Integrand size = 19, antiderivative size = 151 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {d}{20 x^{20}}-\frac {10 d+e}{19 x^{19}}-\frac {5 (9 d+2 e)}{18 x^{18}}-\frac {15 (8 d+3 e)}{17 x^{17}}-\frac {15 (7 d+4 e)}{8 x^{16}}-\frac {14 (6 d+5 e)}{5 x^{15}}-\frac {3 (5 d+6 e)}{x^{14}}-\frac {30 (4 d+7 e)}{13 x^{13}}-\frac {5 (3 d+8 e)}{4 x^{12}}-\frac {5 (2 d+9 e)}{11 x^{11}}-\frac {d+10 e}{10 x^{10}}-\frac {e}{9 x^9} \]
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Time = 0.05 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {27, 77} \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {10 d+e}{19 x^{19}}-\frac {5 (9 d+2 e)}{18 x^{18}}-\frac {15 (8 d+3 e)}{17 x^{17}}-\frac {15 (7 d+4 e)}{8 x^{16}}-\frac {14 (6 d+5 e)}{5 x^{15}}-\frac {3 (5 d+6 e)}{x^{14}}-\frac {30 (4 d+7 e)}{13 x^{13}}-\frac {5 (3 d+8 e)}{4 x^{12}}-\frac {5 (2 d+9 e)}{11 x^{11}}-\frac {d+10 e}{10 x^{10}}-\frac {d}{20 x^{20}}-\frac {e}{9 x^9} \]
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Rule 27
Rule 77
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1+x)^{10} (d+e x)}{x^{21}} \, dx \\ & = \int \left (\frac {d}{x^{21}}+\frac {10 d+e}{x^{20}}+\frac {5 (9 d+2 e)}{x^{19}}+\frac {15 (8 d+3 e)}{x^{18}}+\frac {30 (7 d+4 e)}{x^{17}}+\frac {42 (6 d+5 e)}{x^{16}}+\frac {42 (5 d+6 e)}{x^{15}}+\frac {30 (4 d+7 e)}{x^{14}}+\frac {15 (3 d+8 e)}{x^{13}}+\frac {5 (2 d+9 e)}{x^{12}}+\frac {d+10 e}{x^{11}}+\frac {e}{x^{10}}\right ) \, dx \\ & = -\frac {d}{20 x^{20}}-\frac {10 d+e}{19 x^{19}}-\frac {5 (9 d+2 e)}{18 x^{18}}-\frac {15 (8 d+3 e)}{17 x^{17}}-\frac {15 (7 d+4 e)}{8 x^{16}}-\frac {14 (6 d+5 e)}{5 x^{15}}-\frac {3 (5 d+6 e)}{x^{14}}-\frac {30 (4 d+7 e)}{13 x^{13}}-\frac {5 (3 d+8 e)}{4 x^{12}}-\frac {5 (2 d+9 e)}{11 x^{11}}-\frac {d+10 e}{10 x^{10}}-\frac {e}{9 x^9} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {d}{20 x^{20}}-\frac {10 d+e}{19 x^{19}}-\frac {5 (9 d+2 e)}{18 x^{18}}-\frac {15 (8 d+3 e)}{17 x^{17}}-\frac {15 (7 d+4 e)}{8 x^{16}}-\frac {14 (6 d+5 e)}{5 x^{15}}-\frac {3 (5 d+6 e)}{x^{14}}-\frac {30 (4 d+7 e)}{13 x^{13}}-\frac {5 (3 d+8 e)}{4 x^{12}}-\frac {5 (2 d+9 e)}{11 x^{11}}-\frac {d+10 e}{10 x^{10}}-\frac {e}{9 x^9} \]
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Time = 0.10 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.81
method | result | size |
norman | \(\frac {-\frac {d}{20}+\left (-\frac {10 d}{19}-\frac {e}{19}\right ) x +\left (-\frac {5 d}{2}-\frac {5 e}{9}\right ) x^{2}+\left (-\frac {120 d}{17}-\frac {45 e}{17}\right ) x^{3}+\left (-\frac {105 d}{8}-\frac {15 e}{2}\right ) x^{4}+\left (-\frac {84 d}{5}-14 e \right ) x^{5}+\left (-15 d -18 e \right ) x^{6}+\left (-\frac {120 d}{13}-\frac {210 e}{13}\right ) x^{7}+\left (-\frac {15 d}{4}-10 e \right ) x^{8}+\left (-\frac {10 d}{11}-\frac {45 e}{11}\right ) x^{9}+\left (-\frac {d}{10}-e \right ) x^{10}-\frac {e \,x^{11}}{9}}{x^{20}}\) | \(123\) |
risch | \(\frac {-\frac {d}{20}+\left (-\frac {10 d}{19}-\frac {e}{19}\right ) x +\left (-\frac {5 d}{2}-\frac {5 e}{9}\right ) x^{2}+\left (-\frac {120 d}{17}-\frac {45 e}{17}\right ) x^{3}+\left (-\frac {105 d}{8}-\frac {15 e}{2}\right ) x^{4}+\left (-\frac {84 d}{5}-14 e \right ) x^{5}+\left (-15 d -18 e \right ) x^{6}+\left (-\frac {120 d}{13}-\frac {210 e}{13}\right ) x^{7}+\left (-\frac {15 d}{4}-10 e \right ) x^{8}+\left (-\frac {10 d}{11}-\frac {45 e}{11}\right ) x^{9}+\left (-\frac {d}{10}-e \right ) x^{10}-\frac {e \,x^{11}}{9}}{x^{20}}\) | \(123\) |
default | \(-\frac {45 d +10 e}{18 x^{18}}-\frac {210 d +252 e}{14 x^{14}}-\frac {d +10 e}{10 x^{10}}-\frac {120 d +210 e}{13 x^{13}}-\frac {e}{9 x^{9}}-\frac {d}{20 x^{20}}-\frac {120 d +45 e}{17 x^{17}}-\frac {10 d +e}{19 x^{19}}-\frac {210 d +120 e}{16 x^{16}}-\frac {252 d +210 e}{15 x^{15}}-\frac {10 d +45 e}{11 x^{11}}-\frac {45 d +120 e}{12 x^{12}}\) | \(130\) |
gosper | \(-\frac {1847560 e \,x^{11}+1662804 d \,x^{10}+16628040 e \,x^{10}+15116400 d \,x^{9}+68023800 e \,x^{9}+62355150 d \,x^{8}+166280400 e \,x^{8}+153489600 d \,x^{7}+268606800 e \,x^{7}+249420600 d \,x^{6}+299304720 e \,x^{6}+279351072 d \,x^{5}+232792560 e \,x^{5}+218243025 d \,x^{4}+124710300 e \,x^{4}+117374400 d \,x^{3}+44015400 e \,x^{3}+41570100 d \,x^{2}+9237800 e \,x^{2}+8751600 d x +875160 e x +831402 d}{16628040 x^{20}}\) | \(132\) |
parallelrisch | \(\frac {-1847560 e \,x^{11}-1662804 d \,x^{10}-16628040 e \,x^{10}-15116400 d \,x^{9}-68023800 e \,x^{9}-62355150 d \,x^{8}-166280400 e \,x^{8}-153489600 d \,x^{7}-268606800 e \,x^{7}-249420600 d \,x^{6}-299304720 e \,x^{6}-279351072 d \,x^{5}-232792560 e \,x^{5}-218243025 d \,x^{4}-124710300 e \,x^{4}-117374400 d \,x^{3}-44015400 e \,x^{3}-41570100 d \,x^{2}-9237800 e \,x^{2}-8751600 d x -875160 e x -831402 d}{16628040 x^{20}}\) | \(132\) |
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Time = 0.33 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {1847560 \, e x^{11} + 1662804 \, {\left (d + 10 \, e\right )} x^{10} + 7558200 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 20785050 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 38372400 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 49884120 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 46558512 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 31177575 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 14671800 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 4618900 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 875160 \, {\left (10 \, d + e\right )} x + 831402 \, d}{16628040 \, x^{20}} \]
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Time = 19.83 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=\frac {- 831402 d - 1847560 e x^{11} + x^{10} \left (- 1662804 d - 16628040 e\right ) + x^{9} \left (- 15116400 d - 68023800 e\right ) + x^{8} \left (- 62355150 d - 166280400 e\right ) + x^{7} \left (- 153489600 d - 268606800 e\right ) + x^{6} \left (- 249420600 d - 299304720 e\right ) + x^{5} \left (- 279351072 d - 232792560 e\right ) + x^{4} \left (- 218243025 d - 124710300 e\right ) + x^{3} \left (- 117374400 d - 44015400 e\right ) + x^{2} \left (- 41570100 d - 9237800 e\right ) + x \left (- 8751600 d - 875160 e\right )}{16628040 x^{20}} \]
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Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.85 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {1847560 \, e x^{11} + 1662804 \, {\left (d + 10 \, e\right )} x^{10} + 7558200 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 20785050 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 38372400 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 49884120 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 46558512 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 31177575 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 14671800 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 4618900 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 875160 \, {\left (10 \, d + e\right )} x + 831402 \, d}{16628040 \, x^{20}} \]
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Time = 0.27 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {1847560 \, e x^{11} + 1662804 \, d x^{10} + 16628040 \, e x^{10} + 15116400 \, d x^{9} + 68023800 \, e x^{9} + 62355150 \, d x^{8} + 166280400 \, e x^{8} + 153489600 \, d x^{7} + 268606800 \, e x^{7} + 249420600 \, d x^{6} + 299304720 \, e x^{6} + 279351072 \, d x^{5} + 232792560 \, e x^{5} + 218243025 \, d x^{4} + 124710300 \, e x^{4} + 117374400 \, d x^{3} + 44015400 \, e x^{3} + 41570100 \, d x^{2} + 9237800 \, e x^{2} + 8751600 \, d x + 875160 \, e x + 831402 \, d}{16628040 \, x^{20}} \]
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Time = 0.13 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{21}} \, dx=-\frac {\frac {e\,x^{11}}{9}+\left (\frac {d}{10}+e\right )\,x^{10}+\left (\frac {10\,d}{11}+\frac {45\,e}{11}\right )\,x^9+\left (\frac {15\,d}{4}+10\,e\right )\,x^8+\left (\frac {120\,d}{13}+\frac {210\,e}{13}\right )\,x^7+\left (15\,d+18\,e\right )\,x^6+\left (\frac {84\,d}{5}+14\,e\right )\,x^5+\left (\frac {105\,d}{8}+\frac {15\,e}{2}\right )\,x^4+\left (\frac {120\,d}{17}+\frac {45\,e}{17}\right )\,x^3+\left (\frac {5\,d}{2}+\frac {5\,e}{9}\right )\,x^2+\left (\frac {10\,d}{19}+\frac {e}{19}\right )\,x+\frac {d}{20}}{x^{20}} \]
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