Optimal. Leaf size=97 \[ -\frac {(x+1)^{12}}{19 x^{19}}+\frac {7 (x+1)^{12}}{342 x^{18}}-\frac {7 (x+1)^{12}}{969 x^{17}}+\frac {35 (x+1)^{12}}{15504 x^{16}}-\frac {7 (x+1)^{12}}{11628 x^{15}}+\frac {(x+1)^{12}}{7752 x^{14}}-\frac {(x+1)^{12}}{50388 x^{13}}+\frac {(x+1)^{12}}{604656 x^{12}} \]
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Rubi [A] time = 0.03, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {27, 45, 37} \[ \frac {(x+1)^{12}}{604656 x^{12}}-\frac {(x+1)^{12}}{50388 x^{13}}+\frac {(x+1)^{12}}{7752 x^{14}}-\frac {7 (x+1)^{12}}{11628 x^{15}}+\frac {35 (x+1)^{12}}{15504 x^{16}}-\frac {7 (x+1)^{12}}{969 x^{17}}+\frac {7 (x+1)^{12}}{342 x^{18}}-\frac {(x+1)^{12}}{19 x^{19}} \]
Antiderivative was successfully verified.
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Rule 27
Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{20}} \, dx &=\int \frac {(1+x)^{11}}{x^{20}} \, dx\\ &=-\frac {(1+x)^{12}}{19 x^{19}}-\frac {7}{19} \int \frac {(1+x)^{11}}{x^{19}} \, dx\\ &=-\frac {(1+x)^{12}}{19 x^{19}}+\frac {7 (1+x)^{12}}{342 x^{18}}+\frac {7}{57} \int \frac {(1+x)^{11}}{x^{18}} \, dx\\ &=-\frac {(1+x)^{12}}{19 x^{19}}+\frac {7 (1+x)^{12}}{342 x^{18}}-\frac {7 (1+x)^{12}}{969 x^{17}}-\frac {35}{969} \int \frac {(1+x)^{11}}{x^{17}} \, dx\\ &=-\frac {(1+x)^{12}}{19 x^{19}}+\frac {7 (1+x)^{12}}{342 x^{18}}-\frac {7 (1+x)^{12}}{969 x^{17}}+\frac {35 (1+x)^{12}}{15504 x^{16}}+\frac {35 \int \frac {(1+x)^{11}}{x^{16}} \, dx}{3876}\\ &=-\frac {(1+x)^{12}}{19 x^{19}}+\frac {7 (1+x)^{12}}{342 x^{18}}-\frac {7 (1+x)^{12}}{969 x^{17}}+\frac {35 (1+x)^{12}}{15504 x^{16}}-\frac {7 (1+x)^{12}}{11628 x^{15}}-\frac {7 \int \frac {(1+x)^{11}}{x^{15}} \, dx}{3876}\\ &=-\frac {(1+x)^{12}}{19 x^{19}}+\frac {7 (1+x)^{12}}{342 x^{18}}-\frac {7 (1+x)^{12}}{969 x^{17}}+\frac {35 (1+x)^{12}}{15504 x^{16}}-\frac {7 (1+x)^{12}}{11628 x^{15}}+\frac {(1+x)^{12}}{7752 x^{14}}+\frac {\int \frac {(1+x)^{11}}{x^{14}} \, dx}{3876}\\ &=-\frac {(1+x)^{12}}{19 x^{19}}+\frac {7 (1+x)^{12}}{342 x^{18}}-\frac {7 (1+x)^{12}}{969 x^{17}}+\frac {35 (1+x)^{12}}{15504 x^{16}}-\frac {7 (1+x)^{12}}{11628 x^{15}}+\frac {(1+x)^{12}}{7752 x^{14}}-\frac {(1+x)^{12}}{50388 x^{13}}-\frac {\int \frac {(1+x)^{11}}{x^{13}} \, dx}{50388}\\ &=-\frac {(1+x)^{12}}{19 x^{19}}+\frac {7 (1+x)^{12}}{342 x^{18}}-\frac {7 (1+x)^{12}}{969 x^{17}}+\frac {35 (1+x)^{12}}{15504 x^{16}}-\frac {7 (1+x)^{12}}{11628 x^{15}}+\frac {(1+x)^{12}}{7752 x^{14}}-\frac {(1+x)^{12}}{50388 x^{13}}+\frac {(1+x)^{12}}{604656 x^{12}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 79, normalized size = 0.81 \[ -\frac {1}{19 x^{19}}-\frac {11}{18 x^{18}}-\frac {55}{17 x^{17}}-\frac {165}{16 x^{16}}-\frac {22}{x^{15}}-\frac {33}{x^{14}}-\frac {462}{13 x^{13}}-\frac {55}{2 x^{12}}-\frac {15}{x^{11}}-\frac {11}{2 x^{10}}-\frac {11}{9 x^9}-\frac {1}{8 x^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 60, normalized size = 0.62 \[ -\frac {75582 \, x^{11} + 739024 \, x^{10} + 3325608 \, x^{9} + 9069840 \, x^{8} + 16628040 \, x^{7} + 21488544 \, x^{6} + 19953648 \, x^{5} + 13302432 \, x^{4} + 6235515 \, x^{3} + 1956240 \, x^{2} + 369512 \, x + 31824}{604656 \, x^{19}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 60, normalized size = 0.62 \[ -\frac {75582 \, x^{11} + 739024 \, x^{10} + 3325608 \, x^{9} + 9069840 \, x^{8} + 16628040 \, x^{7} + 21488544 \, x^{6} + 19953648 \, x^{5} + 13302432 \, x^{4} + 6235515 \, x^{3} + 1956240 \, x^{2} + 369512 \, x + 31824}{604656 \, x^{19}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 62, normalized size = 0.64 \[ -\frac {1}{8 x^{8}}-\frac {11}{9 x^{9}}-\frac {11}{2 x^{10}}-\frac {15}{x^{11}}-\frac {55}{2 x^{12}}-\frac {462}{13 x^{13}}-\frac {33}{x^{14}}-\frac {22}{x^{15}}-\frac {165}{16 x^{16}}-\frac {55}{17 x^{17}}-\frac {11}{18 x^{18}}-\frac {1}{19 x^{19}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 60, normalized size = 0.62 \[ -\frac {75582 \, x^{11} + 739024 \, x^{10} + 3325608 \, x^{9} + 9069840 \, x^{8} + 16628040 \, x^{7} + 21488544 \, x^{6} + 19953648 \, x^{5} + 13302432 \, x^{4} + 6235515 \, x^{3} + 1956240 \, x^{2} + 369512 \, x + 31824}{604656 \, x^{19}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 60, normalized size = 0.62 \[ -\frac {\frac {x^{11}}{8}+\frac {11\,x^{10}}{9}+\frac {11\,x^9}{2}+15\,x^8+\frac {55\,x^7}{2}+\frac {462\,x^6}{13}+33\,x^5+22\,x^4+\frac {165\,x^3}{16}+\frac {55\,x^2}{17}+\frac {11\,x}{18}+\frac {1}{19}}{x^{19}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 61, normalized size = 0.63 \[ \frac {- 75582 x^{11} - 739024 x^{10} - 3325608 x^{9} - 9069840 x^{8} - 16628040 x^{7} - 21488544 x^{6} - 19953648 x^{5} - 13302432 x^{4} - 6235515 x^{3} - 1956240 x^{2} - 369512 x - 31824}{604656 x^{19}} \]
Verification of antiderivative is not currently implemented for this CAS.
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