\(\int x^{-1+5 n} (a+b x^n)^8 \, dx\) [2570]

Optimal result

Integrand size = 17, antiderivative size = 106 \[ \int x^{-1+5 n} \left (a+b x^n\right )^8 \, dx=\frac {a^4 \left (a+b x^n\right )^9}{9 b^5 n}-\frac {2 a^3 \left (a+b x^n\right )^{10}}{5 b^5 n}+\frac {6 a^2 \left (a+b x^n\right )^{11}}{11 b^5 n}-\frac {a \left (a+b x^n\right )^{12}}{3 b^5 n}+\frac {\left (a+b x^n\right )^{13}}{13 b^5 n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 106, 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} \[ \int x^{-1+5 n} \left (a+b x^n\right )^8 \, dx=\frac {a^4 \left (a+b x^n\right )^9}{9 b^5 n}-\frac {2 a^3 \left (a+b x^n\right )^{10}}{5 b^5 n}+\frac {6 a^2 \left (a+b x^n\right )^{11}}{11 b^5 n}+\frac {\left (a+b x^n\right )^{13}}{13 b^5 n}-\frac {a \left (a+b x^n\right )^{12}}{3 b^5 n} \]

[In]

Int[x^(-1 + 5*n)*(a + b*x^n)^8,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.07 \[ \int x^{-1+5 n} \left (a+b x^n\right )^8 \, dx=\frac {x^{5 n} \left (1287 a^8+8580 a^7 b x^n+25740 a^6 b^2 x^{2 n}+45045 a^5 b^3 x^{3 n}+50050 a^4 b^4 x^{4 n}+36036 a^3 b^5 x^{5 n}+16380 a^2 b^6 x^{6 n}+4290 a b^7 x^{7 n}+495 b^8 x^{8 n}\right )}{6435 n} \]

[In]

Integrate[x^(-1 + 5*n)*(a + b*x^n)^8,x]

[Out]

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

Maple [A] (verified)

Time = 9.47 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.28

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

[In]

int(x^(-1+5*n)*(a+b*x^n)^8,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.07 \[ \int x^{-1+5 n} \left (a+b x^n\right )^8 \, dx=\frac {495 \, b^{8} x^{13 \, n} + 4290 \, a b^{7} x^{12 \, n} + 16380 \, a^{2} b^{6} x^{11 \, n} + 36036 \, a^{3} b^{5} x^{10 \, n} + 50050 \, a^{4} b^{4} x^{9 \, n} + 45045 \, a^{5} b^{3} x^{8 \, n} + 25740 \, a^{6} b^{2} x^{7 \, n} + 8580 \, a^{7} b x^{6 \, n} + 1287 \, a^{8} x^{5 \, n}}{6435 \, n} \]

[In]

integrate(x^(-1+5*n)*(a+b*x^n)^8,x, algorithm="fricas")

[Out]

1/6435*(495*b^8*x^(13*n) + 4290*a*b^7*x^(12*n) + 16380*a^2*b^6*x^(11*n) + 36036*a^3*b^5*x^(10*n) + 50050*a^4*b
^4*x^(9*n) + 45045*a^5*b^3*x^(8*n) + 25740*a^6*b^2*x^(7*n) + 8580*a^7*b*x^(6*n) + 1287*a^8*x^(5*n))/n

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (90) = 180\).

Time = 1.99 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.00 \[ \int x^{-1+5 n} \left (a+b x^n\right )^8 \, dx=\begin {cases} \frac {a^{8} x x^{5 n - 1}}{5 n} + \frac {4 a^{7} b x x^{n} x^{5 n - 1}}{3 n} + \frac {4 a^{6} b^{2} x x^{2 n} x^{5 n - 1}}{n} + \frac {7 a^{5} b^{3} x x^{3 n} x^{5 n - 1}}{n} + \frac {70 a^{4} b^{4} x x^{4 n} x^{5 n - 1}}{9 n} + \frac {28 a^{3} b^{5} x x^{5 n} x^{5 n - 1}}{5 n} + \frac {28 a^{2} b^{6} x x^{6 n} x^{5 n - 1}}{11 n} + \frac {2 a b^{7} x x^{7 n} x^{5 n - 1}}{3 n} + \frac {b^{8} x x^{8 n} x^{5 n - 1}}{13 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(x**(-1+5*n)*(a+b*x**n)**8,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.27 \[ \int x^{-1+5 n} \left (a+b x^n\right )^8 \, dx=\frac {b^{8} x^{13 \, n}}{13 \, n} + \frac {2 \, a b^{7} x^{12 \, n}}{3 \, n} + \frac {28 \, a^{2} b^{6} x^{11 \, n}}{11 \, n} + \frac {28 \, a^{3} b^{5} x^{10 \, n}}{5 \, n} + \frac {70 \, a^{4} b^{4} x^{9 \, n}}{9 \, n} + \frac {7 \, a^{5} b^{3} x^{8 \, n}}{n} + \frac {4 \, a^{6} b^{2} x^{7 \, n}}{n} + \frac {4 \, a^{7} b x^{6 \, n}}{3 \, n} + \frac {a^{8} x^{5 \, n}}{5 \, n} \]

[In]

integrate(x^(-1+5*n)*(a+b*x^n)^8,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int x^{-1+5 n} \left (a+b x^n\right )^8 \, dx=\int { {\left (b x^{n} + a\right )}^{8} x^{5 \, n - 1} \,d x } \]

[In]

integrate(x^(-1+5*n)*(a+b*x^n)^8,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^8*x^(5*n - 1), x)

Mupad [B] (verification not implemented)

Time = 5.89 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.27 \[ \int x^{-1+5 n} \left (a+b x^n\right )^8 \, dx=\frac {a^8\,x^{5\,n}}{5\,n}+\frac {b^8\,x^{13\,n}}{13\,n}+\frac {4\,a^6\,b^2\,x^{7\,n}}{n}+\frac {7\,a^5\,b^3\,x^{8\,n}}{n}+\frac {70\,a^4\,b^4\,x^{9\,n}}{9\,n}+\frac {28\,a^3\,b^5\,x^{10\,n}}{5\,n}+\frac {28\,a^2\,b^6\,x^{11\,n}}{11\,n}+\frac {4\,a^7\,b\,x^{6\,n}}{3\,n}+\frac {2\,a\,b^7\,x^{12\,n}}{3\,n} \]

[In]

int(x^(5*n - 1)*(a + b*x^n)^8,x)

[Out]

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