Integrand size = 11, antiderivative size = 154 \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {a^9}{9 b^{10} (a+b x)^9}-\frac {9 a^8}{8 b^{10} (a+b x)^8}+\frac {36 a^7}{7 b^{10} (a+b x)^7}-\frac {14 a^6}{b^{10} (a+b x)^6}+\frac {126 a^5}{5 b^{10} (a+b x)^5}-\frac {63 a^4}{2 b^{10} (a+b x)^4}+\frac {28 a^3}{b^{10} (a+b x)^3}-\frac {18 a^2}{b^{10} (a+b x)^2}+\frac {9 a}{b^{10} (a+b x)}+\frac {\log (a+b x)}{b^{10}} \]
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Time = 0.08 (sec) , antiderivative size = 154, 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} \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {a^9}{9 b^{10} (a+b x)^9}-\frac {9 a^8}{8 b^{10} (a+b x)^8}+\frac {36 a^7}{7 b^{10} (a+b x)^7}-\frac {14 a^6}{b^{10} (a+b x)^6}+\frac {126 a^5}{5 b^{10} (a+b x)^5}-\frac {63 a^4}{2 b^{10} (a+b x)^4}+\frac {28 a^3}{b^{10} (a+b x)^3}-\frac {18 a^2}{b^{10} (a+b x)^2}+\frac {9 a}{b^{10} (a+b x)}+\frac {\log (a+b x)}{b^{10}} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^9}{b^9 (a+b x)^{10}}+\frac {9 a^8}{b^9 (a+b x)^9}-\frac {36 a^7}{b^9 (a+b x)^8}+\frac {84 a^6}{b^9 (a+b x)^7}-\frac {126 a^5}{b^9 (a+b x)^6}+\frac {126 a^4}{b^9 (a+b x)^5}-\frac {84 a^3}{b^9 (a+b x)^4}+\frac {36 a^2}{b^9 (a+b x)^3}-\frac {9 a}{b^9 (a+b x)^2}+\frac {1}{b^9 (a+b x)}\right ) \, dx \\ & = \frac {a^9}{9 b^{10} (a+b x)^9}-\frac {9 a^8}{8 b^{10} (a+b x)^8}+\frac {36 a^7}{7 b^{10} (a+b x)^7}-\frac {14 a^6}{b^{10} (a+b x)^6}+\frac {126 a^5}{5 b^{10} (a+b x)^5}-\frac {63 a^4}{2 b^{10} (a+b x)^4}+\frac {28 a^3}{b^{10} (a+b x)^3}-\frac {18 a^2}{b^{10} (a+b x)^2}+\frac {9 a}{b^{10} (a+b x)}+\frac {\log (a+b x)}{b^{10}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.72 \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {a \left (7129 a^8+61641 a^7 b x+235224 a^6 b^2 x^2+518616 a^5 b^3 x^3+725004 a^4 b^4 x^4+661500 a^3 b^5 x^5+388080 a^2 b^6 x^6+136080 a b^7 x^7+22680 b^8 x^8\right )}{2520 b^{10} (a+b x)^9}+\frac {\log (a+b x)}{b^{10}} \]
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Time = 0.04 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.73
method | result | size |
norman | \(\frac {\frac {7129 a^{9}}{2520 b^{10}}+\frac {9 a \,x^{8}}{b^{2}}+\frac {54 a^{2} x^{7}}{b^{3}}+\frac {154 a^{3} x^{6}}{b^{4}}+\frac {525 a^{4} x^{5}}{2 b^{5}}+\frac {2877 a^{5} x^{4}}{10 b^{6}}+\frac {1029 a^{6} x^{3}}{5 b^{7}}+\frac {3267 a^{7} x^{2}}{35 b^{8}}+\frac {6849 a^{8} x}{280 b^{9}}}{\left (b x +a \right )^{9}}+\frac {\ln \left (b x +a \right )}{b^{10}}\) | \(113\) |
risch | \(\frac {\frac {7129 a^{9}}{2520 b^{10}}+\frac {9 a \,x^{8}}{b^{2}}+\frac {54 a^{2} x^{7}}{b^{3}}+\frac {154 a^{3} x^{6}}{b^{4}}+\frac {525 a^{4} x^{5}}{2 b^{5}}+\frac {2877 a^{5} x^{4}}{10 b^{6}}+\frac {1029 a^{6} x^{3}}{5 b^{7}}+\frac {3267 a^{7} x^{2}}{35 b^{8}}+\frac {6849 a^{8} x}{280 b^{9}}}{\left (b x +a \right )^{9}}+\frac {\ln \left (b x +a \right )}{b^{10}}\) | \(113\) |
default | \(\frac {a^{9}}{9 b^{10} \left (b x +a \right )^{9}}-\frac {9 a^{8}}{8 b^{10} \left (b x +a \right )^{8}}+\frac {36 a^{7}}{7 b^{10} \left (b x +a \right )^{7}}-\frac {14 a^{6}}{b^{10} \left (b x +a \right )^{6}}+\frac {126 a^{5}}{5 b^{10} \left (b x +a \right )^{5}}-\frac {63 a^{4}}{2 b^{10} \left (b x +a \right )^{4}}+\frac {28 a^{3}}{b^{10} \left (b x +a \right )^{3}}-\frac {18 a^{2}}{b^{10} \left (b x +a \right )^{2}}+\frac {9 a}{b^{10} \left (b x +a \right )}+\frac {\ln \left (b x +a \right )}{b^{10}}\) | \(145\) |
parallelrisch | \(\frac {7129 a^{9}+2520 \ln \left (b x +a \right ) a^{9}+22680 \ln \left (b x +a \right ) x^{8} a \,b^{8}+90720 \ln \left (b x +a \right ) x^{7} a^{2} b^{7}+22680 \ln \left (b x +a \right ) x \,a^{8} b +211680 \ln \left (b x +a \right ) x^{6} a^{3} b^{6}+317520 \ln \left (b x +a \right ) x^{5} a^{4} b^{5}+317520 \ln \left (b x +a \right ) x^{4} a^{5} b^{4}+211680 \ln \left (b x +a \right ) x^{3} a^{6} b^{3}+90720 \ln \left (b x +a \right ) x^{2} a^{7} b^{2}+661500 a^{4} x^{5} b^{5}+2520 \ln \left (b x +a \right ) x^{9} b^{9}+388080 x^{6} a^{3} b^{6}+22680 a \,x^{8} b^{8}+725004 a^{5} b^{4} x^{4}+518616 a^{6} b^{3} x^{3}+235224 a^{7} b^{2} x^{2}+61641 a^{8} b x +136080 a^{2} x^{7} b^{7}}{2520 b^{10} \left (b x +a \right )^{9}}\) | \(256\) |
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Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (144) = 288\).
Time = 0.22 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.90 \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {22680 \, a b^{8} x^{8} + 136080 \, a^{2} b^{7} x^{7} + 388080 \, a^{3} b^{6} x^{6} + 661500 \, a^{4} b^{5} x^{5} + 725004 \, a^{5} b^{4} x^{4} + 518616 \, a^{6} b^{3} x^{3} + 235224 \, a^{7} b^{2} x^{2} + 61641 \, a^{8} b x + 7129 \, a^{9} + 2520 \, {\left (b^{9} x^{9} + 9 \, a b^{8} x^{8} + 36 \, a^{2} b^{7} x^{7} + 84 \, a^{3} b^{6} x^{6} + 126 \, a^{4} b^{5} x^{5} + 126 \, a^{5} b^{4} x^{4} + 84 \, a^{6} b^{3} x^{3} + 36 \, a^{7} b^{2} x^{2} + 9 \, a^{8} b x + a^{9}\right )} \log \left (b x + a\right )}{2520 \, {\left (b^{19} x^{9} + 9 \, a b^{18} x^{8} + 36 \, a^{2} b^{17} x^{7} + 84 \, a^{3} b^{16} x^{6} + 126 \, a^{4} b^{15} x^{5} + 126 \, a^{5} b^{14} x^{4} + 84 \, a^{6} b^{13} x^{3} + 36 \, a^{7} b^{12} x^{2} + 9 \, a^{8} b^{11} x + a^{9} b^{10}\right )}} \]
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Time = 0.57 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.38 \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {7129 a^{9} + 61641 a^{8} b x + 235224 a^{7} b^{2} x^{2} + 518616 a^{6} b^{3} x^{3} + 725004 a^{5} b^{4} x^{4} + 661500 a^{4} b^{5} x^{5} + 388080 a^{3} b^{6} x^{6} + 136080 a^{2} b^{7} x^{7} + 22680 a b^{8} x^{8}}{2520 a^{9} b^{10} + 22680 a^{8} b^{11} x + 90720 a^{7} b^{12} x^{2} + 211680 a^{6} b^{13} x^{3} + 317520 a^{5} b^{14} x^{4} + 317520 a^{4} b^{15} x^{5} + 211680 a^{3} b^{16} x^{6} + 90720 a^{2} b^{17} x^{7} + 22680 a b^{18} x^{8} + 2520 b^{19} x^{9}} + \frac {\log {\left (a + b x \right )}}{b^{10}} \]
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Time = 0.23 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.31 \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {22680 \, a b^{8} x^{8} + 136080 \, a^{2} b^{7} x^{7} + 388080 \, a^{3} b^{6} x^{6} + 661500 \, a^{4} b^{5} x^{5} + 725004 \, a^{5} b^{4} x^{4} + 518616 \, a^{6} b^{3} x^{3} + 235224 \, a^{7} b^{2} x^{2} + 61641 \, a^{8} b x + 7129 \, a^{9}}{2520 \, {\left (b^{19} x^{9} + 9 \, a b^{18} x^{8} + 36 \, a^{2} b^{17} x^{7} + 84 \, a^{3} b^{16} x^{6} + 126 \, a^{4} b^{15} x^{5} + 126 \, a^{5} b^{14} x^{4} + 84 \, a^{6} b^{13} x^{3} + 36 \, a^{7} b^{12} x^{2} + 9 \, a^{8} b^{11} x + a^{9} b^{10}\right )}} + \frac {\log \left (b x + a\right )}{b^{10}} \]
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Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.73 \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {\log \left ({\left | b x + a \right |}\right )}{b^{10}} + \frac {22680 \, a b^{7} x^{8} + 136080 \, a^{2} b^{6} x^{7} + 388080 \, a^{3} b^{5} x^{6} + 661500 \, a^{4} b^{4} x^{5} + 725004 \, a^{5} b^{3} x^{4} + 518616 \, a^{6} b^{2} x^{3} + 235224 \, a^{7} b x^{2} + 61641 \, a^{8} x + \frac {7129 \, a^{9}}{b}}{2520 \, {\left (b x + a\right )}^{9} b^{9}} \]
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Time = 0.22 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.76 \[ \int \frac {x^9}{(a+b x)^{10}} \, dx=\frac {\ln \left (a+b\,x\right )+\frac {9\,a}{a+b\,x}-\frac {18\,a^2}{{\left (a+b\,x\right )}^2}+\frac {28\,a^3}{{\left (a+b\,x\right )}^3}-\frac {63\,a^4}{2\,{\left (a+b\,x\right )}^4}+\frac {126\,a^5}{5\,{\left (a+b\,x\right )}^5}-\frac {14\,a^6}{{\left (a+b\,x\right )}^6}+\frac {36\,a^7}{7\,{\left (a+b\,x\right )}^7}-\frac {9\,a^8}{8\,{\left (a+b\,x\right )}^8}+\frac {a^9}{9\,{\left (a+b\,x\right )}^9}}{b^{10}} \]
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