n-588 a^6 b^2 x^{2 n}-1470 a^5 b^3 x^{3 n}-2450 a^4 b^4 x^{4 n}-2940 a^3 b^5 x^{5 n}-2940 a^2 b^6 x^{6 n}+105 b^8 x^{8 n}\right )}{105 n}+\frac {8 a b^7 \log \left (x^n\right )}{n} \]

[In]

Integrate[x^(-1 - 7*n)*(a + b*x^n)^8,x]

[Out]

(-15*a^8 - 140*a^7*b*x^n - 588*a^6*b^2*x^(2*n) - 1470*a^5*b^3*x^(3*n) - 2450*a^4*b^4*x^(4*n) - 2940*a^3*b^5*x^
(5*n) - 2940*a^2*b^6*x^(6*n) + 105*b^8*x^(8*n))/(105*n*x^(7*n)) + (8*a*b^7*Log[x^n])/n

Maple [A] (verified)

Time = 4.45 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.95

method	result	size
risch	\(8 a \,b^{7} \ln \left (x \right )+\frac {b^{8} x^{n}}{n}-\frac {28 a^{2} b^{6} x^{-n}}{n}-\frac {28 a^{3} b^{5} x^{-2 n}}{n}-\frac {70 a^{4} b^{4} x^{-3 n}}{3 n}-\frac {14 a^{5} b^{3} x^{-4 n}}{n}-\frac {28 a^{6} b^{2} x^{-5 n}}{5 n}-\frac {4 a^{7} b \,x^{-6 n}}{3 n}-\frac {a^{8} x^{-7 n}}{7 n}\)	\(127\)

[In]

int(x^(-1-7*n)*(a+b*x^n)^8,x,method=_RETURNVERBOSE)

[Out]

8*a*b^7*ln(x)+b^8*x^n/n-28*a^2*b^6/n/(x^n)-28*a^3*b^5/n/(x^n)^2-70/3*a^4*b^4/n/(x^n)^3-14*a^5*b^3/n/(x^n)^4-28
/5*a^6*b^2/n/(x^n)^5-4/3*a^7*b/n/(x^n)^6-1/7*a^8/n/(x^n)^7

Fricas [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87 \[ \int x^{-1-7 n} \left (a+b x^n\right )^8 \, dx=\frac {840 \, a b^{7} n x^{7 \, n} \log \left (x\right ) + 105 \, b^{8} x^{8 \, n} - 2940 \, a^{2} b^{6} x^{6 \, n} - 2940 \, a^{3} b^{5} x^{5 \, n} - 2450 \, a^{4} b^{4} x^{4 \, n} - 1470 \, a^{5} b^{3} x^{3 \, n} - 588 \, a^{6} b^{2} x^{2 \, n} - 140 \, a^{7} b x^{n} - 15 \, a^{8}}{105 \, n x^{7 \, n}} \]

[In]

integrate(x^(-1-7*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

1/105*(840*a*b^7*n*x^(7*n)*log(x) + 105*b^8*x^(8*n) - 2940*a^2*b^6*x^(6*n) - 2940*a^3*b^5*x^(5*n) - 2450*a^4*b
^4*x^(4*n) - 1470*a^5*b^3*x^(3*n) - 588*a^6*b^2*x^(2*n) - 140*a^7*b*x^n - 15*a^8)/(n*x^(7*n))

Sympy [A] (verification not implemented)

Time = 18.53 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.98 \[ \int x^{-1-7 n} \left (a+b x^n\right )^8 \, dx=\begin {cases} - \frac {a^{8} x^{- 7 n}}{7 n} - \frac {4 a^{7} b x^{- 6 n}}{3 n} - \frac {28 a^{6} b^{2} x^{- 5 n}}{5 n} - \frac {14 a^{5} b^{3} x^{- 4 n}}{n} - \frac {70 a^{4} b^{4} x^{- 3 n}}{3 n} - \frac {28 a^{3} b^{5} x^{- 2 n}}{n} - \frac {28 a^{2} b^{6} x^{- n}}{n} + 8 a b^{7} \log {\left (x \right )} + \frac {b^{8} x^{n}}{n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(x**(-1-7*n)*(a+b*x**n)**8,x)

[Out]

Piecewise((-a**8/(7*n*x**(7*n)) - 4*a**7*b/(3*n*x**(6*n)) - 28*a**6*b**2/(5*n*x**(5*n)) - 14*a**5*b**3/(n*x**(
4*n)) - 70*a**4*b**4/(3*n*x**(3*n)) - 28*a**3*b**5/(n*x**(2*n)) - 28*a**2*b**6/(n*x**n) + 8*a*b**7*log(x) + b*
*8*x**n/n, Ne(n, 0)), ((a + b)**8*log(x), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.03 \[ \int x^{-1-7 n} \left (a+b x^n\right )^8 \, dx=8 \, a b^{7} \log \left (x\right ) + \frac {b^{8} x^{n}}{n} - \frac {a^{8}}{7 \, n x^{7 \, n}} - \frac {4 \, a^{7} b}{3 \, n x^{6 \, n}} - \frac {28 \, a^{6} b^{2}}{5 \, n x^{5 \, n}} - \frac {14 \, a^{5} b^{3}}{n x^{4 \, n}} - \frac {70 \, a^{4} b^{4}}{3 \, n x^{3 \, n}} - \frac {28 \, a^{3} b^{5}}{n x^{2 \, n}} - \frac {28 \, a^{2} b^{6}}{n x^{n}} \]

[In]

integrate(x^(-1-7*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

8*a*b^7*log(x) + b^8*x^n/n - 1/7*a^8/(n*x^(7*n)) - 4/3*a^7*b/(n*x^(6*n)) - 28/5*a^6*b^2/(n*x^(5*n)) - 14*a^5*b
^3/(n*x^(4*n)) - 70/3*a^4*b^4/(n*x^(3*n)) - 28*a^3*b^5/(n*x^(2*n)) - 28*a^2*b^6/(n*x^n)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87 \[ \int x^{-1-7 n} \left (a+b x^n\right )^8 \, dx=\frac {840 \, a b^{7} n x^{7 \, n} \log \left (x\right ) + 105 \, b^{8} x^{8 \, n} - 2940 \, a^{2} b^{6} x^{6 \, n} - 2940 \, a^{3} b^{5} x^{5 \, n} - 2450 \, a^{4} b^{4} x^{4 \, n} - 1470 \, a^{5} b^{3} x^{3 \, n} - 588 \, a^{6} b^{2} x^{2 \, n} - 140 \, a^{7} b x^{n} - 15 \, a^{8}}{105 \, n x^{7 \, n}} \]

[In]

integrate(x^(-1-7*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

1/105*(840*a*b^7*n*x^(7*n)*log(x) + 105*b^8*x^(8*n) - 2940*a^2*b^6*x^(6*n) - 2940*a^3*b^5*x^(5*n) - 2450*a^4*b
^4*x^(4*n) - 1470*a^5*b^3*x^(3*n) - 588*a^6*b^2*x^(2*n) - 140*a^7*b*x^n - 15*a^8)/(n*x^(7*n))

Mupad [B] (verification not implemented)

Time = 5.97 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.03 \[ \int x^{-1-7 n} \left (a+b x^n\right )^8 \, dx=\frac {b^8\,x^n}{n}+8\,a\,b^7\,\ln \left (x\right )-\frac {a^8}{7\,n\,x^{7\,n}}-\frac {28\,a^2\,b^6}{n\,x^n}-\frac {28\,a^3\,b^5}{n\,x^{2\,n}}-\frac {70\,a^4\,b^4}{3\,n\,x^{3\,n}}-\frac {14\,a^5\,b^3}{n\,x^{4\,n}}-\frac {28\,a^6\,b^2}{5\,n\,x^{5\,n}}-\frac {4\,a^7\,b}{3\,n\,x^{6\,n}} \]

[In]

int((a + b*x^n)^8/x^(7*n + 1),x)

[Out]

(b^8*x^n)/n + 8*a*b^7*log(x) - a^8/(7*n*x^(7*n)) - (28*a^2*b^6)/(n*x^n) - (28*a^3*b^5)/(n*x^(2*n)) - (70*a^4*b
^4)/(3*n*x^(3*n)) - (14*a^5*b^3)/(n*x^(4*n)) - (28*a^6*b^2)/(5*n*x^(5*n)) - (4*a^7*b)/(3*n*x^(6*n))