Integrand size = 17, antiderivative size = 150 \[ \int x^{-1+7 n} \left (a+b x^n\right )^8 \, dx=\frac {a^6 \left (a+b x^n\right )^9}{9 b^7 n}-\frac {3 a^5 \left (a+b x^n\right )^{10}}{5 b^7 n}+\frac {15 a^4 \left (a+b x^n\right )^{11}}{11 b^7 n}-\frac {5 a^3 \left (a+b x^n\right )^{12}}{3 b^7 n}+\frac {15 a^2 \left (a+b x^n\right )^{13}}{13 b^7 n}-\frac {3 a \left (a+b x^n\right )^{14}}{7 b^7 n}+\frac {\left (a+b x^n\right )^{15}}{15 b^7 n} \] Output:
1/9*a^6*(a+b*x^n)^9/b^7/n-3/5*a^5*(a+b*x^n)^10/b^7/n+15/11*a^4*(a+b*x^n)^1 1/b^7/n-5/3*a^3*(a+b*x^n)^12/b^7/n+15/13*a^2*(a+b*x^n)^13/b^7/n-3/7*a*(a+b *x^n)^14/b^7/n+1/15*(a+b*x^n)^15/b^7/n
Time = 0.06 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75 \[ \int x^{-1+7 n} \left (a+b x^n\right )^8 \, dx=\frac {x^{7 n} \left (6435 a^8+45045 a^7 b x^n+140140 a^6 b^2 x^{2 n}+252252 a^5 b^3 x^{3 n}+286650 a^4 b^4 x^{4 n}+210210 a^3 b^5 x^{5 n}+97020 a^2 b^6 x^{6 n}+25740 a b^7 x^{7 n}+3003 b^8 x^{8 n}\right )}{45045 n} \] Input:
Integrate[x^(-1 + 7*n)*(a + b*x^n)^8,x]
Output:
(x^(7*n)*(6435*a^8 + 45045*a^7*b*x^n + 140140*a^6*b^2*x^(2*n) + 252252*a^5 *b^3*x^(3*n) + 286650*a^4*b^4*x^(4*n) + 210210*a^3*b^5*x^(5*n) + 97020*a^2 *b^6*x^(6*n) + 25740*a*b^7*x^(7*n) + 3003*b^8*x^(8*n)))/(45045*n)
Time = 0.43 (sec) , antiderivative size = 133, 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.
\(\displaystyle \int x^{7 n-1} \left (a+b x^n\right )^8 \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int x^{6 n} \left (b x^n+a\right )^8dx^n}{n}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (\frac {\left (b x^n+a\right )^{14}}{b^6}-\frac {6 a \left (b x^n+a\right )^{13}}{b^6}+\frac {15 a^2 \left (b x^n+a\right )^{12}}{b^6}-\frac {20 a^3 \left (b x^n+a\right )^{11}}{b^6}+\frac {15 a^4 \left (b x^n+a\right )^{10}}{b^6}-\frac {6 a^5 \left (b x^n+a\right )^9}{b^6}+\frac {a^6 \left (b x^n+a\right )^8}{b^6}\right )dx^n}{n}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {a^6 \left (a+b x^n\right )^9}{9 b^7}-\frac {3 a^5 \left (a+b x^n\right )^{10}}{5 b^7}+\frac {15 a^4 \left (a+b x^n\right )^{11}}{11 b^7}-\frac {5 a^3 \left (a+b x^n\right )^{12}}{3 b^7}+\frac {15 a^2 \left (a+b x^n\right )^{13}}{13 b^7}+\frac {\left (a+b x^n\right )^{15}}{15 b^7}-\frac {3 a \left (a+b x^n\right )^{14}}{7 b^7}}{n}\) |
Input:
Int[x^(-1 + 7*n)*(a + b*x^n)^8,x]
Output:
((a^6*(a + b*x^n)^9)/(9*b^7) - (3*a^5*(a + b*x^n)^10)/(5*b^7) + (15*a^4*(a + b*x^n)^11)/(11*b^7) - (5*a^3*(a + b*x^n)^12)/(3*b^7) + (15*a^2*(a + b*x ^n)^13)/(13*b^7) - (3*a*(a + b*x^n)^14)/(7*b^7) + (a + b*x^n)^15/(15*b^7)) /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 = 8.57 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {b^{8} x^{15 n}}{15 n}+\frac {4 a \,b^{7} x^{14 n}}{7 n}+\frac {28 a^{2} b^{6} x^{13 n}}{13 n}+\frac {14 a^{3} b^{5} x^{12 n}}{3 n}+\frac {70 a^{4} b^{4} x^{11 n}}{11 n}+\frac {28 a^{5} b^{3} x^{10 n}}{5 n}+\frac {28 a^{6} b^{2} x^{9 n}}{9 n}+\frac {a^{7} b \,x^{8 n}}{n}+\frac {a^{8} x^{7 n}}{7 n}\) | \(135\) |
parallelrisch | \(\frac {3003 x \,x^{8 n} x^{-1+7 n} b^{8}+25740 x \,x^{7 n} x^{-1+7 n} a \,b^{7}+97020 x \,x^{6 n} x^{-1+7 n} a^{2} b^{6}+210210 x \,x^{5 n} x^{-1+7 n} a^{3} b^{5}+286650 x \,x^{4 n} x^{-1+7 n} a^{4} b^{4}+252252 x \,x^{3 n} x^{-1+7 n} a^{5} b^{3}+140140 x \,x^{2 n} x^{-1+7 n} a^{6} b^{2}+45045 x \,x^{n} x^{-1+7 n} a^{7} b +6435 x \,x^{-1+7 n} a^{8}}{45045 n}\) | \(179\) |
orering | \(\text {Expression too large to display}\) | \(12553\) |
Input:
int(x^(-1+7*n)*(a+b*x^n)^8,x,method=_RETURNVERBOSE)
Output:
1/15*b^8/n*(x^n)^15+4/7*a*b^7/n*(x^n)^14+28/13*a^2*b^6/n*(x^n)^13+14/3*a^3 *b^5/n*(x^n)^12+70/11*a^4*b^4/n*(x^n)^11+28/5*a^5*b^3/n*(x^n)^10+28/9*a^6* b^2/n*(x^n)^9+a^7*b/n*(x^n)^8+1/7*a^8/n*(x^n)^7
Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75 \[ \int x^{-1+7 n} \left (a+b x^n\right )^8 \, dx=\frac {3003 \, b^{8} x^{15 \, n} + 25740 \, a b^{7} x^{14 \, n} + 97020 \, a^{2} b^{6} x^{13 \, n} + 210210 \, a^{3} b^{5} x^{12 \, n} + 286650 \, a^{4} b^{4} x^{11 \, n} + 252252 \, a^{5} b^{3} x^{10 \, n} + 140140 \, a^{6} b^{2} x^{9 \, n} + 45045 \, a^{7} b x^{8 \, n} + 6435 \, a^{8} x^{7 \, n}}{45045 \, n} \] Input:
integrate(x^(-1+7*n)*(a+b*x^n)^8,x, algorithm="fricas")
Output:
1/45045*(3003*b^8*x^(15*n) + 25740*a*b^7*x^(14*n) + 97020*a^2*b^6*x^(13*n) + 210210*a^3*b^5*x^(12*n) + 286650*a^4*b^4*x^(11*n) + 252252*a^5*b^3*x^(1 0*n) + 140140*a^6*b^2*x^(9*n) + 45045*a^7*b*x^(8*n) + 6435*a^8*x^(7*n))/n
Time = 2.27 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.41 \[ \int x^{-1+7 n} \left (a+b x^n\right )^8 \, dx=\begin {cases} \frac {a^{8} x x^{7 n - 1}}{7 n} + \frac {a^{7} b x x^{n} x^{7 n - 1}}{n} + \frac {28 a^{6} b^{2} x x^{2 n} x^{7 n - 1}}{9 n} + \frac {28 a^{5} b^{3} x x^{3 n} x^{7 n - 1}}{5 n} + \frac {70 a^{4} b^{4} x x^{4 n} x^{7 n - 1}}{11 n} + \frac {14 a^{3} b^{5} x x^{5 n} x^{7 n - 1}}{3 n} + \frac {28 a^{2} b^{6} x x^{6 n} x^{7 n - 1}}{13 n} + \frac {4 a b^{7} x x^{7 n} x^{7 n - 1}}{7 n} + \frac {b^{8} x x^{8 n} x^{7 n - 1}}{15 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \] Input:
integrate(x**(-1+7*n)*(a+b*x**n)**8,x)
Output:
Piecewise((a**8*x*x**(7*n - 1)/(7*n) + a**7*b*x*x**n*x**(7*n - 1)/n + 28*a **6*b**2*x*x**(2*n)*x**(7*n - 1)/(9*n) + 28*a**5*b**3*x*x**(3*n)*x**(7*n - 1)/(5*n) + 70*a**4*b**4*x*x**(4*n)*x**(7*n - 1)/(11*n) + 14*a**3*b**5*x*x **(5*n)*x**(7*n - 1)/(3*n) + 28*a**2*b**6*x*x**(6*n)*x**(7*n - 1)/(13*n) + 4*a*b**7*x*x**(7*n)*x**(7*n - 1)/(7*n) + b**8*x*x**(8*n)*x**(7*n - 1)/(15 *n), Ne(n, 0)), ((a + b)**8*log(x), True))
Time = 0.04 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.89 \[ \int x^{-1+7 n} \left (a+b x^n\right )^8 \, dx=\frac {b^{8} x^{15 \, n}}{15 \, n} + \frac {4 \, a b^{7} x^{14 \, n}}{7 \, n} + \frac {28 \, a^{2} b^{6} x^{13 \, n}}{13 \, n} + \frac {14 \, a^{3} b^{5} x^{12 \, n}}{3 \, n} + \frac {70 \, a^{4} b^{4} x^{11 \, n}}{11 \, n} + \frac {28 \, a^{5} b^{3} x^{10 \, n}}{5 \, n} + \frac {28 \, a^{6} b^{2} x^{9 \, n}}{9 \, n} + \frac {a^{7} b x^{8 \, n}}{n} + \frac {a^{8} x^{7 \, n}}{7 \, n} \] Input:
integrate(x^(-1+7*n)*(a+b*x^n)^8,x, algorithm="maxima")
Output:
1/15*b^8*x^(15*n)/n + 4/7*a*b^7*x^(14*n)/n + 28/13*a^2*b^6*x^(13*n)/n + 14 /3*a^3*b^5*x^(12*n)/n + 70/11*a^4*b^4*x^(11*n)/n + 28/5*a^5*b^3*x^(10*n)/n + 28/9*a^6*b^2*x^(9*n)/n + a^7*b*x^(8*n)/n + 1/7*a^8*x^(7*n)/n
\[ \int x^{-1+7 n} \left (a+b x^n\right )^8 \, dx=\int { {\left (b x^{n} + a\right )}^{8} x^{7 \, n - 1} \,d x } \] Input:
integrate(x^(-1+7*n)*(a+b*x^n)^8,x, algorithm="giac")
Output:
integrate((b*x^n + a)^8*x^(7*n - 1), x)
Time = 0.67 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.89 \[ \int x^{-1+7 n} \left (a+b x^n\right )^8 \, dx=\frac {a^8\,x^{7\,n}}{7\,n}+\frac {b^8\,x^{15\,n}}{15\,n}+\frac {28\,a^6\,b^2\,x^{9\,n}}{9\,n}+\frac {28\,a^5\,b^3\,x^{10\,n}}{5\,n}+\frac {70\,a^4\,b^4\,x^{11\,n}}{11\,n}+\frac {14\,a^3\,b^5\,x^{12\,n}}{3\,n}+\frac {28\,a^2\,b^6\,x^{13\,n}}{13\,n}+\frac {a^7\,b\,x^{8\,n}}{n}+\frac {4\,a\,b^7\,x^{14\,n}}{7\,n} \] Input:
int(x^(7*n - 1)*(a + b*x^n)^8,x)
Output:
(a^8*x^(7*n))/(7*n) + (b^8*x^(15*n))/(15*n) + (28*a^6*b^2*x^(9*n))/(9*n) + (28*a^5*b^3*x^(10*n))/(5*n) + (70*a^4*b^4*x^(11*n))/(11*n) + (14*a^3*b^5* x^(12*n))/(3*n) + (28*a^2*b^6*x^(13*n))/(13*n) + (a^7*b*x^(8*n))/n + (4*a* b^7*x^(14*n))/(7*n)
Time = 0.21 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.74 \[ \int x^{-1+7 n} \left (a+b x^n\right )^8 \, dx=\frac {x^{7 n} \left (3003 x^{8 n} b^{8}+25740 x^{7 n} a \,b^{7}+97020 x^{6 n} a^{2} b^{6}+210210 x^{5 n} a^{3} b^{5}+286650 x^{4 n} a^{4} b^{4}+252252 x^{3 n} a^{5} b^{3}+140140 x^{2 n} a^{6} b^{2}+45045 x^{n} a^{7} b +6435 a^{8}\right )}{45045 n} \] Input:
int(x^(-1+7*n)*(a+b*x^n)^8,x)
Output:
(x**(7*n)*(3003*x**(8*n)*b**8 + 25740*x**(7*n)*a*b**7 + 97020*x**(6*n)*a** 2*b**6 + 210210*x**(5*n)*a**3*b**5 + 286650*x**(4*n)*a**4*b**4 + 252252*x* *(3*n)*a**5*b**3 + 140140*x**(2*n)*a**6*b**2 + 45045*x**n*a**7*b + 6435*a* *8))/(45045*n)