Integrand size = 13, antiderivative size = 129 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=-\frac {a^6 \left (a+\frac {b}{x}\right )^9}{9 b^7}+\frac {3 a^5 \left (a+\frac {b}{x}\right )^{10}}{5 b^7}-\frac {15 a^4 \left (a+\frac {b}{x}\right )^{11}}{11 b^7}+\frac {5 a^3 \left (a+\frac {b}{x}\right )^{12}}{3 b^7}-\frac {15 a^2 \left (a+\frac {b}{x}\right )^{13}}{13 b^7}+\frac {3 a \left (a+\frac {b}{x}\right )^{14}}{7 b^7}-\frac {\left (a+\frac {b}{x}\right )^{15}}{15 b^7} \] Output:
-1/9*a^6*(a+b/x)^9/b^7+3/5*a^5*(a+b/x)^10/b^7-15/11*a^4*(a+b/x)^11/b^7+5/3 *a^3*(a+b/x)^12/b^7-15/13*a^2*(a+b/x)^13/b^7+3/7*a*(a+b/x)^14/b^7-1/15*(a+ b/x)^15/b^7
Time = 0.01 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=-\frac {b^8}{15 x^{15}}-\frac {4 a b^7}{7 x^{14}}-\frac {28 a^2 b^6}{13 x^{13}}-\frac {14 a^3 b^5}{3 x^{12}}-\frac {70 a^4 b^4}{11 x^{11}}-\frac {28 a^5 b^3}{5 x^{10}}-\frac {28 a^6 b^2}{9 x^9}-\frac {a^7 b}{x^8}-\frac {a^8}{7 x^7} \] Input:
Integrate[(a + b/x)^8/x^8,x]
Output:
-1/15*b^8/x^15 - (4*a*b^7)/(7*x^14) - (28*a^2*b^6)/(13*x^13) - (14*a^3*b^5 )/(3*x^12) - (70*a^4*b^4)/(11*x^11) - (28*a^5*b^3)/(5*x^10) - (28*a^6*b^2) /(9*x^9) - (a^7*b)/x^8 - a^8/(7*x^7)
Time = 0.37 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.82, 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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {(a x+b)^8}{x^{16}}dx\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \int \left (\frac {a^8}{x^8}+\frac {8 a^7 b}{x^9}+\frac {28 a^6 b^2}{x^{10}}+\frac {56 a^5 b^3}{x^{11}}+\frac {70 a^4 b^4}{x^{12}}+\frac {56 a^3 b^5}{x^{13}}+\frac {28 a^2 b^6}{x^{14}}+\frac {8 a b^7}{x^{15}}+\frac {b^8}{x^{16}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^8}{7 x^7}-\frac {a^7 b}{x^8}-\frac {28 a^6 b^2}{9 x^9}-\frac {28 a^5 b^3}{5 x^{10}}-\frac {70 a^4 b^4}{11 x^{11}}-\frac {14 a^3 b^5}{3 x^{12}}-\frac {28 a^2 b^6}{13 x^{13}}-\frac {4 a b^7}{7 x^{14}}-\frac {b^8}{15 x^{15}}\) |
Input:
Int[(a + b/x)^8/x^8,x]
Output:
-1/15*b^8/x^15 - (4*a*b^7)/(7*x^14) - (28*a^2*b^6)/(13*x^13) - (14*a^3*b^5 )/(3*x^12) - (70*a^4*b^4)/(11*x^11) - (28*a^5*b^3)/(5*x^10) - (28*a^6*b^2) /(9*x^9) - (a^7*b)/x^8 - a^8/(7*x^7)
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 = 0.70
method | result | size |
norman | \(\frac {-\frac {1}{15} b^{8}-\frac {4}{7} a \,b^{7} x -\frac {28}{13} a^{2} b^{6} x^{2}-\frac {14}{3} a^{3} b^{5} x^{3}-\frac {70}{11} a^{4} x^{4} b^{4}-\frac {28}{5} a^{5} b^{3} x^{5}-\frac {28}{9} a^{6} b^{2} x^{6}-a^{7} b \,x^{7}-\frac {1}{7} a^{8} x^{8}}{x^{15}}\) | \(90\) |
risch | \(\frac {-\frac {1}{15} b^{8}-\frac {4}{7} a \,b^{7} x -\frac {28}{13} a^{2} b^{6} x^{2}-\frac {14}{3} a^{3} b^{5} x^{3}-\frac {70}{11} a^{4} x^{4} b^{4}-\frac {28}{5} a^{5} b^{3} x^{5}-\frac {28}{9} a^{6} b^{2} x^{6}-a^{7} b \,x^{7}-\frac {1}{7} a^{8} x^{8}}{x^{15}}\) | \(90\) |
gosper | \(-\frac {6435 a^{8} x^{8}+45045 a^{7} b \,x^{7}+140140 a^{6} b^{2} x^{6}+252252 a^{5} b^{3} x^{5}+286650 a^{4} x^{4} b^{4}+210210 a^{3} b^{5} x^{3}+97020 a^{2} b^{6} x^{2}+25740 a \,b^{7} x +3003 b^{8}}{45045 x^{15}}\) | \(91\) |
default | \(-\frac {70 a^{4} b^{4}}{11 x^{11}}-\frac {b^{8}}{15 x^{15}}-\frac {a^{8}}{7 x^{7}}-\frac {a^{7} b}{x^{8}}-\frac {28 a^{2} b^{6}}{13 x^{13}}-\frac {28 a^{5} b^{3}}{5 x^{10}}-\frac {4 a \,b^{7}}{7 x^{14}}-\frac {28 a^{6} b^{2}}{9 x^{9}}-\frac {14 a^{3} b^{5}}{3 x^{12}}\) | \(91\) |
parallelrisch | \(\frac {-6435 a^{8} x^{8}-45045 a^{7} b \,x^{7}-140140 a^{6} b^{2} x^{6}-252252 a^{5} b^{3} x^{5}-286650 a^{4} x^{4} b^{4}-210210 a^{3} b^{5} x^{3}-97020 a^{2} b^{6} x^{2}-25740 a \,b^{7} x -3003 b^{8}}{45045 x^{15}}\) | \(91\) |
orering | \(-\frac {\left (6435 a^{8} x^{8}+45045 a^{7} b \,x^{7}+140140 a^{6} b^{2} x^{6}+252252 a^{5} b^{3} x^{5}+286650 a^{4} x^{4} b^{4}+210210 a^{3} b^{5} x^{3}+97020 a^{2} b^{6} x^{2}+25740 a \,b^{7} x +3003 b^{8}\right ) \left (a +\frac {b}{x}\right )^{8}}{45045 x^{7} \left (a x +b \right )^{8}}\) | \(107\) |
Input:
int((a+b/x)^8/x^8,x,method=_RETURNVERBOSE)
Output:
(-1/15*b^8-4/7*a*b^7*x-28/13*a^2*b^6*x^2-14/3*a^3*b^5*x^3-70/11*a^4*x^4*b^ 4-28/5*a^5*b^3*x^5-28/9*a^6*b^2*x^6-a^7*b*x^7-1/7*a^8*x^8)/x^15
Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=-\frac {6435 \, a^{8} x^{8} + 45045 \, a^{7} b x^{7} + 140140 \, a^{6} b^{2} x^{6} + 252252 \, a^{5} b^{3} x^{5} + 286650 \, a^{4} b^{4} x^{4} + 210210 \, a^{3} b^{5} x^{3} + 97020 \, a^{2} b^{6} x^{2} + 25740 \, a b^{7} x + 3003 \, b^{8}}{45045 \, x^{15}} \] Input:
integrate((a+b/x)^8/x^8,x, algorithm="fricas")
Output:
-1/45045*(6435*a^8*x^8 + 45045*a^7*b*x^7 + 140140*a^6*b^2*x^6 + 252252*a^5 *b^3*x^5 + 286650*a^4*b^4*x^4 + 210210*a^3*b^5*x^3 + 97020*a^2*b^6*x^2 + 2 5740*a*b^7*x + 3003*b^8)/x^15
Time = 0.52 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=\frac {- 6435 a^{8} x^{8} - 45045 a^{7} b x^{7} - 140140 a^{6} b^{2} x^{6} - 252252 a^{5} b^{3} x^{5} - 286650 a^{4} b^{4} x^{4} - 210210 a^{3} b^{5} x^{3} - 97020 a^{2} b^{6} x^{2} - 25740 a b^{7} x - 3003 b^{8}}{45045 x^{15}} \] Input:
integrate((a+b/x)**8/x**8,x)
Output:
(-6435*a**8*x**8 - 45045*a**7*b*x**7 - 140140*a**6*b**2*x**6 - 252252*a**5 *b**3*x**5 - 286650*a**4*b**4*x**4 - 210210*a**3*b**5*x**3 - 97020*a**2*b* *6*x**2 - 25740*a*b**7*x - 3003*b**8)/(45045*x**15)
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=-\frac {6435 \, a^{8} x^{8} + 45045 \, a^{7} b x^{7} + 140140 \, a^{6} b^{2} x^{6} + 252252 \, a^{5} b^{3} x^{5} + 286650 \, a^{4} b^{4} x^{4} + 210210 \, a^{3} b^{5} x^{3} + 97020 \, a^{2} b^{6} x^{2} + 25740 \, a b^{7} x + 3003 \, b^{8}}{45045 \, x^{15}} \] Input:
integrate((a+b/x)^8/x^8,x, algorithm="maxima")
Output:
-1/45045*(6435*a^8*x^8 + 45045*a^7*b*x^7 + 140140*a^6*b^2*x^6 + 252252*a^5 *b^3*x^5 + 286650*a^4*b^4*x^4 + 210210*a^3*b^5*x^3 + 97020*a^2*b^6*x^2 + 2 5740*a*b^7*x + 3003*b^8)/x^15
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=-\frac {6435 \, a^{8} x^{8} + 45045 \, a^{7} b x^{7} + 140140 \, a^{6} b^{2} x^{6} + 252252 \, a^{5} b^{3} x^{5} + 286650 \, a^{4} b^{4} x^{4} + 210210 \, a^{3} b^{5} x^{3} + 97020 \, a^{2} b^{6} x^{2} + 25740 \, a b^{7} x + 3003 \, b^{8}}{45045 \, x^{15}} \] Input:
integrate((a+b/x)^8/x^8,x, algorithm="giac")
Output:
-1/45045*(6435*a^8*x^8 + 45045*a^7*b*x^7 + 140140*a^6*b^2*x^6 + 252252*a^5 *b^3*x^5 + 286650*a^4*b^4*x^4 + 210210*a^3*b^5*x^3 + 97020*a^2*b^6*x^2 + 2 5740*a*b^7*x + 3003*b^8)/x^15
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=-\frac {\frac {a^8\,x^8}{7}+a^7\,b\,x^7+\frac {28\,a^6\,b^2\,x^6}{9}+\frac {28\,a^5\,b^3\,x^5}{5}+\frac {70\,a^4\,b^4\,x^4}{11}+\frac {14\,a^3\,b^5\,x^3}{3}+\frac {28\,a^2\,b^6\,x^2}{13}+\frac {4\,a\,b^7\,x}{7}+\frac {b^8}{15}}{x^{15}} \] Input:
int((a + b/x)^8/x^8,x)
Output:
-(b^8/15 + (a^8*x^8)/7 + a^7*b*x^7 + (28*a^2*b^6*x^2)/13 + (14*a^3*b^5*x^3 )/3 + (70*a^4*b^4*x^4)/11 + (28*a^5*b^3*x^5)/5 + (28*a^6*b^2*x^6)/9 + (4*a *b^7*x)/7)/x^15
Time = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=\frac {-6435 a^{8} x^{8}-45045 a^{7} b \,x^{7}-140140 a^{6} b^{2} x^{6}-252252 a^{5} b^{3} x^{5}-286650 a^{4} b^{4} x^{4}-210210 a^{3} b^{5} x^{3}-97020 a^{2} b^{6} x^{2}-25740 a \,b^{7} x -3003 b^{8}}{45045 x^{15}} \] Input:
int((a+b/x)^8/x^8,x)
Output:
( - 6435*a**8*x**8 - 45045*a**7*b*x**7 - 140140*a**6*b**2*x**6 - 252252*a* *5*b**3*x**5 - 286650*a**4*b**4*x**4 - 210210*a**3*b**5*x**3 - 97020*a**2* b**6*x**2 - 25740*a*b**7*x - 3003*b**8)/(45045*x**15)