Integrand size = 13, antiderivative size = 110 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^7} \, dx=\frac {a^5 \left (a+\frac {b}{x}\right )^9}{9 b^6}-\frac {a^4 \left (a+\frac {b}{x}\right )^{10}}{2 b^6}+\frac {10 a^3 \left (a+\frac {b}{x}\right )^{11}}{11 b^6}-\frac {5 a^2 \left (a+\frac {b}{x}\right )^{12}}{6 b^6}+\frac {5 a \left (a+\frac {b}{x}\right )^{13}}{13 b^6}-\frac {\left (a+\frac {b}{x}\right )^{14}}{14 b^6} \] Output:
1/9*a^5*(a+b/x)^9/b^6-1/2*a^4*(a+b/x)^10/b^6+10/11*a^3*(a+b/x)^11/b^6-5/6* a^2*(a+b/x)^12/b^6+5/13*a*(a+b/x)^13/b^6-1/14*(a+b/x)^14/b^6
Time = 0.00 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^7} \, dx=-\frac {b^8}{14 x^{14}}-\frac {8 a b^7}{13 x^{13}}-\frac {7 a^2 b^6}{3 x^{12}}-\frac {56 a^3 b^5}{11 x^{11}}-\frac {7 a^4 b^4}{x^{10}}-\frac {56 a^5 b^3}{9 x^9}-\frac {7 a^6 b^2}{2 x^8}-\frac {8 a^7 b}{7 x^7}-\frac {a^8}{6 x^6} \] Input:
Integrate[(a + b/x)^8/x^7,x]
Output:
-1/14*b^8/x^14 - (8*a*b^7)/(13*x^13) - (7*a^2*b^6)/(3*x^12) - (56*a^3*b^5) /(11*x^11) - (7*a^4*b^4)/x^10 - (56*a^5*b^3)/(9*x^9) - (7*a^6*b^2)/(2*x^8) - (8*a^7*b)/(7*x^7) - a^8/(6*x^6)
Time = 0.36 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {795, 53, 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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^8}{x^7} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {(a x+b)^8}{x^{15}}dx\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \int \left (\frac {a^8}{x^7}+\frac {8 a^7 b}{x^8}+\frac {28 a^6 b^2}{x^9}+\frac {56 a^5 b^3}{x^{10}}+\frac {70 a^4 b^4}{x^{11}}+\frac {56 a^3 b^5}{x^{12}}+\frac {28 a^2 b^6}{x^{13}}+\frac {8 a b^7}{x^{14}}+\frac {b^8}{x^{15}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^8}{6 x^6}-\frac {8 a^7 b}{7 x^7}-\frac {7 a^6 b^2}{2 x^8}-\frac {56 a^5 b^3}{9 x^9}-\frac {7 a^4 b^4}{x^{10}}-\frac {56 a^3 b^5}{11 x^{11}}-\frac {7 a^2 b^6}{3 x^{12}}-\frac {8 a b^7}{13 x^{13}}-\frac {b^8}{14 x^{14}}\) |
Input:
Int[(a + b/x)^8/x^7,x]
Output:
-1/14*b^8/x^14 - (8*a*b^7)/(13*x^13) - (7*a^2*b^6)/(3*x^12) - (56*a^3*b^5) /(11*x^11) - (7*a^4*b^4)/x^10 - (56*a^5*b^3)/(9*x^9) - (7*a^6*b^2)/(2*x^8) - (8*a^7*b)/(7*x^7) - a^8/(6*x^6)
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] && 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])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82
method | result | size |
norman | \(\frac {-\frac {1}{14} b^{8}-\frac {8}{13} a \,b^{7} x -\frac {7}{3} a^{2} b^{6} x^{2}-\frac {56}{11} a^{3} b^{5} x^{3}-7 a^{4} x^{4} b^{4}-\frac {56}{9} a^{5} b^{3} x^{5}-\frac {7}{2} a^{6} b^{2} x^{6}-\frac {8}{7} a^{7} b \,x^{7}-\frac {1}{6} a^{8} x^{8}}{x^{14}}\) | \(90\) |
risch | \(\frac {-\frac {1}{14} b^{8}-\frac {8}{13} a \,b^{7} x -\frac {7}{3} a^{2} b^{6} x^{2}-\frac {56}{11} a^{3} b^{5} x^{3}-7 a^{4} x^{4} b^{4}-\frac {56}{9} a^{5} b^{3} x^{5}-\frac {7}{2} a^{6} b^{2} x^{6}-\frac {8}{7} a^{7} b \,x^{7}-\frac {1}{6} a^{8} x^{8}}{x^{14}}\) | \(90\) |
gosper | \(-\frac {3003 a^{8} x^{8}+20592 a^{7} b \,x^{7}+63063 a^{6} b^{2} x^{6}+112112 a^{5} b^{3} x^{5}+126126 a^{4} x^{4} b^{4}+91728 a^{3} b^{5} x^{3}+42042 a^{2} b^{6} x^{2}+11088 a \,b^{7} x +1287 b^{8}}{18018 x^{14}}\) | \(91\) |
default | \(-\frac {56 a^{3} b^{5}}{11 x^{11}}-\frac {8 a^{7} b}{7 x^{7}}-\frac {7 a^{6} b^{2}}{2 x^{8}}-\frac {8 a \,b^{7}}{13 x^{13}}-\frac {7 a^{4} b^{4}}{x^{10}}-\frac {b^{8}}{14 x^{14}}-\frac {a^{8}}{6 x^{6}}-\frac {56 a^{5} b^{3}}{9 x^{9}}-\frac {7 a^{2} b^{6}}{3 x^{12}}\) | \(91\) |
parallelrisch | \(\frac {-3003 a^{8} x^{8}-20592 a^{7} b \,x^{7}-63063 a^{6} b^{2} x^{6}-112112 a^{5} b^{3} x^{5}-126126 a^{4} x^{4} b^{4}-91728 a^{3} b^{5} x^{3}-42042 a^{2} b^{6} x^{2}-11088 a \,b^{7} x -1287 b^{8}}{18018 x^{14}}\) | \(91\) |
orering | \(-\frac {\left (3003 a^{8} x^{8}+20592 a^{7} b \,x^{7}+63063 a^{6} b^{2} x^{6}+112112 a^{5} b^{3} x^{5}+126126 a^{4} x^{4} b^{4}+91728 a^{3} b^{5} x^{3}+42042 a^{2} b^{6} x^{2}+11088 a \,b^{7} x +1287 b^{8}\right ) \left (a +\frac {b}{x}\right )^{8}}{18018 x^{6} \left (a x +b \right )^{8}}\) | \(107\) |
Input:
int((a+b/x)^8/x^7,x,method=_RETURNVERBOSE)
Output:
(-1/14*b^8-8/13*a*b^7*x-7/3*a^2*b^6*x^2-56/11*a^3*b^5*x^3-7*a^4*x^4*b^4-56 /9*a^5*b^3*x^5-7/2*a^6*b^2*x^6-8/7*a^7*b*x^7-1/6*a^8*x^8)/x^14
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^7} \, dx=-\frac {3003 \, a^{8} x^{8} + 20592 \, a^{7} b x^{7} + 63063 \, a^{6} b^{2} x^{6} + 112112 \, a^{5} b^{3} x^{5} + 126126 \, a^{4} b^{4} x^{4} + 91728 \, a^{3} b^{5} x^{3} + 42042 \, a^{2} b^{6} x^{2} + 11088 \, a b^{7} x + 1287 \, b^{8}}{18018 \, x^{14}} \] Input:
integrate((a+b/x)^8/x^7,x, algorithm="fricas")
Output:
-1/18018*(3003*a^8*x^8 + 20592*a^7*b*x^7 + 63063*a^6*b^2*x^6 + 112112*a^5* b^3*x^5 + 126126*a^4*b^4*x^4 + 91728*a^3*b^5*x^3 + 42042*a^2*b^6*x^2 + 110 88*a*b^7*x + 1287*b^8)/x^14
Time = 0.46 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^7} \, dx=\frac {- 3003 a^{8} x^{8} - 20592 a^{7} b x^{7} - 63063 a^{6} b^{2} x^{6} - 112112 a^{5} b^{3} x^{5} - 126126 a^{4} b^{4} x^{4} - 91728 a^{3} b^{5} x^{3} - 42042 a^{2} b^{6} x^{2} - 11088 a b^{7} x - 1287 b^{8}}{18018 x^{14}} \] Input:
integrate((a+b/x)**8/x**7,x)
Output:
(-3003*a**8*x**8 - 20592*a**7*b*x**7 - 63063*a**6*b**2*x**6 - 112112*a**5* b**3*x**5 - 126126*a**4*b**4*x**4 - 91728*a**3*b**5*x**3 - 42042*a**2*b**6 *x**2 - 11088*a*b**7*x - 1287*b**8)/(18018*x**14)
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^7} \, dx=-\frac {3003 \, a^{8} x^{8} + 20592 \, a^{7} b x^{7} + 63063 \, a^{6} b^{2} x^{6} + 112112 \, a^{5} b^{3} x^{5} + 126126 \, a^{4} b^{4} x^{4} + 91728 \, a^{3} b^{5} x^{3} + 42042 \, a^{2} b^{6} x^{2} + 11088 \, a b^{7} x + 1287 \, b^{8}}{18018 \, x^{14}} \] Input:
integrate((a+b/x)^8/x^7,x, algorithm="maxima")
Output:
-1/18018*(3003*a^8*x^8 + 20592*a^7*b*x^7 + 63063*a^6*b^2*x^6 + 112112*a^5* b^3*x^5 + 126126*a^4*b^4*x^4 + 91728*a^3*b^5*x^3 + 42042*a^2*b^6*x^2 + 110 88*a*b^7*x + 1287*b^8)/x^14
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^7} \, dx=-\frac {3003 \, a^{8} x^{8} + 20592 \, a^{7} b x^{7} + 63063 \, a^{6} b^{2} x^{6} + 112112 \, a^{5} b^{3} x^{5} + 126126 \, a^{4} b^{4} x^{4} + 91728 \, a^{3} b^{5} x^{3} + 42042 \, a^{2} b^{6} x^{2} + 11088 \, a b^{7} x + 1287 \, b^{8}}{18018 \, x^{14}} \] Input:
integrate((a+b/x)^8/x^7,x, algorithm="giac")
Output:
-1/18018*(3003*a^8*x^8 + 20592*a^7*b*x^7 + 63063*a^6*b^2*x^6 + 112112*a^5* b^3*x^5 + 126126*a^4*b^4*x^4 + 91728*a^3*b^5*x^3 + 42042*a^2*b^6*x^2 + 110 88*a*b^7*x + 1287*b^8)/x^14
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^7} \, dx=-\frac {\frac {a^8\,x^8}{6}+\frac {8\,a^7\,b\,x^7}{7}+\frac {7\,a^6\,b^2\,x^6}{2}+\frac {56\,a^5\,b^3\,x^5}{9}+7\,a^4\,b^4\,x^4+\frac {56\,a^3\,b^5\,x^3}{11}+\frac {7\,a^2\,b^6\,x^2}{3}+\frac {8\,a\,b^7\,x}{13}+\frac {b^8}{14}}{x^{14}} \] Input:
int((a + b/x)^8/x^7,x)
Output:
-(b^8/14 + (a^8*x^8)/6 + (8*a^7*b*x^7)/7 + (7*a^2*b^6*x^2)/3 + (56*a^3*b^5 *x^3)/11 + 7*a^4*b^4*x^4 + (56*a^5*b^3*x^5)/9 + (7*a^6*b^2*x^6)/2 + (8*a*b ^7*x)/13)/x^14
Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^7} \, dx=\frac {-3003 a^{8} x^{8}-20592 a^{7} b \,x^{7}-63063 a^{6} b^{2} x^{6}-112112 a^{5} b^{3} x^{5}-126126 a^{4} b^{4} x^{4}-91728 a^{3} b^{5} x^{3}-42042 a^{2} b^{6} x^{2}-11088 a \,b^{7} x -1287 b^{8}}{18018 x^{14}} \] Input:
int((a+b/x)^8/x^7,x)
Output:
( - 3003*a**8*x**8 - 20592*a**7*b*x**7 - 63063*a**6*b**2*x**6 - 112112*a** 5*b**3*x**5 - 126126*a**4*b**4*x**4 - 91728*a**3*b**5*x**3 - 42042*a**2*b* *6*x**2 - 11088*a*b**7*x - 1287*b**8)/(18018*x**14)