Integrand size = 11, antiderivative size = 158 \[ \int \frac {1}{x^2 (a+b x)^{10}} \, dx=-\frac {1}{a^{10} x}-\frac {b}{9 a^2 (a+b x)^9}-\frac {b}{4 a^3 (a+b x)^8}-\frac {3 b}{7 a^4 (a+b x)^7}-\frac {2 b}{3 a^5 (a+b x)^6}-\frac {b}{a^6 (a+b x)^5}-\frac {3 b}{2 a^7 (a+b x)^4}-\frac {7 b}{3 a^8 (a+b x)^3}-\frac {4 b}{a^9 (a+b x)^2}-\frac {9 b}{a^{10} (a+b x)}-\frac {10 b \log (x)}{a^{11}}+\frac {10 b \log (a+b x)}{a^{11}} \]
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Time = 0.09 (sec) , antiderivative size = 158, 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 = {46} \[ \int \frac {1}{x^2 (a+b x)^{10}} \, dx=-\frac {10 b \log (x)}{a^{11}}+\frac {10 b \log (a+b x)}{a^{11}}-\frac {9 b}{a^{10} (a+b x)}-\frac {1}{a^{10} x}-\frac {4 b}{a^9 (a+b x)^2}-\frac {7 b}{3 a^8 (a+b x)^3}-\frac {3 b}{2 a^7 (a+b x)^4}-\frac {b}{a^6 (a+b x)^5}-\frac {2 b}{3 a^5 (a+b x)^6}-\frac {3 b}{7 a^4 (a+b x)^7}-\frac {b}{4 a^3 (a+b x)^8}-\frac {b}{9 a^2 (a+b x)^9} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^{10} x^2}-\frac {10 b}{a^{11} x}+\frac {b^2}{a^2 (a+b x)^{10}}+\frac {2 b^2}{a^3 (a+b x)^9}+\frac {3 b^2}{a^4 (a+b x)^8}+\frac {4 b^2}{a^5 (a+b x)^7}+\frac {5 b^2}{a^6 (a+b x)^6}+\frac {6 b^2}{a^7 (a+b x)^5}+\frac {7 b^2}{a^8 (a+b x)^4}+\frac {8 b^2}{a^9 (a+b x)^3}+\frac {9 b^2}{a^{10} (a+b x)^2}+\frac {10 b^2}{a^{11} (a+b x)}\right ) \, dx \\ & = -\frac {1}{a^{10} x}-\frac {b}{9 a^2 (a+b x)^9}-\frac {b}{4 a^3 (a+b x)^8}-\frac {3 b}{7 a^4 (a+b x)^7}-\frac {2 b}{3 a^5 (a+b x)^6}-\frac {b}{a^6 (a+b x)^5}-\frac {3 b}{2 a^7 (a+b x)^4}-\frac {7 b}{3 a^8 (a+b x)^3}-\frac {4 b}{a^9 (a+b x)^2}-\frac {9 b}{a^{10} (a+b x)}-\frac {10 b \log (x)}{a^{11}}+\frac {10 b \log (a+b x)}{a^{11}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^2 (a+b x)^{10}} \, dx=-\frac {\frac {a \left (252 a^9+7129 a^8 b x+41481 a^7 b^2 x^2+120564 a^6 b^3 x^3+210756 a^5 b^4 x^4+236754 a^4 b^5 x^5+173250 a^3 b^6 x^6+80220 a^2 b^7 x^7+21420 a b^8 x^8+2520 b^9 x^9\right )}{x (a+b x)^9}+2520 b \log (x)-2520 b \log (a+b x)}{252 a^{11}} \]
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Time = 0.06 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.87
method | result | size |
risch | \(\frac {-\frac {10 b^{9} x^{9}}{a^{10}}-\frac {85 b^{8} x^{8}}{a^{9}}-\frac {955 b^{7} x^{7}}{3 a^{8}}-\frac {1375 b^{6} x^{6}}{2 a^{7}}-\frac {1879 b^{5} x^{5}}{2 a^{6}}-\frac {2509 b^{4} x^{4}}{3 a^{5}}-\frac {3349 b^{3} x^{3}}{7 a^{4}}-\frac {4609 b^{2} x^{2}}{28 a^{3}}-\frac {7129 b x}{252 a^{2}}-\frac {1}{a}}{x \left (b x +a \right )^{9}}+\frac {10 b \ln \left (-b x -a \right )}{a^{11}}-\frac {10 b \ln \left (x \right )}{a^{11}}\) | \(137\) |
norman | \(\frac {-\frac {1}{a}+\frac {90 b^{2} x^{2}}{a^{3}}+\frac {540 b^{3} x^{3}}{a^{4}}+\frac {1540 b^{4} x^{4}}{a^{5}}+\frac {2625 b^{5} x^{5}}{a^{6}}+\frac {2877 b^{6} x^{6}}{a^{7}}+\frac {2058 b^{7} x^{7}}{a^{8}}+\frac {6534 b^{8} x^{8}}{7 a^{9}}+\frac {6849 b^{9} x^{9}}{28 a^{10}}+\frac {7129 b^{10} x^{10}}{252 a^{11}}}{x \left (b x +a \right )^{9}}-\frac {10 b \ln \left (x \right )}{a^{11}}+\frac {10 b \ln \left (b x +a \right )}{a^{11}}\) | \(138\) |
default | \(-\frac {1}{a^{10} x}-\frac {b}{9 a^{2} \left (b x +a \right )^{9}}-\frac {b}{4 a^{3} \left (b x +a \right )^{8}}-\frac {3 b}{7 a^{4} \left (b x +a \right )^{7}}-\frac {2 b}{3 a^{5} \left (b x +a \right )^{6}}-\frac {b}{a^{6} \left (b x +a \right )^{5}}-\frac {3 b}{2 a^{7} \left (b x +a \right )^{4}}-\frac {7 b}{3 a^{8} \left (b x +a \right )^{3}}-\frac {4 b}{a^{9} \left (b x +a \right )^{2}}-\frac {9 b}{a^{10} \left (b x +a \right )}-\frac {10 b \ln \left (x \right )}{a^{11}}+\frac {10 b \ln \left (b x +a \right )}{a^{11}}\) | \(147\) |
parallelrisch | \(-\frac {252 a^{10}-7129 b^{10} x^{10}-22680 \ln \left (b x +a \right ) x^{9} a \,b^{9}-90720 \ln \left (b x +a \right ) x^{8} a^{2} b^{8}-211680 \ln \left (b x +a \right ) x^{7} a^{3} b^{7}-90720 \ln \left (b x +a \right ) x^{3} a^{7} b^{3}-22680 \ln \left (b x +a \right ) x^{2} a^{8} b^{2}-2520 \ln \left (b x +a \right ) x \,a^{9} b -317520 \ln \left (b x +a \right ) x^{6} a^{4} b^{6}-317520 \ln \left (b x +a \right ) x^{5} a^{5} b^{5}-211680 \ln \left (b x +a \right ) x^{4} a^{6} b^{4}+90720 a^{7} b^{3} \ln \left (x \right ) x^{3}+211680 a^{6} b^{4} \ln \left (x \right ) x^{4}+22680 a \,b^{9} \ln \left (x \right ) x^{9}+211680 a^{3} b^{7} \ln \left (x \right ) x^{7}+90720 a^{2} b^{8} \ln \left (x \right ) x^{8}+317520 a^{4} b^{6} \ln \left (x \right ) x^{6}+317520 a^{5} b^{5} \ln \left (x \right ) x^{5}+22680 a^{8} b^{2} \ln \left (x \right ) x^{2}+2520 a^{9} b \ln \left (x \right ) x -22680 a^{8} b^{2} x^{2}-136080 a^{7} b^{3} x^{3}-388080 a^{6} b^{4} x^{4}-661500 a^{5} b^{5} x^{5}-725004 a^{4} b^{6} x^{6}-518616 a^{3} b^{7} x^{7}-235224 a^{2} b^{8} x^{8}-61641 a \,b^{9} x^{9}+2520 b^{10} \ln \left (x \right ) x^{10}-2520 \ln \left (b x +a \right ) x^{10} b^{10}}{252 a^{11} x \left (b x +a \right )^{9}}\) | \(398\) |
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Leaf count of result is larger than twice the leaf count of optimal. 417 vs. \(2 (146) = 292\).
Time = 0.24 (sec) , antiderivative size = 417, normalized size of antiderivative = 2.64 \[ \int \frac {1}{x^2 (a+b x)^{10}} \, dx=-\frac {2520 \, a b^{9} x^{9} + 21420 \, a^{2} b^{8} x^{8} + 80220 \, a^{3} b^{7} x^{7} + 173250 \, a^{4} b^{6} x^{6} + 236754 \, a^{5} b^{5} x^{5} + 210756 \, a^{6} b^{4} x^{4} + 120564 \, a^{7} b^{3} x^{3} + 41481 \, a^{8} b^{2} x^{2} + 7129 \, a^{9} b x + 252 \, a^{10} - 2520 \, {\left (b^{10} x^{10} + 9 \, a b^{9} x^{9} + 36 \, a^{2} b^{8} x^{8} + 84 \, a^{3} b^{7} x^{7} + 126 \, a^{4} b^{6} x^{6} + 126 \, a^{5} b^{5} x^{5} + 84 \, a^{6} b^{4} x^{4} + 36 \, a^{7} b^{3} x^{3} + 9 \, a^{8} b^{2} x^{2} + a^{9} b x\right )} \log \left (b x + a\right ) + 2520 \, {\left (b^{10} x^{10} + 9 \, a b^{9} x^{9} + 36 \, a^{2} b^{8} x^{8} + 84 \, a^{3} b^{7} x^{7} + 126 \, a^{4} b^{6} x^{6} + 126 \, a^{5} b^{5} x^{5} + 84 \, a^{6} b^{4} x^{4} + 36 \, a^{7} b^{3} x^{3} + 9 \, a^{8} b^{2} x^{2} + a^{9} b x\right )} \log \left (x\right )}{252 \, {\left (a^{11} b^{9} x^{10} + 9 \, a^{12} b^{8} x^{9} + 36 \, a^{13} b^{7} x^{8} + 84 \, a^{14} b^{6} x^{7} + 126 \, a^{15} b^{5} x^{6} + 126 \, a^{16} b^{4} x^{5} + 84 \, a^{17} b^{3} x^{4} + 36 \, a^{18} b^{2} x^{3} + 9 \, a^{19} b x^{2} + a^{20} x\right )}} \]
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Time = 0.66 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.47 \[ \int \frac {1}{x^2 (a+b x)^{10}} \, dx=\frac {- 252 a^{9} - 7129 a^{8} b x - 41481 a^{7} b^{2} x^{2} - 120564 a^{6} b^{3} x^{3} - 210756 a^{5} b^{4} x^{4} - 236754 a^{4} b^{5} x^{5} - 173250 a^{3} b^{6} x^{6} - 80220 a^{2} b^{7} x^{7} - 21420 a b^{8} x^{8} - 2520 b^{9} x^{9}}{252 a^{19} x + 2268 a^{18} b x^{2} + 9072 a^{17} b^{2} x^{3} + 21168 a^{16} b^{3} x^{4} + 31752 a^{15} b^{4} x^{5} + 31752 a^{14} b^{5} x^{6} + 21168 a^{13} b^{6} x^{7} + 9072 a^{12} b^{7} x^{8} + 2268 a^{11} b^{8} x^{9} + 252 a^{10} b^{9} x^{10}} + \frac {10 b \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{11}} \]
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Time = 0.22 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.41 \[ \int \frac {1}{x^2 (a+b x)^{10}} \, dx=-\frac {2520 \, b^{9} x^{9} + 21420 \, a b^{8} x^{8} + 80220 \, a^{2} b^{7} x^{7} + 173250 \, a^{3} b^{6} x^{6} + 236754 \, a^{4} b^{5} x^{5} + 210756 \, a^{5} b^{4} x^{4} + 120564 \, a^{6} b^{3} x^{3} + 41481 \, a^{7} b^{2} x^{2} + 7129 \, a^{8} b x + 252 \, a^{9}}{252 \, {\left (a^{10} b^{9} x^{10} + 9 \, a^{11} b^{8} x^{9} + 36 \, a^{12} b^{7} x^{8} + 84 \, a^{13} b^{6} x^{7} + 126 \, a^{14} b^{5} x^{6} + 126 \, a^{15} b^{4} x^{5} + 84 \, a^{16} b^{3} x^{4} + 36 \, a^{17} b^{2} x^{3} + 9 \, a^{18} b x^{2} + a^{19} x\right )}} + \frac {10 \, b \log \left (b x + a\right )}{a^{11}} - \frac {10 \, b \log \left (x\right )}{a^{11}} \]
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Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^2 (a+b x)^{10}} \, dx=\frac {10 \, b \log \left ({\left | b x + a \right |}\right )}{a^{11}} - \frac {10 \, b \log \left ({\left | x \right |}\right )}{a^{11}} - \frac {2520 \, a b^{9} x^{9} + 21420 \, a^{2} b^{8} x^{8} + 80220 \, a^{3} b^{7} x^{7} + 173250 \, a^{4} b^{6} x^{6} + 236754 \, a^{5} b^{5} x^{5} + 210756 \, a^{6} b^{4} x^{4} + 120564 \, a^{7} b^{3} x^{3} + 41481 \, a^{8} b^{2} x^{2} + 7129 \, a^{9} b x + 252 \, a^{10}}{252 \, {\left (b x + a\right )}^{9} a^{11} x} \]
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Time = 0.40 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x^2 (a+b x)^{10}} \, dx=\frac {20\,b\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^{11}}-\frac {\frac {1}{a}+\frac {4609\,b^2\,x^2}{28\,a^3}+\frac {3349\,b^3\,x^3}{7\,a^4}+\frac {2509\,b^4\,x^4}{3\,a^5}+\frac {1879\,b^5\,x^5}{2\,a^6}+\frac {1375\,b^6\,x^6}{2\,a^7}+\frac {955\,b^7\,x^7}{3\,a^8}+\frac {85\,b^8\,x^8}{a^9}+\frac {10\,b^9\,x^9}{a^{10}}+\frac {7129\,b\,x}{252\,a^2}}{a^9\,x+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^3+84\,a^6\,b^3\,x^4+126\,a^5\,b^4\,x^5+126\,a^4\,b^5\,x^6+84\,a^3\,b^6\,x^7+36\,a^2\,b^7\,x^8+9\,a\,b^8\,x^9+b^9\,x^{10}} \]
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