Integrand size = 11, antiderivative size = 163 \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=-\frac {1}{9 a^3 x^9}+\frac {3 b}{8 a^4 x^8}-\frac {6 b^2}{7 a^5 x^7}+\frac {5 b^3}{3 a^6 x^6}-\frac {3 b^4}{a^7 x^5}+\frac {21 b^5}{4 a^8 x^4}-\frac {28 b^6}{3 a^9 x^3}+\frac {18 b^7}{a^{10} x^2}-\frac {45 b^8}{a^{11} x}-\frac {b^9}{2 a^{10} (a+b x)^2}-\frac {10 b^9}{a^{11} (a+b x)}-\frac {55 b^9 \log (x)}{a^{12}}+\frac {55 b^9 \log (a+b x)}{a^{12}} \]
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Time = 0.08 (sec) , antiderivative size = 163, 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} \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=-\frac {55 b^9 \log (x)}{a^{12}}+\frac {55 b^9 \log (a+b x)}{a^{12}}-\frac {10 b^9}{a^{11} (a+b x)}-\frac {45 b^8}{a^{11} x}-\frac {b^9}{2 a^{10} (a+b x)^2}+\frac {18 b^7}{a^{10} x^2}-\frac {28 b^6}{3 a^9 x^3}+\frac {21 b^5}{4 a^8 x^4}-\frac {3 b^4}{a^7 x^5}+\frac {5 b^3}{3 a^6 x^6}-\frac {6 b^2}{7 a^5 x^7}+\frac {3 b}{8 a^4 x^8}-\frac {1}{9 a^3 x^9} \]
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Rule 46
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{a^3 x^{10}}-\frac {3 b}{a^4 x^9}+\frac {6 b^2}{a^5 x^8}-\frac {10 b^3}{a^6 x^7}+\frac {15 b^4}{a^7 x^6}-\frac {21 b^5}{a^8 x^5}+\frac {28 b^6}{a^9 x^4}-\frac {36 b^7}{a^{10} x^3}+\frac {45 b^8}{a^{11} x^2}-\frac {55 b^9}{a^{12} x}+\frac {b^{10}}{a^{10} (a+b x)^3}+\frac {10 b^{10}}{a^{11} (a+b x)^2}+\frac {55 b^{10}}{a^{12} (a+b x)}\right ) \, dx \\ & = -\frac {1}{9 a^3 x^9}+\frac {3 b}{8 a^4 x^8}-\frac {6 b^2}{7 a^5 x^7}+\frac {5 b^3}{3 a^6 x^6}-\frac {3 b^4}{a^7 x^5}+\frac {21 b^5}{4 a^8 x^4}-\frac {28 b^6}{3 a^9 x^3}+\frac {18 b^7}{a^{10} x^2}-\frac {45 b^8}{a^{11} x}-\frac {b^9}{2 a^{10} (a+b x)^2}-\frac {10 b^9}{a^{11} (a+b x)}-\frac {55 b^9 \log (x)}{a^{12}}+\frac {55 b^9 \log (a+b x)}{a^{12}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=-\frac {\frac {a \left (56 a^{10}-77 a^9 b x+110 a^8 b^2 x^2-165 a^7 b^3 x^3+264 a^6 b^4 x^4-462 a^5 b^5 x^5+924 a^4 b^6 x^6-2310 a^3 b^7 x^7+9240 a^2 b^8 x^8+41580 a b^9 x^9+27720 b^{10} x^{10}\right )}{x^9 (a+b x)^2}+27720 b^9 \log (x)-27720 b^9 \log (a+b x)}{504 a^{12}} \]
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Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.91
method | result | size |
norman | \(\frac {-\frac {1}{9 a}+\frac {11 b x}{72 a^{2}}-\frac {55 b^{2} x^{2}}{252 a^{3}}+\frac {55 b^{3} x^{3}}{168 a^{4}}-\frac {11 b^{4} x^{4}}{21 a^{5}}+\frac {11 b^{5} x^{5}}{12 a^{6}}-\frac {11 b^{6} x^{6}}{6 a^{7}}+\frac {55 b^{7} x^{7}}{12 a^{8}}-\frac {55 b^{8} x^{8}}{3 a^{9}}+\frac {110 b^{10} x^{10}}{a^{11}}+\frac {165 b^{11} x^{11}}{2 a^{12}}}{x^{9} \left (b x +a \right )^{2}}-\frac {55 b^{9} \ln \left (x \right )}{a^{12}}+\frac {55 b^{9} \ln \left (b x +a \right )}{a^{12}}\) | \(149\) |
default | \(-\frac {1}{9 a^{3} x^{9}}+\frac {3 b}{8 a^{4} x^{8}}-\frac {6 b^{2}}{7 a^{5} x^{7}}+\frac {5 b^{3}}{3 a^{6} x^{6}}-\frac {3 b^{4}}{a^{7} x^{5}}+\frac {21 b^{5}}{4 a^{8} x^{4}}-\frac {28 b^{6}}{3 a^{9} x^{3}}+\frac {18 b^{7}}{a^{10} x^{2}}-\frac {45 b^{8}}{a^{11} x}-\frac {b^{9}}{2 a^{10} \left (b x +a \right )^{2}}-\frac {10 b^{9}}{a^{11} \left (b x +a \right )}-\frac {55 b^{9} \ln \left (x \right )}{a^{12}}+\frac {55 b^{9} \ln \left (b x +a \right )}{a^{12}}\) | \(150\) |
risch | \(\frac {-\frac {55 b^{10} x^{10}}{a^{11}}-\frac {165 b^{9} x^{9}}{2 a^{10}}-\frac {55 b^{8} x^{8}}{3 a^{9}}+\frac {55 b^{7} x^{7}}{12 a^{8}}-\frac {11 b^{6} x^{6}}{6 a^{7}}+\frac {11 b^{5} x^{5}}{12 a^{6}}-\frac {11 b^{4} x^{4}}{21 a^{5}}+\frac {55 b^{3} x^{3}}{168 a^{4}}-\frac {55 b^{2} x^{2}}{252 a^{3}}+\frac {11 b x}{72 a^{2}}-\frac {1}{9 a}}{x^{9} \left (b x +a \right )^{2}}+\frac {55 b^{9} \ln \left (-b x -a \right )}{a^{12}}-\frac {55 b^{9} \ln \left (x \right )}{a^{12}}\) | \(152\) |
parallelrisch | \(-\frac {27720 \ln \left (x \right ) x^{11} b^{11}-27720 \ln \left (b x +a \right ) x^{11} b^{11}+55440 \ln \left (x \right ) x^{10} a \,b^{10}-55440 \ln \left (b x +a \right ) x^{10} a \,b^{10}-41580 b^{11} x^{11}+27720 \ln \left (x \right ) x^{9} a^{2} b^{9}-27720 \ln \left (b x +a \right ) x^{9} a^{2} b^{9}-55440 a \,x^{10} b^{10}+9240 a^{3} x^{8} b^{8}-2310 a^{4} b^{7} x^{7}+924 a^{5} b^{6} x^{6}-462 a^{6} b^{5} x^{5}+264 a^{7} b^{4} x^{4}-165 b^{3} a^{8} x^{3}+110 b^{2} a^{9} x^{2}-77 a^{10} b x +56 a^{11}}{504 a^{12} x^{9} \left (b x +a \right )^{2}}\) | \(203\) |
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Time = 0.23 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=-\frac {27720 \, a b^{10} x^{10} + 41580 \, a^{2} b^{9} x^{9} + 9240 \, a^{3} b^{8} x^{8} - 2310 \, a^{4} b^{7} x^{7} + 924 \, a^{5} b^{6} x^{6} - 462 \, a^{6} b^{5} x^{5} + 264 \, a^{7} b^{4} x^{4} - 165 \, a^{8} b^{3} x^{3} + 110 \, a^{9} b^{2} x^{2} - 77 \, a^{10} b x + 56 \, a^{11} - 27720 \, {\left (b^{11} x^{11} + 2 \, a b^{10} x^{10} + a^{2} b^{9} x^{9}\right )} \log \left (b x + a\right ) + 27720 \, {\left (b^{11} x^{11} + 2 \, a b^{10} x^{10} + a^{2} b^{9} x^{9}\right )} \log \left (x\right )}{504 \, {\left (a^{12} b^{2} x^{11} + 2 \, a^{13} b x^{10} + a^{14} x^{9}\right )}} \]
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Time = 0.37 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=\frac {- 56 a^{10} + 77 a^{9} b x - 110 a^{8} b^{2} x^{2} + 165 a^{7} b^{3} x^{3} - 264 a^{6} b^{4} x^{4} + 462 a^{5} b^{5} x^{5} - 924 a^{4} b^{6} x^{6} + 2310 a^{3} b^{7} x^{7} - 9240 a^{2} b^{8} x^{8} - 41580 a b^{9} x^{9} - 27720 b^{10} x^{10}}{504 a^{13} x^{9} + 1008 a^{12} b x^{10} + 504 a^{11} b^{2} x^{11}} + \frac {55 b^{9} \left (- \log {\left (x \right )} + \log {\left (\frac {a}{b} + x \right )}\right )}{a^{12}} \]
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Time = 0.21 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=-\frac {27720 \, b^{10} x^{10} + 41580 \, a b^{9} x^{9} + 9240 \, a^{2} b^{8} x^{8} - 2310 \, a^{3} b^{7} x^{7} + 924 \, a^{4} b^{6} x^{6} - 462 \, a^{5} b^{5} x^{5} + 264 \, a^{6} b^{4} x^{4} - 165 \, a^{7} b^{3} x^{3} + 110 \, a^{8} b^{2} x^{2} - 77 \, a^{9} b x + 56 \, a^{10}}{504 \, {\left (a^{11} b^{2} x^{11} + 2 \, a^{12} b x^{10} + a^{13} x^{9}\right )}} + \frac {55 \, b^{9} \log \left (b x + a\right )}{a^{12}} - \frac {55 \, b^{9} \log \left (x\right )}{a^{12}} \]
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Time = 0.29 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=\frac {55 \, b^{9} \log \left ({\left | b x + a \right |}\right )}{a^{12}} - \frac {55 \, b^{9} \log \left ({\left | x \right |}\right )}{a^{12}} - \frac {27720 \, a b^{10} x^{10} + 41580 \, a^{2} b^{9} x^{9} + 9240 \, a^{3} b^{8} x^{8} - 2310 \, a^{4} b^{7} x^{7} + 924 \, a^{5} b^{6} x^{6} - 462 \, a^{6} b^{5} x^{5} + 264 \, a^{7} b^{4} x^{4} - 165 \, a^{8} b^{3} x^{3} + 110 \, a^{9} b^{2} x^{2} - 77 \, a^{10} b x + 56 \, a^{11}}{504 \, {\left (b x + a\right )}^{2} a^{12} x^{9}} \]
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Time = 0.25 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^{10} (a+b x)^3} \, dx=\frac {110\,b^9\,\mathrm {atanh}\left (\frac {2\,b\,x}{a}+1\right )}{a^{12}}-\frac {\frac {1}{9\,a}+\frac {55\,b^2\,x^2}{252\,a^3}-\frac {55\,b^3\,x^3}{168\,a^4}+\frac {11\,b^4\,x^4}{21\,a^5}-\frac {11\,b^5\,x^5}{12\,a^6}+\frac {11\,b^6\,x^6}{6\,a^7}-\frac {55\,b^7\,x^7}{12\,a^8}+\frac {55\,b^8\,x^8}{3\,a^9}+\frac {165\,b^9\,x^9}{2\,a^{10}}+\frac {55\,b^{10}\,x^{10}}{a^{11}}-\frac {11\,b\,x}{72\,a^2}}{a^2\,x^9+2\,a\,b\,x^{10}+b^2\,x^{11}} \]
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