Integrand size = 11, antiderivative size = 126 \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=-\frac {a^{10}}{19 x^{19}}-\frac {5 a^9 b}{9 x^{18}}-\frac {45 a^8 b^2}{17 x^{17}}-\frac {15 a^7 b^3}{2 x^{16}}-\frac {14 a^6 b^4}{x^{15}}-\frac {18 a^5 b^5}{x^{14}}-\frac {210 a^4 b^6}{13 x^{13}}-\frac {10 a^3 b^7}{x^{12}}-\frac {45 a^2 b^8}{11 x^{11}}-\frac {a b^9}{x^{10}}-\frac {b^{10}}{9 x^9} \]
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Time = 0.03 (sec) , antiderivative size = 126, 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} \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=-\frac {a^{10}}{19 x^{19}}-\frac {5 a^9 b}{9 x^{18}}-\frac {45 a^8 b^2}{17 x^{17}}-\frac {15 a^7 b^3}{2 x^{16}}-\frac {14 a^6 b^4}{x^{15}}-\frac {18 a^5 b^5}{x^{14}}-\frac {210 a^4 b^6}{13 x^{13}}-\frac {10 a^3 b^7}{x^{12}}-\frac {45 a^2 b^8}{11 x^{11}}-\frac {a b^9}{x^{10}}-\frac {b^{10}}{9 x^9} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^{10}}{x^{20}}+\frac {10 a^9 b}{x^{19}}+\frac {45 a^8 b^2}{x^{18}}+\frac {120 a^7 b^3}{x^{17}}+\frac {210 a^6 b^4}{x^{16}}+\frac {252 a^5 b^5}{x^{15}}+\frac {210 a^4 b^6}{x^{14}}+\frac {120 a^3 b^7}{x^{13}}+\frac {45 a^2 b^8}{x^{12}}+\frac {10 a b^9}{x^{11}}+\frac {b^{10}}{x^{10}}\right ) \, dx \\ & = -\frac {a^{10}}{19 x^{19}}-\frac {5 a^9 b}{9 x^{18}}-\frac {45 a^8 b^2}{17 x^{17}}-\frac {15 a^7 b^3}{2 x^{16}}-\frac {14 a^6 b^4}{x^{15}}-\frac {18 a^5 b^5}{x^{14}}-\frac {210 a^4 b^6}{13 x^{13}}-\frac {10 a^3 b^7}{x^{12}}-\frac {45 a^2 b^8}{11 x^{11}}-\frac {a b^9}{x^{10}}-\frac {b^{10}}{9 x^9} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=-\frac {a^{10}}{19 x^{19}}-\frac {5 a^9 b}{9 x^{18}}-\frac {45 a^8 b^2}{17 x^{17}}-\frac {15 a^7 b^3}{2 x^{16}}-\frac {14 a^6 b^4}{x^{15}}-\frac {18 a^5 b^5}{x^{14}}-\frac {210 a^4 b^6}{13 x^{13}}-\frac {10 a^3 b^7}{x^{12}}-\frac {45 a^2 b^8}{11 x^{11}}-\frac {a b^9}{x^{10}}-\frac {b^{10}}{9 x^9} \]
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Time = 0.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.89
method | result | size |
norman | \(\frac {-\frac {1}{19} a^{10}-\frac {5}{9} a^{9} b x -\frac {45}{17} a^{8} b^{2} x^{2}-\frac {15}{2} a^{7} b^{3} x^{3}-14 a^{6} b^{4} x^{4}-18 a^{5} b^{5} x^{5}-\frac {210}{13} a^{4} b^{6} x^{6}-10 a^{3} b^{7} x^{7}-\frac {45}{11} a^{2} b^{8} x^{8}-a \,b^{9} x^{9}-\frac {1}{9} b^{10} x^{10}}{x^{19}}\) | \(112\) |
risch | \(\frac {-\frac {1}{19} a^{10}-\frac {5}{9} a^{9} b x -\frac {45}{17} a^{8} b^{2} x^{2}-\frac {15}{2} a^{7} b^{3} x^{3}-14 a^{6} b^{4} x^{4}-18 a^{5} b^{5} x^{5}-\frac {210}{13} a^{4} b^{6} x^{6}-10 a^{3} b^{7} x^{7}-\frac {45}{11} a^{2} b^{8} x^{8}-a \,b^{9} x^{9}-\frac {1}{9} b^{10} x^{10}}{x^{19}}\) | \(112\) |
gosper | \(-\frac {92378 b^{10} x^{10}+831402 a \,b^{9} x^{9}+3401190 a^{2} b^{8} x^{8}+8314020 a^{3} b^{7} x^{7}+13430340 a^{4} b^{6} x^{6}+14965236 a^{5} b^{5} x^{5}+11639628 a^{6} b^{4} x^{4}+6235515 a^{7} b^{3} x^{3}+2200770 a^{8} b^{2} x^{2}+461890 a^{9} b x +43758 a^{10}}{831402 x^{19}}\) | \(113\) |
default | \(-\frac {a^{10}}{19 x^{19}}-\frac {5 a^{9} b}{9 x^{18}}-\frac {45 a^{8} b^{2}}{17 x^{17}}-\frac {15 a^{7} b^{3}}{2 x^{16}}-\frac {14 a^{6} b^{4}}{x^{15}}-\frac {18 a^{5} b^{5}}{x^{14}}-\frac {210 a^{4} b^{6}}{13 x^{13}}-\frac {10 a^{3} b^{7}}{x^{12}}-\frac {45 a^{2} b^{8}}{11 x^{11}}-\frac {a \,b^{9}}{x^{10}}-\frac {b^{10}}{9 x^{9}}\) | \(113\) |
parallelrisch | \(\frac {-92378 b^{10} x^{10}-831402 a \,b^{9} x^{9}-3401190 a^{2} b^{8} x^{8}-8314020 a^{3} b^{7} x^{7}-13430340 a^{4} b^{6} x^{6}-14965236 a^{5} b^{5} x^{5}-11639628 a^{6} b^{4} x^{4}-6235515 a^{7} b^{3} x^{3}-2200770 a^{8} b^{2} x^{2}-461890 a^{9} b x -43758 a^{10}}{831402 x^{19}}\) | \(113\) |
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Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=-\frac {92378 \, b^{10} x^{10} + 831402 \, a b^{9} x^{9} + 3401190 \, a^{2} b^{8} x^{8} + 8314020 \, a^{3} b^{7} x^{7} + 13430340 \, a^{4} b^{6} x^{6} + 14965236 \, a^{5} b^{5} x^{5} + 11639628 \, a^{6} b^{4} x^{4} + 6235515 \, a^{7} b^{3} x^{3} + 2200770 \, a^{8} b^{2} x^{2} + 461890 \, a^{9} b x + 43758 \, a^{10}}{831402 \, x^{19}} \]
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Time = 0.70 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=\frac {- 43758 a^{10} - 461890 a^{9} b x - 2200770 a^{8} b^{2} x^{2} - 6235515 a^{7} b^{3} x^{3} - 11639628 a^{6} b^{4} x^{4} - 14965236 a^{5} b^{5} x^{5} - 13430340 a^{4} b^{6} x^{6} - 8314020 a^{3} b^{7} x^{7} - 3401190 a^{2} b^{8} x^{8} - 831402 a b^{9} x^{9} - 92378 b^{10} x^{10}}{831402 x^{19}} \]
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Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=-\frac {92378 \, b^{10} x^{10} + 831402 \, a b^{9} x^{9} + 3401190 \, a^{2} b^{8} x^{8} + 8314020 \, a^{3} b^{7} x^{7} + 13430340 \, a^{4} b^{6} x^{6} + 14965236 \, a^{5} b^{5} x^{5} + 11639628 \, a^{6} b^{4} x^{4} + 6235515 \, a^{7} b^{3} x^{3} + 2200770 \, a^{8} b^{2} x^{2} + 461890 \, a^{9} b x + 43758 \, a^{10}}{831402 \, x^{19}} \]
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Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=-\frac {92378 \, b^{10} x^{10} + 831402 \, a b^{9} x^{9} + 3401190 \, a^{2} b^{8} x^{8} + 8314020 \, a^{3} b^{7} x^{7} + 13430340 \, a^{4} b^{6} x^{6} + 14965236 \, a^{5} b^{5} x^{5} + 11639628 \, a^{6} b^{4} x^{4} + 6235515 \, a^{7} b^{3} x^{3} + 2200770 \, a^{8} b^{2} x^{2} + 461890 \, a^{9} b x + 43758 \, a^{10}}{831402 \, x^{19}} \]
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Time = 0.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=-\frac {\frac {a^{10}}{19}+\frac {5\,a^9\,b\,x}{9}+\frac {45\,a^8\,b^2\,x^2}{17}+\frac {15\,a^7\,b^3\,x^3}{2}+14\,a^6\,b^4\,x^4+18\,a^5\,b^5\,x^5+\frac {210\,a^4\,b^6\,x^6}{13}+10\,a^3\,b^7\,x^7+\frac {45\,a^2\,b^8\,x^8}{11}+a\,b^9\,x^9+\frac {b^{10}\,x^{10}}{9}}{x^{19}} \]
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