Integrand size = 11, antiderivative size = 109 \[ \int \frac {(a+b x)^9}{x^{10}} \, dx=-\frac {a^9}{9 x^9}-\frac {9 a^8 b}{8 x^8}-\frac {36 a^7 b^2}{7 x^7}-\frac {14 a^6 b^3}{x^6}-\frac {126 a^5 b^4}{5 x^5}-\frac {63 a^4 b^5}{2 x^4}-\frac {28 a^3 b^6}{x^3}-\frac {18 a^2 b^7}{x^2}-\frac {9 a b^8}{x}+b^9 \log (x) \]
[Out]
Time = 0.04 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {(a+b x)^9}{x^{10}} \, dx=-\frac {a^9}{9 x^9}-\frac {9 a^8 b}{8 x^8}-\frac {36 a^7 b^2}{7 x^7}-\frac {14 a^6 b^3}{x^6}-\frac {126 a^5 b^4}{5 x^5}-\frac {63 a^4 b^5}{2 x^4}-\frac {28 a^3 b^6}{x^3}-\frac {18 a^2 b^7}{x^2}-\frac {9 a b^8}{x}+b^9 \log (x) \]
[In]
[Out]
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^9}{x^{10}}+\frac {9 a^8 b}{x^9}+\frac {36 a^7 b^2}{x^8}+\frac {84 a^6 b^3}{x^7}+\frac {126 a^5 b^4}{x^6}+\frac {126 a^4 b^5}{x^5}+\frac {84 a^3 b^6}{x^4}+\frac {36 a^2 b^7}{x^3}+\frac {9 a b^8}{x^2}+\frac {b^9}{x}\right ) \, dx \\ & = -\frac {a^9}{9 x^9}-\frac {9 a^8 b}{8 x^8}-\frac {36 a^7 b^2}{7 x^7}-\frac {14 a^6 b^3}{x^6}-\frac {126 a^5 b^4}{5 x^5}-\frac {63 a^4 b^5}{2 x^4}-\frac {28 a^3 b^6}{x^3}-\frac {18 a^2 b^7}{x^2}-\frac {9 a b^8}{x}+b^9 \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^9}{x^{10}} \, dx=-\frac {a^9}{9 x^9}-\frac {9 a^8 b}{8 x^8}-\frac {36 a^7 b^2}{7 x^7}-\frac {14 a^6 b^3}{x^6}-\frac {126 a^5 b^4}{5 x^5}-\frac {63 a^4 b^5}{2 x^4}-\frac {28 a^3 b^6}{x^3}-\frac {18 a^2 b^7}{x^2}-\frac {9 a b^8}{x}+b^9 \log (x) \]
[In]
[Out]
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {a^{9}}{9 x^{9}}-\frac {9 a^{8} b}{8 x^{8}}-\frac {36 a^{7} b^{2}}{7 x^{7}}-\frac {14 a^{6} b^{3}}{x^{6}}-\frac {126 a^{5} b^{4}}{5 x^{5}}-\frac {63 a^{4} b^{5}}{2 x^{4}}-\frac {28 a^{3} b^{6}}{x^{3}}-\frac {18 a^{2} b^{7}}{x^{2}}-\frac {9 a \,b^{8}}{x}+b^{9} \ln \left (x \right )\) | \(100\) |
norman | \(\frac {-\frac {1}{9} a^{9}-9 a \,x^{8} b^{8}-18 a^{2} x^{7} b^{7}-\frac {63}{2} a^{4} x^{5} b^{5}-\frac {126}{5} a^{5} b^{4} x^{4}-14 a^{6} b^{3} x^{3}-\frac {36}{7} a^{7} b^{2} x^{2}-\frac {9}{8} a^{8} b x -28 x^{6} a^{3} b^{6}}{x^{9}}+b^{9} \ln \left (x \right )\) | \(100\) |
risch | \(\frac {-\frac {1}{9} a^{9}-9 a \,x^{8} b^{8}-18 a^{2} x^{7} b^{7}-\frac {63}{2} a^{4} x^{5} b^{5}-\frac {126}{5} a^{5} b^{4} x^{4}-14 a^{6} b^{3} x^{3}-\frac {36}{7} a^{7} b^{2} x^{2}-\frac {9}{8} a^{8} b x -28 x^{6} a^{3} b^{6}}{x^{9}}+b^{9} \ln \left (x \right )\) | \(100\) |
parallelrisch | \(\frac {2520 \ln \left (x \right ) x^{9} b^{9}-22680 a \,x^{8} b^{8}-45360 a^{2} x^{7} b^{7}-70560 x^{6} a^{3} b^{6}-79380 a^{4} x^{5} b^{5}-63504 a^{5} b^{4} x^{4}-35280 a^{6} b^{3} x^{3}-12960 a^{7} b^{2} x^{2}-2835 a^{8} b x -280 a^{9}}{2520 x^{9}}\) | \(104\) |
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x)^9}{x^{10}} \, dx=\frac {2520 \, b^{9} x^{9} \log \left (x\right ) - 22680 \, a b^{8} x^{8} - 45360 \, a^{2} b^{7} x^{7} - 70560 \, a^{3} b^{6} x^{6} - 79380 \, a^{4} b^{5} x^{5} - 63504 \, a^{5} b^{4} x^{4} - 35280 \, a^{6} b^{3} x^{3} - 12960 \, a^{7} b^{2} x^{2} - 2835 \, a^{8} b x - 280 \, a^{9}}{2520 \, x^{9}} \]
[In]
[Out]
Time = 0.51 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x)^9}{x^{10}} \, dx=b^{9} \log {\left (x \right )} + \frac {- 280 a^{9} - 2835 a^{8} b x - 12960 a^{7} b^{2} x^{2} - 35280 a^{6} b^{3} x^{3} - 63504 a^{5} b^{4} x^{4} - 79380 a^{4} b^{5} x^{5} - 70560 a^{3} b^{6} x^{6} - 45360 a^{2} b^{7} x^{7} - 22680 a b^{8} x^{8}}{2520 x^{9}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^9}{x^{10}} \, dx=b^{9} \log \left (x\right ) - \frac {22680 \, a b^{8} x^{8} + 45360 \, a^{2} b^{7} x^{7} + 70560 \, a^{3} b^{6} x^{6} + 79380 \, a^{4} b^{5} x^{5} + 63504 \, a^{5} b^{4} x^{4} + 35280 \, a^{6} b^{3} x^{3} + 12960 \, a^{7} b^{2} x^{2} + 2835 \, a^{8} b x + 280 \, a^{9}}{2520 \, x^{9}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^9}{x^{10}} \, dx=b^{9} \log \left ({\left | x \right |}\right ) - \frac {22680 \, a b^{8} x^{8} + 45360 \, a^{2} b^{7} x^{7} + 70560 \, a^{3} b^{6} x^{6} + 79380 \, a^{4} b^{5} x^{5} + 63504 \, a^{5} b^{4} x^{4} + 35280 \, a^{6} b^{3} x^{3} + 12960 \, a^{7} b^{2} x^{2} + 2835 \, a^{8} b x + 280 \, a^{9}}{2520 \, x^{9}} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^9}{x^{10}} \, dx=b^9\,\ln \left (x\right )-\frac {\frac {a^9}{9}+\frac {9\,a^8\,b\,x}{8}+\frac {36\,a^7\,b^2\,x^2}{7}+14\,a^6\,b^3\,x^3+\frac {126\,a^5\,b^4\,x^4}{5}+\frac {63\,a^4\,b^5\,x^5}{2}+28\,a^3\,b^6\,x^6+18\,a^2\,b^7\,x^7+9\,a\,b^8\,x^8}{x^9} \]
[In]
[Out]