Integrand size = 11, antiderivative size = 186 \[ \int \frac {x^{12}}{(a+b x)^{10}} \, dx=\frac {55 a^2 x}{b^{12}}-\frac {5 a x^2}{b^{11}}+\frac {x^3}{3 b^{10}}-\frac {a^{12}}{9 b^{13} (a+b x)^9}+\frac {3 a^{11}}{2 b^{13} (a+b x)^8}-\frac {66 a^{10}}{7 b^{13} (a+b x)^7}+\frac {110 a^9}{3 b^{13} (a+b x)^6}-\frac {99 a^8}{b^{13} (a+b x)^5}+\frac {198 a^7}{b^{13} (a+b x)^4}-\frac {308 a^6}{b^{13} (a+b x)^3}+\frac {396 a^5}{b^{13} (a+b x)^2}-\frac {495 a^4}{b^{13} (a+b x)}-\frac {220 a^3 \log (a+b x)}{b^{13}} \]
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Time = 0.13 (sec) , antiderivative size = 186, 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} \[ \int \frac {x^{12}}{(a+b x)^{10}} \, dx=-\frac {a^{12}}{9 b^{13} (a+b x)^9}+\frac {3 a^{11}}{2 b^{13} (a+b x)^8}-\frac {66 a^{10}}{7 b^{13} (a+b x)^7}+\frac {110 a^9}{3 b^{13} (a+b x)^6}-\frac {99 a^8}{b^{13} (a+b x)^5}+\frac {198 a^7}{b^{13} (a+b x)^4}-\frac {308 a^6}{b^{13} (a+b x)^3}+\frac {396 a^5}{b^{13} (a+b x)^2}-\frac {495 a^4}{b^{13} (a+b x)}-\frac {220 a^3 \log (a+b x)}{b^{13}}+\frac {55 a^2 x}{b^{12}}-\frac {5 a x^2}{b^{11}}+\frac {x^3}{3 b^{10}} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {55 a^2}{b^{12}}-\frac {10 a x}{b^{11}}+\frac {x^2}{b^{10}}+\frac {a^{12}}{b^{12} (a+b x)^{10}}-\frac {12 a^{11}}{b^{12} (a+b x)^9}+\frac {66 a^{10}}{b^{12} (a+b x)^8}-\frac {220 a^9}{b^{12} (a+b x)^7}+\frac {495 a^8}{b^{12} (a+b x)^6}-\frac {792 a^7}{b^{12} (a+b x)^5}+\frac {924 a^6}{b^{12} (a+b x)^4}-\frac {792 a^5}{b^{12} (a+b x)^3}+\frac {495 a^4}{b^{12} (a+b x)^2}-\frac {220 a^3}{b^{12} (a+b x)}\right ) \, dx \\ & = \frac {55 a^2 x}{b^{12}}-\frac {5 a x^2}{b^{11}}+\frac {x^3}{3 b^{10}}-\frac {a^{12}}{9 b^{13} (a+b x)^9}+\frac {3 a^{11}}{2 b^{13} (a+b x)^8}-\frac {66 a^{10}}{7 b^{13} (a+b x)^7}+\frac {110 a^9}{3 b^{13} (a+b x)^6}-\frac {99 a^8}{b^{13} (a+b x)^5}+\frac {198 a^7}{b^{13} (a+b x)^4}-\frac {308 a^6}{b^{13} (a+b x)^3}+\frac {396 a^5}{b^{13} (a+b x)^2}-\frac {495 a^4}{b^{13} (a+b x)}-\frac {220 a^3 \log (a+b x)}{b^{13}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.87 \[ \int \frac {x^{12}}{(a+b x)^{10}} \, dx=-\frac {35201 a^{12}+289089 a^{11} b x+1031616 a^{10} b^2 x^2+2074464 a^9 b^3 x^3+2529576 a^8 b^4 x^4+1831032 a^7 b^5 x^5+638568 a^6 b^6 x^6-58968 a^5 b^7 x^7-139482 a^4 b^8 x^8-43218 a^3 b^9 x^9-2772 a^2 b^{10} x^{10}+252 a b^{11} x^{11}-42 b^{12} x^{12}+27720 a^3 (a+b x)^9 \log (a+b x)}{126 b^{13} (a+b x)^9} \]
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Time = 0.05 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.77
method | result | size |
risch | \(\frac {x^{3}}{3 b^{10}}-\frac {5 a \,x^{2}}{b^{11}}+\frac {55 a^{2} x}{b^{12}}+\frac {-495 a^{4} b^{7} x^{8}-3564 a^{5} b^{6} x^{7}-11396 a^{6} b^{5} x^{6}-21054 a^{7} b^{4} x^{5}-24519 b^{3} a^{8} x^{4}-\frac {55198 b^{2} a^{9} x^{3}}{3}-\frac {60742 a^{10} b \,x^{2}}{7}-\frac {32891 a^{11} x}{14}-\frac {35201 a^{12}}{126 b}}{b^{12} \left (b x +a \right )^{9}}-\frac {220 a^{3} \ln \left (b x +a \right )}{b^{13}}\) | \(143\) |
norman | \(\frac {\frac {x^{12}}{3 b}-\frac {2 a \,x^{11}}{b^{2}}+\frac {22 a^{2} x^{10}}{b^{3}}-\frac {78419 a^{12}}{126 b^{13}}-\frac {1980 a^{4} x^{8}}{b^{5}}-\frac {11880 a^{5} x^{7}}{b^{6}}-\frac {33880 a^{6} x^{6}}{b^{7}}-\frac {57750 a^{7} x^{5}}{b^{8}}-\frac {63294 a^{8} x^{4}}{b^{9}}-\frac {45276 a^{9} x^{3}}{b^{10}}-\frac {143748 a^{10} x^{2}}{7 b^{11}}-\frac {75339 a^{11} x}{14 b^{12}}}{\left (b x +a \right )^{9}}-\frac {220 a^{3} \ln \left (b x +a \right )}{b^{13}}\) | \(147\) |
default | \(\frac {\frac {1}{3} b^{2} x^{3}-5 a b \,x^{2}+55 a^{2} x}{b^{12}}-\frac {a^{12}}{9 b^{13} \left (b x +a \right )^{9}}-\frac {220 a^{3} \ln \left (b x +a \right )}{b^{13}}+\frac {110 a^{9}}{3 b^{13} \left (b x +a \right )^{6}}+\frac {198 a^{7}}{b^{13} \left (b x +a \right )^{4}}-\frac {308 a^{6}}{b^{13} \left (b x +a \right )^{3}}+\frac {3 a^{11}}{2 b^{13} \left (b x +a \right )^{8}}-\frac {99 a^{8}}{b^{13} \left (b x +a \right )^{5}}-\frac {66 a^{10}}{7 b^{13} \left (b x +a \right )^{7}}+\frac {396 a^{5}}{b^{13} \left (b x +a \right )^{2}}-\frac {495 a^{4}}{b^{13} \left (b x +a \right )}\) | \(177\) |
parallelrisch | \(-\frac {78419 a^{12}+27720 \ln \left (b x +a \right ) a^{12}+27720 \ln \left (b x +a \right ) x^{9} a^{3} b^{9}+249480 \ln \left (b x +a \right ) x^{8} a^{4} b^{8}+997920 \ln \left (b x +a \right ) x^{7} a^{5} b^{7}+2328480 \ln \left (b x +a \right ) x^{6} a^{6} b^{6}+3492720 \ln \left (b x +a \right ) x^{5} a^{7} b^{5}+3492720 \ln \left (b x +a \right ) x^{4} a^{8} b^{4}+2328480 \ln \left (b x +a \right ) x^{3} a^{9} b^{3}+997920 \ln \left (b x +a \right ) x^{2} a^{10} b^{2}+249480 \ln \left (b x +a \right ) x \,a^{11} b -42 b^{12} x^{12}+249480 a^{4} x^{8} b^{8}+1496880 a^{5} x^{7} b^{7}+4268880 a^{6} x^{6} b^{6}+7276500 a^{7} x^{5} b^{5}+7975044 a^{8} x^{4} b^{4}+5704776 a^{9} x^{3} b^{3}+2587464 a^{10} x^{2} b^{2}+678051 a^{11} x b +252 a \,x^{11} b^{11}-2772 a^{2} x^{10} b^{10}}{126 b^{13} \left (b x +a \right )^{9}}\) | \(291\) |
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Time = 0.22 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.82 \[ \int \frac {x^{12}}{(a+b x)^{10}} \, dx=\frac {42 \, b^{12} x^{12} - 252 \, a b^{11} x^{11} + 2772 \, a^{2} b^{10} x^{10} + 43218 \, a^{3} b^{9} x^{9} + 139482 \, a^{4} b^{8} x^{8} + 58968 \, a^{5} b^{7} x^{7} - 638568 \, a^{6} b^{6} x^{6} - 1831032 \, a^{7} b^{5} x^{5} - 2529576 \, a^{8} b^{4} x^{4} - 2074464 \, a^{9} b^{3} x^{3} - 1031616 \, a^{10} b^{2} x^{2} - 289089 \, a^{11} b x - 35201 \, a^{12} - 27720 \, {\left (a^{3} b^{9} x^{9} + 9 \, a^{4} b^{8} x^{8} + 36 \, a^{5} b^{7} x^{7} + 84 \, a^{6} b^{6} x^{6} + 126 \, a^{7} b^{5} x^{5} + 126 \, a^{8} b^{4} x^{4} + 84 \, a^{9} b^{3} x^{3} + 36 \, a^{10} b^{2} x^{2} + 9 \, a^{11} b x + a^{12}\right )} \log \left (b x + a\right )}{126 \, {\left (b^{22} x^{9} + 9 \, a b^{21} x^{8} + 36 \, a^{2} b^{20} x^{7} + 84 \, a^{3} b^{19} x^{6} + 126 \, a^{4} b^{18} x^{5} + 126 \, a^{5} b^{17} x^{4} + 84 \, a^{6} b^{16} x^{3} + 36 \, a^{7} b^{15} x^{2} + 9 \, a^{8} b^{14} x + a^{9} b^{13}\right )}} \]
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Time = 0.76 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.34 \[ \int \frac {x^{12}}{(a+b x)^{10}} \, dx=- \frac {220 a^{3} \log {\left (a + b x \right )}}{b^{13}} + \frac {55 a^{2} x}{b^{12}} - \frac {5 a x^{2}}{b^{11}} + \frac {- 35201 a^{12} - 296019 a^{11} b x - 1093356 a^{10} b^{2} x^{2} - 2318316 a^{9} b^{3} x^{3} - 3089394 a^{8} b^{4} x^{4} - 2652804 a^{7} b^{5} x^{5} - 1435896 a^{6} b^{6} x^{6} - 449064 a^{5} b^{7} x^{7} - 62370 a^{4} b^{8} x^{8}}{126 a^{9} b^{13} + 1134 a^{8} b^{14} x + 4536 a^{7} b^{15} x^{2} + 10584 a^{6} b^{16} x^{3} + 15876 a^{5} b^{17} x^{4} + 15876 a^{4} b^{18} x^{5} + 10584 a^{3} b^{19} x^{6} + 4536 a^{2} b^{20} x^{7} + 1134 a b^{21} x^{8} + 126 b^{22} x^{9}} + \frac {x^{3}}{3 b^{10}} \]
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Time = 0.22 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.26 \[ \int \frac {x^{12}}{(a+b x)^{10}} \, dx=-\frac {62370 \, a^{4} b^{8} x^{8} + 449064 \, a^{5} b^{7} x^{7} + 1435896 \, a^{6} b^{6} x^{6} + 2652804 \, a^{7} b^{5} x^{5} + 3089394 \, a^{8} b^{4} x^{4} + 2318316 \, a^{9} b^{3} x^{3} + 1093356 \, a^{10} b^{2} x^{2} + 296019 \, a^{11} b x + 35201 \, a^{12}}{126 \, {\left (b^{22} x^{9} + 9 \, a b^{21} x^{8} + 36 \, a^{2} b^{20} x^{7} + 84 \, a^{3} b^{19} x^{6} + 126 \, a^{4} b^{18} x^{5} + 126 \, a^{5} b^{17} x^{4} + 84 \, a^{6} b^{16} x^{3} + 36 \, a^{7} b^{15} x^{2} + 9 \, a^{8} b^{14} x + a^{9} b^{13}\right )}} - \frac {220 \, a^{3} \log \left (b x + a\right )}{b^{13}} + \frac {b^{2} x^{3} - 15 \, a b x^{2} + 165 \, a^{2} x}{3 \, b^{12}} \]
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Time = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.80 \[ \int \frac {x^{12}}{(a+b x)^{10}} \, dx=-\frac {220 \, a^{3} \log \left ({\left | b x + a \right |}\right )}{b^{13}} - \frac {62370 \, a^{4} b^{8} x^{8} + 449064 \, a^{5} b^{7} x^{7} + 1435896 \, a^{6} b^{6} x^{6} + 2652804 \, a^{7} b^{5} x^{5} + 3089394 \, a^{8} b^{4} x^{4} + 2318316 \, a^{9} b^{3} x^{3} + 1093356 \, a^{10} b^{2} x^{2} + 296019 \, a^{11} b x + 35201 \, a^{12}}{126 \, {\left (b x + a\right )}^{9} b^{13}} + \frac {b^{20} x^{3} - 15 \, a b^{19} x^{2} + 165 \, a^{2} b^{18} x}{3 \, b^{30}} \]
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Time = 1.35 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.81 \[ \int \frac {x^{12}}{(a+b x)^{10}} \, dx=-\frac {6\,a\,{\left (a+b\,x\right )}^2-\frac {{\left (a+b\,x\right )}^3}{3}+\frac {495\,a^4}{a+b\,x}-\frac {396\,a^5}{{\left (a+b\,x\right )}^2}+\frac {308\,a^6}{{\left (a+b\,x\right )}^3}-\frac {198\,a^7}{{\left (a+b\,x\right )}^4}+\frac {99\,a^8}{{\left (a+b\,x\right )}^5}-\frac {110\,a^9}{3\,{\left (a+b\,x\right )}^6}+\frac {66\,a^{10}}{7\,{\left (a+b\,x\right )}^7}-\frac {3\,a^{11}}{2\,{\left (a+b\,x\right )}^8}+\frac {a^{12}}{9\,{\left (a+b\,x\right )}^9}+220\,a^3\,\ln \left (a+b\,x\right )-66\,a^2\,b\,x}{b^{13}} \]
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