Optimal. Leaf size=97 \[ \frac{(x+1)^{12}}{604656 x^{12}}-\frac{(x+1)^{12}}{50388 x^{13}}+\frac{(x+1)^{12}}{7752 x^{14}}-\frac{7 (x+1)^{12}}{11628 x^{15}}+\frac{35 (x+1)^{12}}{15504 x^{16}}-\frac{7 (x+1)^{12}}{969 x^{17}}+\frac{7 (x+1)^{12}}{342 x^{18}}-\frac{(x+1)^{12}}{19 x^{19}} \]
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Rubi [A] time = 0.0268267, antiderivative size = 97, normalized size of antiderivative = 1., 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 45
Rule 37
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*}
Mathematica [A] time = 0.0023949, size = 79, normalized size = 0.81 \[ -\frac{1}{8 x^8}-\frac{11}{9 x^9}-\frac{11}{2 x^{10}}-\frac{15}{x^{11}}-\frac{55}{2 x^{12}}-\frac{462}{13 x^{13}}-\frac{33}{x^{14}}-\frac{22}{x^{15}}-\frac{165}{16 x^{16}}-\frac{55}{17 x^{17}}-\frac{11}{18 x^{18}}-\frac{1}{19 x^{19}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 62, normalized size = 0.6 \begin{align*} -{\frac{11}{9\,{x}^{9}}}-{\frac{165}{16\,{x}^{16}}}-22\,{x}^{-15}-{\frac{55}{17\,{x}^{17}}}-33\,{x}^{-14}-{\frac{1}{8\,{x}^{8}}}-{\frac{11}{2\,{x}^{10}}}-{\frac{55}{2\,{x}^{12}}}-{\frac{462}{13\,{x}^{13}}}-{\frac{1}{19\,{x}^{19}}}-15\,{x}^{-11}-{\frac{11}{18\,{x}^{18}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02666, size = 81, normalized size = 0.84 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23851, size = 240, normalized size = 2.47 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.229472, size = 61, normalized size = 0.63 \begin{align*} - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08878, size = 81, normalized size = 0.84 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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