Integrand size = 17, antiderivative size = 128 \[ \int x^{-1+6 n} \left (a+b x^n\right )^8 \, dx=-\frac {a^5 \left (a+b x^n\right )^9}{9 b^6 n}+\frac {a^4 \left (a+b x^n\right )^{10}}{2 b^6 n}-\frac {10 a^3 \left (a+b x^n\right )^{11}}{11 b^6 n}+\frac {5 a^2 \left (a+b x^n\right )^{12}}{6 b^6 n}-\frac {5 a \left (a+b x^n\right )^{13}}{13 b^6 n}+\frac {\left (a+b x^n\right )^{14}}{14 b^6 n} \] Output:
-1/9*a^5*(a+b*x^n)^9/b^6/n+1/2*a^4*(a+b*x^n)^10/b^6/n-10/11*a^3*(a+b*x^n)^ 11/b^6/n+5/6*a^2*(a+b*x^n)^12/b^6/n-5/13*a*(a+b*x^n)^13/b^6/n+1/14*(a+b*x^ n)^14/b^6/n
Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.88 \[ \int x^{-1+6 n} \left (a+b x^n\right )^8 \, dx=\frac {x^{6 n} \left (3003 a^8+20592 a^7 b x^n+63063 a^6 b^2 x^{2 n}+112112 a^5 b^3 x^{3 n}+126126 a^4 b^4 x^{4 n}+91728 a^3 b^5 x^{5 n}+42042 a^2 b^6 x^{6 n}+11088 a b^7 x^{7 n}+1287 b^8 x^{8 n}\right )}{18018 n} \] Input:
Integrate[x^(-1 + 6*n)*(a + b*x^n)^8,x]
Output:
(x^(6*n)*(3003*a^8 + 20592*a^7*b*x^n + 63063*a^6*b^2*x^(2*n) + 112112*a^5* b^3*x^(3*n) + 126126*a^4*b^4*x^(4*n) + 91728*a^3*b^5*x^(5*n) + 42042*a^2*b ^6*x^(6*n) + 11088*a*b^7*x^(7*n) + 1287*b^8*x^(8*n)))/(18018*n)
Time = 0.40 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.89, 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.
\(\displaystyle \int x^{6 n-1} \left (a+b x^n\right )^8 \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int x^{5 n} \left (b x^n+a\right )^8dx^n}{n}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (\frac {\left (b x^n+a\right )^{13}}{b^5}-\frac {5 a \left (b x^n+a\right )^{12}}{b^5}+\frac {10 a^2 \left (b x^n+a\right )^{11}}{b^5}-\frac {10 a^3 \left (b x^n+a\right )^{10}}{b^5}+\frac {5 a^4 \left (b x^n+a\right )^9}{b^5}-\frac {a^5 \left (b x^n+a\right )^8}{b^5}\right )dx^n}{n}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {a^5 \left (a+b x^n\right )^9}{9 b^6}+\frac {a^4 \left (a+b x^n\right )^{10}}{2 b^6}-\frac {10 a^3 \left (a+b x^n\right )^{11}}{11 b^6}+\frac {5 a^2 \left (a+b x^n\right )^{12}}{6 b^6}+\frac {\left (a+b x^n\right )^{14}}{14 b^6}-\frac {5 a \left (a+b x^n\right )^{13}}{13 b^6}}{n}\) |
Input:
Int[x^(-1 + 6*n)*(a + b*x^n)^8,x]
Output:
(-1/9*(a^5*(a + b*x^n)^9)/b^6 + (a^4*(a + b*x^n)^10)/(2*b^6) - (10*a^3*(a + b*x^n)^11)/(11*b^6) + (5*a^2*(a + b*x^n)^12)/(6*b^6) - (5*a*(a + b*x^n)^ 13)/(13*b^6) + (a + b*x^n)^14/(14*b^6))/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 = 8.63 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.06
method | result | size |
risch | \(\frac {b^{8} x^{14 n}}{14 n}+\frac {8 a \,b^{7} x^{13 n}}{13 n}+\frac {7 a^{2} b^{6} x^{12 n}}{3 n}+\frac {56 a^{3} b^{5} x^{11 n}}{11 n}+\frac {7 a^{4} b^{4} x^{10 n}}{n}+\frac {56 a^{5} b^{3} x^{9 n}}{9 n}+\frac {7 a^{6} b^{2} x^{8 n}}{2 n}+\frac {8 a^{7} b \,x^{7 n}}{7 n}+\frac {a^{8} x^{6 n}}{6 n}\) | \(136\) |
parallelrisch | \(\frac {1287 x \,x^{8 n} x^{-1+6 n} b^{8}+11088 x \,x^{7 n} x^{-1+6 n} a \,b^{7}+42042 x \,x^{6 n} x^{-1+6 n} a^{2} b^{6}+91728 x \,x^{5 n} x^{-1+6 n} a^{3} b^{5}+126126 x \,x^{4 n} x^{-1+6 n} a^{4} b^{4}+112112 x \,x^{3 n} x^{-1+6 n} a^{5} b^{3}+63063 x \,x^{2 n} x^{-1+6 n} a^{6} b^{2}+20592 x \,x^{n} x^{-1+6 n} a^{7} b +3003 x \,x^{-1+6 n} a^{8}}{18018 n}\) | \(179\) |
orering | \(\text {Expression too large to display}\) | \(12553\) |
Input:
int(x^(-1+6*n)*(a+b*x^n)^8,x,method=_RETURNVERBOSE)
Output:
1/14*b^8/n*(x^n)^14+8/13*a*b^7/n*(x^n)^13+7/3*a^2*b^6/n*(x^n)^12+56/11*a^3 *b^5/n*(x^n)^11+7*a^4*b^4/n*(x^n)^10+56/9*a^5*b^3/n*(x^n)^9+7/2*a^6*b^2/n* (x^n)^8+8/7*a^7*b/n*(x^n)^7+1/6*a^8/n*(x^n)^6
Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.88 \[ \int x^{-1+6 n} \left (a+b x^n\right )^8 \, dx=\frac {1287 \, b^{8} x^{14 \, n} + 11088 \, a b^{7} x^{13 \, n} + 42042 \, a^{2} b^{6} x^{12 \, n} + 91728 \, a^{3} b^{5} x^{11 \, n} + 126126 \, a^{4} b^{4} x^{10 \, n} + 112112 \, a^{5} b^{3} x^{9 \, n} + 63063 \, a^{6} b^{2} x^{8 \, n} + 20592 \, a^{7} b x^{7 \, n} + 3003 \, a^{8} x^{6 \, n}}{18018 \, n} \] Input:
integrate(x^(-1+6*n)*(a+b*x^n)^8,x, algorithm="fricas")
Output:
1/18018*(1287*b^8*x^(14*n) + 11088*a*b^7*x^(13*n) + 42042*a^2*b^6*x^(12*n) + 91728*a^3*b^5*x^(11*n) + 126126*a^4*b^4*x^(10*n) + 112112*a^5*b^3*x^(9* n) + 63063*a^6*b^2*x^(8*n) + 20592*a^7*b*x^(7*n) + 3003*a^8*x^(6*n))/n
Time = 2.16 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.67 \[ \int x^{-1+6 n} \left (a+b x^n\right )^8 \, dx=\begin {cases} \frac {a^{8} x x^{6 n - 1}}{6 n} + \frac {8 a^{7} b x x^{n} x^{6 n - 1}}{7 n} + \frac {7 a^{6} b^{2} x x^{2 n} x^{6 n - 1}}{2 n} + \frac {56 a^{5} b^{3} x x^{3 n} x^{6 n - 1}}{9 n} + \frac {7 a^{4} b^{4} x x^{4 n} x^{6 n - 1}}{n} + \frac {56 a^{3} b^{5} x x^{5 n} x^{6 n - 1}}{11 n} + \frac {7 a^{2} b^{6} x x^{6 n} x^{6 n - 1}}{3 n} + \frac {8 a b^{7} x x^{7 n} x^{6 n - 1}}{13 n} + \frac {b^{8} x x^{8 n} x^{6 n - 1}}{14 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \] Input:
integrate(x**(-1+6*n)*(a+b*x**n)**8,x)
Output:
Piecewise((a**8*x*x**(6*n - 1)/(6*n) + 8*a**7*b*x*x**n*x**(6*n - 1)/(7*n) + 7*a**6*b**2*x*x**(2*n)*x**(6*n - 1)/(2*n) + 56*a**5*b**3*x*x**(3*n)*x**( 6*n - 1)/(9*n) + 7*a**4*b**4*x*x**(4*n)*x**(6*n - 1)/n + 56*a**3*b**5*x*x* *(5*n)*x**(6*n - 1)/(11*n) + 7*a**2*b**6*x*x**(6*n)*x**(6*n - 1)/(3*n) + 8 *a*b**7*x*x**(7*n)*x**(6*n - 1)/(13*n) + b**8*x*x**(8*n)*x**(6*n - 1)/(14* n), Ne(n, 0)), ((a + b)**8*log(x), True))
Time = 0.04 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.05 \[ \int x^{-1+6 n} \left (a+b x^n\right )^8 \, dx=\frac {b^{8} x^{14 \, n}}{14 \, n} + \frac {8 \, a b^{7} x^{13 \, n}}{13 \, n} + \frac {7 \, a^{2} b^{6} x^{12 \, n}}{3 \, n} + \frac {56 \, a^{3} b^{5} x^{11 \, n}}{11 \, n} + \frac {7 \, a^{4} b^{4} x^{10 \, n}}{n} + \frac {56 \, a^{5} b^{3} x^{9 \, n}}{9 \, n} + \frac {7 \, a^{6} b^{2} x^{8 \, n}}{2 \, n} + \frac {8 \, a^{7} b x^{7 \, n}}{7 \, n} + \frac {a^{8} x^{6 \, n}}{6 \, n} \] Input:
integrate(x^(-1+6*n)*(a+b*x^n)^8,x, algorithm="maxima")
Output:
1/14*b^8*x^(14*n)/n + 8/13*a*b^7*x^(13*n)/n + 7/3*a^2*b^6*x^(12*n)/n + 56/ 11*a^3*b^5*x^(11*n)/n + 7*a^4*b^4*x^(10*n)/n + 56/9*a^5*b^3*x^(9*n)/n + 7/ 2*a^6*b^2*x^(8*n)/n + 8/7*a^7*b*x^(7*n)/n + 1/6*a^8*x^(6*n)/n
\[ \int x^{-1+6 n} \left (a+b x^n\right )^8 \, dx=\int { {\left (b x^{n} + a\right )}^{8} x^{6 \, n - 1} \,d x } \] Input:
integrate(x^(-1+6*n)*(a+b*x^n)^8,x, algorithm="giac")
Output:
integrate((b*x^n + a)^8*x^(6*n - 1), x)
Time = 0.70 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.05 \[ \int x^{-1+6 n} \left (a+b x^n\right )^8 \, dx=\frac {a^8\,x^{6\,n}}{6\,n}+\frac {b^8\,x^{14\,n}}{14\,n}+\frac {7\,a^6\,b^2\,x^{8\,n}}{2\,n}+\frac {56\,a^5\,b^3\,x^{9\,n}}{9\,n}+\frac {7\,a^4\,b^4\,x^{10\,n}}{n}+\frac {56\,a^3\,b^5\,x^{11\,n}}{11\,n}+\frac {7\,a^2\,b^6\,x^{12\,n}}{3\,n}+\frac {8\,a^7\,b\,x^{7\,n}}{7\,n}+\frac {8\,a\,b^7\,x^{13\,n}}{13\,n} \] Input:
int(x^(6*n - 1)*(a + b*x^n)^8,x)
Output:
(a^8*x^(6*n))/(6*n) + (b^8*x^(14*n))/(14*n) + (7*a^6*b^2*x^(8*n))/(2*n) + (56*a^5*b^3*x^(9*n))/(9*n) + (7*a^4*b^4*x^(10*n))/n + (56*a^3*b^5*x^(11*n) )/(11*n) + (7*a^2*b^6*x^(12*n))/(3*n) + (8*a^7*b*x^(7*n))/(7*n) + (8*a*b^7 *x^(13*n))/(13*n)
Time = 0.22 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.87 \[ \int x^{-1+6 n} \left (a+b x^n\right )^8 \, dx=\frac {x^{6 n} \left (1287 x^{8 n} b^{8}+11088 x^{7 n} a \,b^{7}+42042 x^{6 n} a^{2} b^{6}+91728 x^{5 n} a^{3} b^{5}+126126 x^{4 n} a^{4} b^{4}+112112 x^{3 n} a^{5} b^{3}+63063 x^{2 n} a^{6} b^{2}+20592 x^{n} a^{7} b +3003 a^{8}\right )}{18018 n} \] Input:
int(x^(-1+6*n)*(a+b*x^n)^8,x)
Output:
(x**(6*n)*(1287*x**(8*n)*b**8 + 11088*x**(7*n)*a*b**7 + 42042*x**(6*n)*a** 2*b**6 + 91728*x**(5*n)*a**3*b**5 + 126126*x**(4*n)*a**4*b**4 + 112112*x** (3*n)*a**5*b**3 + 63063*x**(2*n)*a**6*b**2 + 20592*x**n*a**7*b + 3003*a**8 ))/(18018*n)