Integrand size = 17, antiderivative size = 151 \[ \int x^{-1+8 n} \left (a+b x^n\right )^8 \, dx=\frac {a^8 x^{8 n}}{8 n}+\frac {8 a^7 b x^{9 n}}{9 n}+\frac {14 a^6 b^2 x^{10 n}}{5 n}+\frac {56 a^5 b^3 x^{11 n}}{11 n}+\frac {35 a^4 b^4 x^{12 n}}{6 n}+\frac {56 a^3 b^5 x^{13 n}}{13 n}+\frac {2 a^2 b^6 x^{14 n}}{n}+\frac {8 a b^7 x^{15 n}}{15 n}+\frac {b^8 x^{16 n}}{16 n} \] Output:
1/8*a^8*x^(8*n)/n+8/9*a^7*b*x^(9*n)/n+14/5*a^6*b^2*x^(10*n)/n+56/11*a^5*b^ 3*x^(11*n)/n+35/6*a^4*b^4*x^(12*n)/n+56/13*a^3*b^5*x^(13*n)/n+2*a^2*b^6*x^ (14*n)/n+8/15*a*b^7*x^(15*n)/n+1/16*b^8*x^(16*n)/n
Time = 0.06 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75 \[ \int x^{-1+8 n} \left (a+b x^n\right )^8 \, dx=\frac {x^{8 n} \left (12870 a^8+91520 a^7 b x^n+288288 a^6 b^2 x^{2 n}+524160 a^5 b^3 x^{3 n}+600600 a^4 b^4 x^{4 n}+443520 a^3 b^5 x^{5 n}+205920 a^2 b^6 x^{6 n}+54912 a b^7 x^{7 n}+6435 b^8 x^{8 n}\right )}{102960 n} \] Input:
Integrate[x^(-1 + 8*n)*(a + b*x^n)^8,x]
Output:
(x^(8*n)*(12870*a^8 + 91520*a^7*b*x^n + 288288*a^6*b^2*x^(2*n) + 524160*a^ 5*b^3*x^(3*n) + 600600*a^4*b^4*x^(4*n) + 443520*a^3*b^5*x^(5*n) + 205920*a ^2*b^6*x^(6*n) + 54912*a*b^7*x^(7*n) + 6435*b^8*x^(8*n)))/(102960*n)
Time = 0.42 (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.
\(\displaystyle \int x^{8 n-1} \left (a+b x^n\right )^8 \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int x^{7 n} \left (b x^n+a\right )^8dx^n}{n}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (a^8 x^{7 n}+8 a^7 b x^{8 n}+28 a^6 b^2 x^{9 n}+56 a^5 b^3 x^{10 n}+70 a^4 b^4 x^{11 n}+56 a^3 b^5 x^{12 n}+28 a^2 b^6 x^{13 n}+8 a b^7 x^{14 n}+b^8 x^{15 n}\right )dx^n}{n}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{8} a^8 x^{8 n}+\frac {8}{9} a^7 b x^{9 n}+\frac {14}{5} a^6 b^2 x^{10 n}+\frac {56}{11} a^5 b^3 x^{11 n}+\frac {35}{6} a^4 b^4 x^{12 n}+\frac {56}{13} a^3 b^5 x^{13 n}+2 a^2 b^6 x^{14 n}+\frac {8}{15} a b^7 x^{15 n}+\frac {1}{16} b^8 x^{16 n}}{n}\) |
Input:
Int[x^(-1 + 8*n)*(a + b*x^n)^8,x]
Output:
((a^8*x^(8*n))/8 + (8*a^7*b*x^(9*n))/9 + (14*a^6*b^2*x^(10*n))/5 + (56*a^5 *b^3*x^(11*n))/11 + (35*a^4*b^4*x^(12*n))/6 + (56*a^3*b^5*x^(13*n))/13 + 2 *a^2*b^6*x^(14*n) + (8*a*b^7*x^(15*n))/15 + (b^8*x^(16*n))/16)/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 = 8.92 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.90
method | result | size |
risch | \(\frac {a^{8} x^{8 n}}{8 n}+\frac {8 a^{7} b \,x^{9 n}}{9 n}+\frac {14 a^{6} b^{2} x^{10 n}}{5 n}+\frac {56 a^{5} b^{3} x^{11 n}}{11 n}+\frac {35 a^{4} b^{4} x^{12 n}}{6 n}+\frac {56 a^{3} b^{5} x^{13 n}}{13 n}+\frac {2 a^{2} b^{6} x^{14 n}}{n}+\frac {8 a \,b^{7} x^{15 n}}{15 n}+\frac {b^{8} x^{16 n}}{16 n}\) | \(136\) |
parallelrisch | \(\frac {6435 x \,x^{8 n} x^{-1+8 n} b^{8}+54912 x \,x^{7 n} x^{-1+8 n} a \,b^{7}+205920 x \,x^{6 n} x^{-1+8 n} a^{2} b^{6}+443520 x \,x^{5 n} x^{-1+8 n} a^{3} b^{5}+600600 x \,x^{4 n} x^{-1+8 n} a^{4} b^{4}+524160 x \,x^{3 n} x^{-1+8 n} a^{5} b^{3}+288288 x \,x^{2 n} x^{-1+8 n} a^{6} b^{2}+91520 x \,x^{n} x^{-1+8 n} a^{7} b +12870 x \,x^{-1+8 n} a^{8}}{102960 n}\) | \(179\) |
orering | \(\text {Expression too large to display}\) | \(12553\) |
Input:
int(x^(-1+8*n)*(a+b*x^n)^8,x,method=_RETURNVERBOSE)
Output:
1/16*b^8/n*(x^n)^16+8/15*a*b^7/n*(x^n)^15+2*a^2*b^6/n*(x^n)^14+56/13*a^3*b ^5/n*(x^n)^13+35/6*a^4*b^4/n*(x^n)^12+56/11*a^5*b^3/n*(x^n)^11+14/5*a^6*b^ 2/n*(x^n)^10+8/9*a^7*b/n*(x^n)^9+1/8*a^8/n*(x^n)^8
Time = 0.08 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75 \[ \int x^{-1+8 n} \left (a+b x^n\right )^8 \, dx=\frac {6435 \, b^{8} x^{16 \, n} + 54912 \, a b^{7} x^{15 \, n} + 205920 \, a^{2} b^{6} x^{14 \, n} + 443520 \, a^{3} b^{5} x^{13 \, n} + 600600 \, a^{4} b^{4} x^{12 \, n} + 524160 \, a^{5} b^{3} x^{11 \, n} + 288288 \, a^{6} b^{2} x^{10 \, n} + 91520 \, a^{7} b x^{9 \, n} + 12870 \, a^{8} x^{8 \, n}}{102960 \, n} \] Input:
integrate(x^(-1+8*n)*(a+b*x^n)^8,x, algorithm="fricas")
Output:
1/102960*(6435*b^8*x^(16*n) + 54912*a*b^7*x^(15*n) + 205920*a^2*b^6*x^(14* n) + 443520*a^3*b^5*x^(13*n) + 600600*a^4*b^4*x^(12*n) + 524160*a^5*b^3*x^ (11*n) + 288288*a^6*b^2*x^(10*n) + 91520*a^7*b*x^(9*n) + 12870*a^8*x^(8*n) )/n
Time = 2.11 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.42 \[ \int x^{-1+8 n} \left (a+b x^n\right )^8 \, dx=\begin {cases} \frac {a^{8} x x^{8 n - 1}}{8 n} + \frac {8 a^{7} b x x^{n} x^{8 n - 1}}{9 n} + \frac {14 a^{6} b^{2} x x^{2 n} x^{8 n - 1}}{5 n} + \frac {56 a^{5} b^{3} x x^{3 n} x^{8 n - 1}}{11 n} + \frac {35 a^{4} b^{4} x x^{4 n} x^{8 n - 1}}{6 n} + \frac {56 a^{3} b^{5} x x^{5 n} x^{8 n - 1}}{13 n} + \frac {2 a^{2} b^{6} x x^{6 n} x^{8 n - 1}}{n} + \frac {8 a b^{7} x x^{7 n} x^{8 n - 1}}{15 n} + \frac {b^{8} x x^{8 n} x^{8 n - 1}}{16 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\left (x \right )} & \text {otherwise} \end {cases} \] Input:
integrate(x**(-1+8*n)*(a+b*x**n)**8,x)
Output:
Piecewise((a**8*x*x**(8*n - 1)/(8*n) + 8*a**7*b*x*x**n*x**(8*n - 1)/(9*n) + 14*a**6*b**2*x*x**(2*n)*x**(8*n - 1)/(5*n) + 56*a**5*b**3*x*x**(3*n)*x** (8*n - 1)/(11*n) + 35*a**4*b**4*x*x**(4*n)*x**(8*n - 1)/(6*n) + 56*a**3*b* *5*x*x**(5*n)*x**(8*n - 1)/(13*n) + 2*a**2*b**6*x*x**(6*n)*x**(8*n - 1)/n + 8*a*b**7*x*x**(7*n)*x**(8*n - 1)/(15*n) + b**8*x*x**(8*n)*x**(8*n - 1)/( 16*n), Ne(n, 0)), ((a + b)**8*log(x), True))
Time = 0.07 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.89 \[ \int x^{-1+8 n} \left (a+b x^n\right )^8 \, dx=\frac {b^{8} x^{16 \, n}}{16 \, n} + \frac {8 \, a b^{7} x^{15 \, n}}{15 \, n} + \frac {2 \, a^{2} b^{6} x^{14 \, n}}{n} + \frac {56 \, a^{3} b^{5} x^{13 \, n}}{13 \, n} + \frac {35 \, a^{4} b^{4} x^{12 \, n}}{6 \, n} + \frac {56 \, a^{5} b^{3} x^{11 \, n}}{11 \, n} + \frac {14 \, a^{6} b^{2} x^{10 \, n}}{5 \, n} + \frac {8 \, a^{7} b x^{9 \, n}}{9 \, n} + \frac {a^{8} x^{8 \, n}}{8 \, n} \] Input:
integrate(x^(-1+8*n)*(a+b*x^n)^8,x, algorithm="maxima")
Output:
1/16*b^8*x^(16*n)/n + 8/15*a*b^7*x^(15*n)/n + 2*a^2*b^6*x^(14*n)/n + 56/13 *a^3*b^5*x^(13*n)/n + 35/6*a^4*b^4*x^(12*n)/n + 56/11*a^5*b^3*x^(11*n)/n + 14/5*a^6*b^2*x^(10*n)/n + 8/9*a^7*b*x^(9*n)/n + 1/8*a^8*x^(8*n)/n
\[ \int x^{-1+8 n} \left (a+b x^n\right )^8 \, dx=\int { {\left (b x^{n} + a\right )}^{8} x^{8 \, n - 1} \,d x } \] Input:
integrate(x^(-1+8*n)*(a+b*x^n)^8,x, algorithm="giac")
Output:
integrate((b*x^n + a)^8*x^(8*n - 1), x)
Time = 0.72 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.89 \[ \int x^{-1+8 n} \left (a+b x^n\right )^8 \, dx=\frac {a^8\,x^{8\,n}}{8\,n}+\frac {b^8\,x^{16\,n}}{16\,n}+\frac {14\,a^6\,b^2\,x^{10\,n}}{5\,n}+\frac {56\,a^5\,b^3\,x^{11\,n}}{11\,n}+\frac {35\,a^4\,b^4\,x^{12\,n}}{6\,n}+\frac {56\,a^3\,b^5\,x^{13\,n}}{13\,n}+\frac {2\,a^2\,b^6\,x^{14\,n}}{n}+\frac {8\,a^7\,b\,x^{9\,n}}{9\,n}+\frac {8\,a\,b^7\,x^{15\,n}}{15\,n} \] Input:
int(x^(8*n - 1)*(a + b*x^n)^8,x)
Output:
(a^8*x^(8*n))/(8*n) + (b^8*x^(16*n))/(16*n) + (14*a^6*b^2*x^(10*n))/(5*n) + (56*a^5*b^3*x^(11*n))/(11*n) + (35*a^4*b^4*x^(12*n))/(6*n) + (56*a^3*b^5 *x^(13*n))/(13*n) + (2*a^2*b^6*x^(14*n))/n + (8*a^7*b*x^(9*n))/(9*n) + (8* a*b^7*x^(15*n))/(15*n)
Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.74 \[ \int x^{-1+8 n} \left (a+b x^n\right )^8 \, dx=\frac {x^{8 n} \left (6435 x^{8 n} b^{8}+54912 x^{7 n} a \,b^{7}+205920 x^{6 n} a^{2} b^{6}+443520 x^{5 n} a^{3} b^{5}+600600 x^{4 n} a^{4} b^{4}+524160 x^{3 n} a^{5} b^{3}+288288 x^{2 n} a^{6} b^{2}+91520 x^{n} a^{7} b +12870 a^{8}\right )}{102960 n} \] Input:
int(x^(-1+8*n)*(a+b*x^n)^8,x)
Output:
(x**(8*n)*(6435*x**(8*n)*b**8 + 54912*x**(7*n)*a*b**7 + 205920*x**(6*n)*a* *2*b**6 + 443520*x**(5*n)*a**3*b**5 + 600600*x**(4*n)*a**4*b**4 + 524160*x **(3*n)*a**5*b**3 + 288288*x**(2*n)*a**6*b**2 + 91520*x**n*a**7*b + 12870* a**8))/(102960*n)