Integrand size = 13, antiderivative size = 106 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=-\frac {b^8}{15 x^{15}}-\frac {4 a b^7}{7 x^{14}}-\frac {28 a^2 b^6}{13 x^{13}}-\frac {14 a^3 b^5}{3 x^{12}}-\frac {70 a^4 b^4}{11 x^{11}}-\frac {28 a^5 b^3}{5 x^{10}}-\frac {28 a^6 b^2}{9 x^9}-\frac {a^7 b}{x^8}-\frac {a^8}{7 x^7} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 45} \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=-\frac {a^8}{7 x^7}-\frac {a^7 b}{x^8}-\frac {28 a^6 b^2}{9 x^9}-\frac {28 a^5 b^3}{5 x^{10}}-\frac {70 a^4 b^4}{11 x^{11}}-\frac {14 a^3 b^5}{3 x^{12}}-\frac {28 a^2 b^6}{13 x^{13}}-\frac {4 a b^7}{7 x^{14}}-\frac {b^8}{15 x^{15}} \]
[In]
[Out]
Rule 45
Rule 269
Rubi steps \begin{align*} \text {integral}& = \int \frac {(b+a x)^8}{x^{16}} \, dx \\ & = \int \left (\frac {b^8}{x^{16}}+\frac {8 a b^7}{x^{15}}+\frac {28 a^2 b^6}{x^{14}}+\frac {56 a^3 b^5}{x^{13}}+\frac {70 a^4 b^4}{x^{12}}+\frac {56 a^5 b^3}{x^{11}}+\frac {28 a^6 b^2}{x^{10}}+\frac {8 a^7 b}{x^9}+\frac {a^8}{x^8}\right ) \, 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} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=-\frac {b^8}{15 x^{15}}-\frac {4 a b^7}{7 x^{14}}-\frac {28 a^2 b^6}{13 x^{13}}-\frac {14 a^3 b^5}{3 x^{12}}-\frac {70 a^4 b^4}{11 x^{11}}-\frac {28 a^5 b^3}{5 x^{10}}-\frac {28 a^6 b^2}{9 x^9}-\frac {a^7 b}{x^8}-\frac {a^8}{7 x^7} \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85
method | result | size |
norman | \(\frac {-\frac {1}{15} b^{8}-\frac {4}{7} a \,b^{7} x -\frac {28}{13} a^{2} b^{6} x^{2}-\frac {14}{3} a^{3} b^{5} x^{3}-\frac {70}{11} a^{4} x^{4} b^{4}-\frac {28}{5} a^{5} b^{3} x^{5}-\frac {28}{9} a^{6} b^{2} x^{6}-x^{7} b \,a^{7}-\frac {1}{7} a^{8} x^{8}}{x^{15}}\) | \(90\) |
risch | \(\frac {-\frac {1}{15} b^{8}-\frac {4}{7} a \,b^{7} x -\frac {28}{13} a^{2} b^{6} x^{2}-\frac {14}{3} a^{3} b^{5} x^{3}-\frac {70}{11} a^{4} x^{4} b^{4}-\frac {28}{5} a^{5} b^{3} x^{5}-\frac {28}{9} a^{6} b^{2} x^{6}-x^{7} b \,a^{7}-\frac {1}{7} a^{8} x^{8}}{x^{15}}\) | \(90\) |
gosper | \(-\frac {6435 a^{8} x^{8}+45045 x^{7} b \,a^{7}+140140 a^{6} b^{2} x^{6}+252252 a^{5} b^{3} x^{5}+286650 a^{4} x^{4} b^{4}+210210 a^{3} b^{5} x^{3}+97020 a^{2} b^{6} x^{2}+25740 a \,b^{7} x +3003 b^{8}}{45045 x^{15}}\) | \(91\) |
default | \(-\frac {b^{8}}{15 x^{15}}-\frac {4 a \,b^{7}}{7 x^{14}}-\frac {28 a^{2} b^{6}}{13 x^{13}}-\frac {14 a^{3} b^{5}}{3 x^{12}}-\frac {70 a^{4} b^{4}}{11 x^{11}}-\frac {28 a^{5} b^{3}}{5 x^{10}}-\frac {28 a^{6} b^{2}}{9 x^{9}}-\frac {a^{7} b}{x^{8}}-\frac {a^{8}}{7 x^{7}}\) | \(91\) |
parallelrisch | \(\frac {-6435 a^{8} x^{8}-45045 x^{7} b \,a^{7}-140140 a^{6} b^{2} x^{6}-252252 a^{5} b^{3} x^{5}-286650 a^{4} x^{4} b^{4}-210210 a^{3} b^{5} x^{3}-97020 a^{2} b^{6} x^{2}-25740 a \,b^{7} x -3003 b^{8}}{45045 x^{15}}\) | \(91\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=-\frac {6435 \, a^{8} x^{8} + 45045 \, a^{7} b x^{7} + 140140 \, a^{6} b^{2} x^{6} + 252252 \, a^{5} b^{3} x^{5} + 286650 \, a^{4} b^{4} x^{4} + 210210 \, a^{3} b^{5} x^{3} + 97020 \, a^{2} b^{6} x^{2} + 25740 \, a b^{7} x + 3003 \, b^{8}}{45045 \, x^{15}} \]
[In]
[Out]
Time = 0.47 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=\frac {- 6435 a^{8} x^{8} - 45045 a^{7} b x^{7} - 140140 a^{6} b^{2} x^{6} - 252252 a^{5} b^{3} x^{5} - 286650 a^{4} b^{4} x^{4} - 210210 a^{3} b^{5} x^{3} - 97020 a^{2} b^{6} x^{2} - 25740 a b^{7} x - 3003 b^{8}}{45045 x^{15}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=-\frac {6435 \, a^{8} x^{8} + 45045 \, a^{7} b x^{7} + 140140 \, a^{6} b^{2} x^{6} + 252252 \, a^{5} b^{3} x^{5} + 286650 \, a^{4} b^{4} x^{4} + 210210 \, a^{3} b^{5} x^{3} + 97020 \, a^{2} b^{6} x^{2} + 25740 \, a b^{7} x + 3003 \, b^{8}}{45045 \, x^{15}} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=-\frac {6435 \, a^{8} x^{8} + 45045 \, a^{7} b x^{7} + 140140 \, a^{6} b^{2} x^{6} + 252252 \, a^{5} b^{3} x^{5} + 286650 \, a^{4} b^{4} x^{4} + 210210 \, a^{3} b^{5} x^{3} + 97020 \, a^{2} b^{6} x^{2} + 25740 \, a b^{7} x + 3003 \, b^{8}}{45045 \, x^{15}} \]
[In]
[Out]
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx=-\frac {\frac {a^8\,x^8}{7}+a^7\,b\,x^7+\frac {28\,a^6\,b^2\,x^6}{9}+\frac {28\,a^5\,b^3\,x^5}{5}+\frac {70\,a^4\,b^4\,x^4}{11}+\frac {14\,a^3\,b^5\,x^3}{3}+\frac {28\,a^2\,b^6\,x^2}{13}+\frac {4\,a\,b^7\,x}{7}+\frac {b^8}{15}}{x^{15}} \]
[In]
[Out]