Integrand size = 17, antiderivative size = 151 \[ \int x^{-1+9 n} \left (a+b x^n\right )^8 \, dx=\frac {a^8 x^{9 n}}{9 n}+\frac {4 a^7 b x^{10 n}}{5 n}+\frac {28 a^6 b^2 x^{11 n}}{11 n}+\frac {14 a^5 b^3 x^{12 n}}{3 n}+\frac {70 a^4 b^4 x^{13 n}}{13 n}+\frac {4 a^3 b^5 x^{14 n}}{n}+\frac {28 a^2 b^6 x^{15 n}}{15 n}+\frac {a b^7 x^{16 n}}{2 n}+\frac {b^8 x^{17 n}}{17 n} \] Output:
1/9*a^8*x^(9*n)/n+4/5*a^7*b*x^(10*n)/n+28/11*a^6*b^2*x^(11*n)/n+14/3*a^5*b ^3*x^(12*n)/n+70/13*a^4*b^4*x^(13*n)/n+4*a^3*b^5*x^(14*n)/n+28/15*a^2*b^6* x^(15*n)/n+1/2*a*b^7*x^(16*n)/n+1/17*b^8*x^(17*n)/n
Time = 0.07 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75 \[ \int x^{-1+9 n} \left (a+b x^n\right )^8 \, dx=\frac {x^{9 n} \left (24310 a^8+175032 a^7 b x^n+556920 a^6 b^2 x^{2 n}+1021020 a^5 b^3 x^{3 n}+1178100 a^4 b^4 x^{4 n}+875160 a^3 b^5 x^{5 n}+408408 a^2 b^6 x^{6 n}+109395 a b^7 x^{7 n}+12870 b^8 x^{8 n}\right )}{218790 n} \] Input:
Integrate[x^(-1 + 9*n)*(a + b*x^n)^8,x]
Output:
(x^(9*n)*(24310*a^8 + 175032*a^7*b*x^n + 556920*a^6*b^2*x^(2*n) + 1021020* a^5*b^3*x^(3*n) + 1178100*a^4*b^4*x^(4*n) + 875160*a^3*b^5*x^(5*n) + 40840 8*a^2*b^6*x^(6*n) + 109395*a*b^7*x^(7*n) + 12870*b^8*x^(8*n)))/(218790*n)
Time = 0.43 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {798, 49, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^{9 n-1} \left (a+b x^n\right )^8 \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int x^{8 n} \left (b x^n+a\right )^8dx^n}{n}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (a^8 x^{8 n}+8 a^7 b x^{9 n}+28 a^6 b^2 x^{10 n}+56 a^5 b^3 x^{11 n}+70 a^4 b^4 x^{12 n}+56 a^3 b^5 x^{13 n}+28 a^2 b^6 x^{14 n}+8 a b^7 x^{15 n}+b^8 x^{16 n}\right )dx^n}{n}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{9} a^8 x^{9 n}+\frac {4}{5} a^7 b x^{10 n}+\frac {28}{11} a^6 b^2 x^{11 n}+\frac {14}{3} a^5 b^3 x^{12 n}+\frac {70}{13} a^4 b^4 x^{13 n}+4 a^3 b^5 x^{14 n}+\frac {28}{15} a^2 b^6 x^{15 n}+\frac {1}{2} a b^7 x^{16 n}+\frac {1}{17} b^8 x^{17 n}}{n}\) |
Input:
Int[x^(-1 + 9*n)*(a + b*x^n)^8,x]
Output:
((a^8*x^(9*n))/9 + (4*a^7*b*x^(10*n))/5 + (28*a^6*b^2*x^(11*n))/11 + (14*a ^5*b^3*x^(12*n))/3 + (70*a^4*b^4*x^(13*n))/13 + 4*a^3*b^5*x^(14*n) + (28*a ^2*b^6*x^(15*n))/15 + (a*b^7*x^(16*n))/2 + (b^8*x^(17*n))/17)/n
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}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Time = 8.53 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {a^{8} x^{9 n}}{9 n}+\frac {4 a^{7} b \,x^{10 n}}{5 n}+\frac {28 a^{6} b^{2} x^{11 n}}{11 n}+\frac {14 a^{5} b^{3} x^{12 n}}{3 n}+\frac {70 a^{4} b^{4} x^{13 n}}{13 n}+\frac {4 a^{3} b^{5} x^{14 n}}{n}+\frac {28 a^{2} b^{6} x^{15 n}}{15 n}+\frac {a \,b^{7} x^{16 n}}{2 n}+\frac {b^{8} x^{17 n}}{17 n}\) | \(136\) |
parallelrisch | \(\frac {12870 x \,x^{8 n} x^{-1+9 n} b^{8}+109395 x \,x^{7 n} x^{-1+9 n} a \,b^{7}+408408 x \,x^{6 n} x^{-1+9 n} a^{2} b^{6}+875160 x \,x^{5 n} x^{-1+9 n} a^{3} b^{5}+1178100 x \,x^{4 n} x^{-1+9 n} a^{4} b^{4}+1021020 x \,x^{3 n} x^{-1+9 n} a^{5} b^{3}+556920 x \,x^{2 n} x^{-1+9 n} a^{6} b^{2}+175032 x \,x^{n} x^{-1+9 n} a^{7} b +24310 x \,x^{-1+9 n} a^{8}}{218790 n}\) | \(179\) |
orering | \(\text {Expression too large to display}\) | \(12553\) |
Input:
int(x^(-1+9*n)*(a+b*x^n)^8,x,method=_RETURNVERBOSE)
Output:
1/17*b^8/n*(x^n)^17+1/2*a*b^7/n*(x^n)^16+28/15*a^2*b^6/n*(x^n)^15+4*a^3*b^ 5/n*(x^n)^14+70/13*a^4*b^4/n*(x^n)^13+14/3*a^5*b^3/n*(x^n)^12+28/11*a^6*b^ 2/n*(x^n)^11+4/5*a^7*b/n*(x^n)^10+1/9*a^8/n*(x^n)^9
Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75 \[ \int x^{-1+9 n} \left (a+b x^n\right )^8 \, dx=\frac {12870 \, b^{8} x^{17 \, n} + 109395 \, a b^{7} x^{16 \, n} + 408408 \, a^{2} b^{6} x^{15 \, n} + 875160 \, a^{3} b^{5} x^{14 \, n} + 1178100 \, a^{4} b^{4} x^{13 \, n} + 1021020 \, a^{5} b^{3} x^{12 \, n} + 556920 \, a^{6} b^{2} x^{11 \, n} + 175032 \, a^{7} b x^{10 \, n} + 24310 \, a^{8} x^{9 \, n}}{218790 \, n} \] Input:
integrate(x^(-1+9*n)*(a+b*x^n)^8,x, algorithm="fricas")
Output:
1/218790*(12870*b^8*x^(17*n) + 109395*a*b^7*x^(16*n) + 408408*a^2*b^6*x^(1 5*n) + 875160*a^3*b^5*x^(14*n) + 1178100*a^4*b^4*x^(13*n) + 1021020*a^5*b^ 3*x^(12*n) + 556920*a^6*b^2*x^(11*n) + 175032*a^7*b*x^(10*n) + 24310*a^8*x ^(9*n))/n
Time = 2.17 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.40 \[ \int x^{-1+9 n} \left (a+b x^n\right )^8 \, dx=\begin {cases} \frac {a^{8} x x^{9 n - 1}}{9 n} + \frac {4 a^{7} b x x^{n} x^{9 n - 1}}{5 n} + \frac {28 a^{6} b^{2} x x^{2 n} x^{9 n - 1}}{11 n} + \frac {14 a^{5} b^{3} x x^{3 n} x^{9 n - 1}}{3 n} + \frac {70 a^{4} b^{4} x x^{4 n} x^{9 n - 1}}{13 n} + \frac {4 a^{3} b^{5} x x^{5 n} x^{9 n - 1}}{n} + \frac {28 a^{2} b^{6} x x^{6 n} x^{9 n - 1}}{15 n} + \frac {a b^{7} x x^{7 n} x^{9 n - 1}}{2 n} + \frac {b^{8} x x^{8 n} x^{9 n - 1}}{17 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \] Input:
integrate(x**(-1+9*n)*(a+b*x**n)**8,x)
Output:
Piecewise((a**8*x*x**(9*n - 1)/(9*n) + 4*a**7*b*x*x**n*x**(9*n - 1)/(5*n) + 28*a**6*b**2*x*x**(2*n)*x**(9*n - 1)/(11*n) + 14*a**5*b**3*x*x**(3*n)*x* *(9*n - 1)/(3*n) + 70*a**4*b**4*x*x**(4*n)*x**(9*n - 1)/(13*n) + 4*a**3*b* *5*x*x**(5*n)*x**(9*n - 1)/n + 28*a**2*b**6*x*x**(6*n)*x**(9*n - 1)/(15*n) + a*b**7*x*x**(7*n)*x**(9*n - 1)/(2*n) + b**8*x*x**(8*n)*x**(9*n - 1)/(17 *n), Ne(n, 0)), ((a + b)**8*log(x), True))
Time = 0.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.89 \[ \int x^{-1+9 n} \left (a+b x^n\right )^8 \, dx=\frac {b^{8} x^{17 \, n}}{17 \, n} + \frac {a b^{7} x^{16 \, n}}{2 \, n} + \frac {28 \, a^{2} b^{6} x^{15 \, n}}{15 \, n} + \frac {4 \, a^{3} b^{5} x^{14 \, n}}{n} + \frac {70 \, a^{4} b^{4} x^{13 \, n}}{13 \, n} + \frac {14 \, a^{5} b^{3} x^{12 \, n}}{3 \, n} + \frac {28 \, a^{6} b^{2} x^{11 \, n}}{11 \, n} + \frac {4 \, a^{7} b x^{10 \, n}}{5 \, n} + \frac {a^{8} x^{9 \, n}}{9 \, n} \] Input:
integrate(x^(-1+9*n)*(a+b*x^n)^8,x, algorithm="maxima")
Output:
1/17*b^8*x^(17*n)/n + 1/2*a*b^7*x^(16*n)/n + 28/15*a^2*b^6*x^(15*n)/n + 4* a^3*b^5*x^(14*n)/n + 70/13*a^4*b^4*x^(13*n)/n + 14/3*a^5*b^3*x^(12*n)/n + 28/11*a^6*b^2*x^(11*n)/n + 4/5*a^7*b*x^(10*n)/n + 1/9*a^8*x^(9*n)/n
\[ \int x^{-1+9 n} \left (a+b x^n\right )^8 \, dx=\int { {\left (b x^{n} + a\right )}^{8} x^{9 \, n - 1} \,d x } \] Input:
integrate(x^(-1+9*n)*(a+b*x^n)^8,x, algorithm="giac")
Output:
integrate((b*x^n + a)^8*x^(9*n - 1), x)
Time = 0.77 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.89 \[ \int x^{-1+9 n} \left (a+b x^n\right )^8 \, dx=\frac {a^8\,x^{9\,n}}{9\,n}+\frac {b^8\,x^{17\,n}}{17\,n}+\frac {28\,a^6\,b^2\,x^{11\,n}}{11\,n}+\frac {14\,a^5\,b^3\,x^{12\,n}}{3\,n}+\frac {70\,a^4\,b^4\,x^{13\,n}}{13\,n}+\frac {4\,a^3\,b^5\,x^{14\,n}}{n}+\frac {28\,a^2\,b^6\,x^{15\,n}}{15\,n}+\frac {4\,a^7\,b\,x^{10\,n}}{5\,n}+\frac {a\,b^7\,x^{16\,n}}{2\,n} \] Input:
int(x^(9*n - 1)*(a + b*x^n)^8,x)
Output:
(a^8*x^(9*n))/(9*n) + (b^8*x^(17*n))/(17*n) + (28*a^6*b^2*x^(11*n))/(11*n) + (14*a^5*b^3*x^(12*n))/(3*n) + (70*a^4*b^4*x^(13*n))/(13*n) + (4*a^3*b^5 *x^(14*n))/n + (28*a^2*b^6*x^(15*n))/(15*n) + (4*a^7*b*x^(10*n))/(5*n) + ( a*b^7*x^(16*n))/(2*n)
Time = 0.20 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.74 \[ \int x^{-1+9 n} \left (a+b x^n\right )^8 \, dx=\frac {x^{9 n} \left (12870 x^{8 n} b^{8}+109395 x^{7 n} a \,b^{7}+408408 x^{6 n} a^{2} b^{6}+875160 x^{5 n} a^{3} b^{5}+1178100 x^{4 n} a^{4} b^{4}+1021020 x^{3 n} a^{5} b^{3}+556920 x^{2 n} a^{6} b^{2}+175032 x^{n} a^{7} b +24310 a^{8}\right )}{218790 n} \] Input:
int(x^(-1+9*n)*(a+b*x^n)^8,x)
Output:
(x**(9*n)*(12870*x**(8*n)*b**8 + 109395*x**(7*n)*a*b**7 + 408408*x**(6*n)* a**2*b**6 + 875160*x**(5*n)*a**3*b**5 + 1178100*x**(4*n)*a**4*b**4 + 10210 20*x**(3*n)*a**5*b**3 + 556920*x**(2*n)*a**6*b**2 + 175032*x**n*a**7*b + 2 4310*a**8))/(218790*n)