Integrand size = 11, antiderivative size = 132 \[ \int \frac {(a+b x)^{11}}{x^{10}} \, dx=-\frac {a^{11}}{9 x^9}-\frac {11 a^{10} b}{8 x^8}-\frac {55 a^9 b^2}{7 x^7}-\frac {55 a^8 b^3}{2 x^6}-\frac {66 a^7 b^4}{x^5}-\frac {231 a^6 b^5}{2 x^4}-\frac {154 a^5 b^6}{x^3}-\frac {165 a^4 b^7}{x^2}-\frac {165 a^3 b^8}{x}+11 a b^{10} x+\frac {b^{11} x^2}{2}+55 a^2 b^9 \log (x) \]
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Time = 0.04 (sec) , antiderivative size = 132, 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} \[ \int \frac {(a+b x)^{11}}{x^{10}} \, dx=-\frac {a^{11}}{9 x^9}-\frac {11 a^{10} b}{8 x^8}-\frac {55 a^9 b^2}{7 x^7}-\frac {55 a^8 b^3}{2 x^6}-\frac {66 a^7 b^4}{x^5}-\frac {231 a^6 b^5}{2 x^4}-\frac {154 a^5 b^6}{x^3}-\frac {165 a^4 b^7}{x^2}-\frac {165 a^3 b^8}{x}+55 a^2 b^9 \log (x)+11 a b^{10} x+\frac {b^{11} x^2}{2} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (11 a b^{10}+\frac {a^{11}}{x^{10}}+\frac {11 a^{10} b}{x^9}+\frac {55 a^9 b^2}{x^8}+\frac {165 a^8 b^3}{x^7}+\frac {330 a^7 b^4}{x^6}+\frac {462 a^6 b^5}{x^5}+\frac {462 a^5 b^6}{x^4}+\frac {330 a^4 b^7}{x^3}+\frac {165 a^3 b^8}{x^2}+\frac {55 a^2 b^9}{x}+b^{11} x\right ) \, dx \\ & = -\frac {a^{11}}{9 x^9}-\frac {11 a^{10} b}{8 x^8}-\frac {55 a^9 b^2}{7 x^7}-\frac {55 a^8 b^3}{2 x^6}-\frac {66 a^7 b^4}{x^5}-\frac {231 a^6 b^5}{2 x^4}-\frac {154 a^5 b^6}{x^3}-\frac {165 a^4 b^7}{x^2}-\frac {165 a^3 b^8}{x}+11 a b^{10} x+\frac {b^{11} x^2}{2}+55 a^2 b^9 \log (x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{11}}{x^{10}} \, dx=-\frac {a^{11}}{9 x^9}-\frac {11 a^{10} b}{8 x^8}-\frac {55 a^9 b^2}{7 x^7}-\frac {55 a^8 b^3}{2 x^6}-\frac {66 a^7 b^4}{x^5}-\frac {231 a^6 b^5}{2 x^4}-\frac {154 a^5 b^6}{x^3}-\frac {165 a^4 b^7}{x^2}-\frac {165 a^3 b^8}{x}+11 a b^{10} x+\frac {b^{11} x^2}{2}+55 a^2 b^9 \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {a^{11}}{9 x^{9}}-\frac {11 a^{10} b}{8 x^{8}}-\frac {55 a^{9} b^{2}}{7 x^{7}}-\frac {55 a^{8} b^{3}}{2 x^{6}}-\frac {66 a^{7} b^{4}}{x^{5}}-\frac {231 a^{6} b^{5}}{2 x^{4}}-\frac {154 a^{5} b^{6}}{x^{3}}-\frac {165 a^{4} b^{7}}{x^{2}}-\frac {165 a^{3} b^{8}}{x}+11 a \,b^{10} x +\frac {b^{11} x^{2}}{2}+55 a^{2} b^{9} \ln \left (x \right )\) | \(121\) |
risch | \(\frac {b^{11} x^{2}}{2}+11 a \,b^{10} x +\frac {-165 a^{3} x^{8} b^{8}-165 a^{4} b^{7} x^{7}-154 a^{5} b^{6} x^{6}-\frac {231}{2} a^{6} b^{5} x^{5}-66 a^{7} b^{4} x^{4}-\frac {55}{2} b^{3} a^{8} x^{3}-\frac {55}{7} b^{2} a^{9} x^{2}-\frac {11}{8} a^{10} b x -\frac {1}{9} a^{11}}{x^{9}}+55 a^{2} b^{9} \ln \left (x \right )\) | \(121\) |
norman | \(\frac {-\frac {1}{9} a^{11}+\frac {1}{2} b^{11} x^{11}+11 a \,x^{10} b^{10}-165 a^{3} x^{8} b^{8}-165 a^{4} b^{7} x^{7}-154 a^{5} b^{6} x^{6}-\frac {231}{2} a^{6} b^{5} x^{5}-66 a^{7} b^{4} x^{4}-\frac {11}{8} a^{10} b x -\frac {55}{7} b^{2} a^{9} x^{2}-\frac {55}{2} b^{3} a^{8} x^{3}}{x^{9}}+55 a^{2} b^{9} \ln \left (x \right )\) | \(123\) |
parallelrisch | \(\frac {252 b^{11} x^{11}+27720 \ln \left (x \right ) x^{9} a^{2} b^{9}+5544 a \,x^{10} b^{10}-83160 a^{3} x^{8} b^{8}-83160 a^{4} b^{7} x^{7}-77616 a^{5} b^{6} x^{6}-58212 a^{6} b^{5} x^{5}-33264 a^{7} b^{4} x^{4}-13860 b^{3} a^{8} x^{3}-3960 b^{2} a^{9} x^{2}-693 a^{10} b x -56 a^{11}}{504 x^{9}}\) | \(126\) |
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Time = 0.22 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.95 \[ \int \frac {(a+b x)^{11}}{x^{10}} \, dx=\frac {252 \, b^{11} x^{11} + 5544 \, a b^{10} x^{10} + 27720 \, a^{2} b^{9} x^{9} \log \left (x\right ) - 83160 \, a^{3} b^{8} x^{8} - 83160 \, a^{4} b^{7} x^{7} - 77616 \, a^{5} b^{6} x^{6} - 58212 \, a^{6} b^{5} x^{5} - 33264 \, a^{7} b^{4} x^{4} - 13860 \, a^{8} b^{3} x^{3} - 3960 \, a^{9} b^{2} x^{2} - 693 \, a^{10} b x - 56 \, a^{11}}{504 \, x^{9}} \]
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Time = 0.40 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x)^{11}}{x^{10}} \, dx=55 a^{2} b^{9} \log {\left (x \right )} + 11 a b^{10} x + \frac {b^{11} x^{2}}{2} + \frac {- 56 a^{11} - 693 a^{10} b x - 3960 a^{9} b^{2} x^{2} - 13860 a^{8} b^{3} x^{3} - 33264 a^{7} b^{4} x^{4} - 58212 a^{6} b^{5} x^{5} - 77616 a^{5} b^{6} x^{6} - 83160 a^{4} b^{7} x^{7} - 83160 a^{3} b^{8} x^{8}}{504 x^{9}} \]
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Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^{11}}{x^{10}} \, dx=\frac {1}{2} \, b^{11} x^{2} + 11 \, a b^{10} x + 55 \, a^{2} b^{9} \log \left (x\right ) - \frac {83160 \, a^{3} b^{8} x^{8} + 83160 \, a^{4} b^{7} x^{7} + 77616 \, a^{5} b^{6} x^{6} + 58212 \, a^{6} b^{5} x^{5} + 33264 \, a^{7} b^{4} x^{4} + 13860 \, a^{8} b^{3} x^{3} + 3960 \, a^{9} b^{2} x^{2} + 693 \, a^{10} b x + 56 \, a^{11}}{504 \, x^{9}} \]
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Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^{11}}{x^{10}} \, dx=\frac {1}{2} \, b^{11} x^{2} + 11 \, a b^{10} x + 55 \, a^{2} b^{9} \log \left ({\left | x \right |}\right ) - \frac {83160 \, a^{3} b^{8} x^{8} + 83160 \, a^{4} b^{7} x^{7} + 77616 \, a^{5} b^{6} x^{6} + 58212 \, a^{6} b^{5} x^{5} + 33264 \, a^{7} b^{4} x^{4} + 13860 \, a^{8} b^{3} x^{3} + 3960 \, a^{9} b^{2} x^{2} + 693 \, a^{10} b x + 56 \, a^{11}}{504 \, x^{9}} \]
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Time = 0.10 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b x)^{11}}{x^{10}} \, dx=\frac {b^{11}\,x^2}{2}-\frac {\frac {a^{11}}{9}+\frac {11\,a^{10}\,b\,x}{8}+\frac {55\,a^9\,b^2\,x^2}{7}+\frac {55\,a^8\,b^3\,x^3}{2}+66\,a^7\,b^4\,x^4+\frac {231\,a^6\,b^5\,x^5}{2}+154\,a^5\,b^6\,x^6+165\,a^4\,b^7\,x^7+165\,a^3\,b^8\,x^8}{x^9}+55\,a^2\,b^9\,\ln \left (x\right )+11\,a\,b^{10}\,x \]
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