Optimal. Leaf size=49 \[ -\frac {(x+1)^{12}}{15 x^{15}}+\frac {(x+1)^{12}}{70 x^{14}}-\frac {(x+1)^{12}}{455 x^{13}}+\frac {(x+1)^{12}}{5460 x^{12}} \]
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Rubi [A] time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {27, 45, 37} \begin {gather*} \frac {(x+1)^{12}}{5460 x^{12}}-\frac {(x+1)^{12}}{455 x^{13}}+\frac {(x+1)^{12}}{70 x^{14}}-\frac {(x+1)^{12}}{15 x^{15}} \end {gather*}
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^{16}} \, dx &=\int \frac {(1+x)^{11}}{x^{16}} \, dx\\ &=-\frac {(1+x)^{12}}{15 x^{15}}-\frac {1}{5} \int \frac {(1+x)^{11}}{x^{15}} \, dx\\ &=-\frac {(1+x)^{12}}{15 x^{15}}+\frac {(1+x)^{12}}{70 x^{14}}+\frac {1}{35} \int \frac {(1+x)^{11}}{x^{14}} \, dx\\ &=-\frac {(1+x)^{12}}{15 x^{15}}+\frac {(1+x)^{12}}{70 x^{14}}-\frac {(1+x)^{12}}{455 x^{13}}-\frac {1}{455} \int \frac {(1+x)^{11}}{x^{13}} \, dx\\ &=-\frac {(1+x)^{12}}{15 x^{15}}+\frac {(1+x)^{12}}{70 x^{14}}-\frac {(1+x)^{12}}{455 x^{13}}+\frac {(1+x)^{12}}{5460 x^{12}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 83, normalized size = 1.69 \begin {gather*} -\frac {1}{15 x^{15}}-\frac {11}{14 x^{14}}-\frac {55}{13 x^{13}}-\frac {55}{4 x^{12}}-\frac {30}{x^{11}}-\frac {231}{5 x^{10}}-\frac {154}{3 x^9}-\frac {165}{4 x^8}-\frac {165}{7 x^7}-\frac {55}{6 x^6}-\frac {11}{5 x^5}-\frac {1}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x^{16}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 60, normalized size = 1.22 \begin {gather*} -\frac {1365 \, x^{11} + 12012 \, x^{10} + 50050 \, x^{9} + 128700 \, x^{8} + 225225 \, x^{7} + 280280 \, x^{6} + 252252 \, x^{5} + 163800 \, x^{4} + 75075 \, x^{3} + 23100 \, x^{2} + 4290 \, x + 364}{5460 \, x^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 60, normalized size = 1.22 \begin {gather*} -\frac {1365 \, x^{11} + 12012 \, x^{10} + 50050 \, x^{9} + 128700 \, x^{8} + 225225 \, x^{7} + 280280 \, x^{6} + 252252 \, x^{5} + 163800 \, x^{4} + 75075 \, x^{3} + 23100 \, x^{2} + 4290 \, x + 364}{5460 \, x^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 62, normalized size = 1.27 \begin {gather*} -\frac {1}{4 x^{4}}-\frac {11}{5 x^{5}}-\frac {55}{6 x^{6}}-\frac {165}{7 x^{7}}-\frac {165}{4 x^{8}}-\frac {154}{3 x^{9}}-\frac {231}{5 x^{10}}-\frac {30}{x^{11}}-\frac {55}{4 x^{12}}-\frac {55}{13 x^{13}}-\frac {11}{14 x^{14}}-\frac {1}{15 x^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 60, normalized size = 1.22 \begin {gather*} -\frac {1365 \, x^{11} + 12012 \, x^{10} + 50050 \, x^{9} + 128700 \, x^{8} + 225225 \, x^{7} + 280280 \, x^{6} + 252252 \, x^{5} + 163800 \, x^{4} + 75075 \, x^{3} + 23100 \, x^{2} + 4290 \, x + 364}{5460 \, x^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 60, normalized size = 1.22 \begin {gather*} -\frac {\frac {x^{11}}{4}+\frac {11\,x^{10}}{5}+\frac {55\,x^9}{6}+\frac {165\,x^8}{7}+\frac {165\,x^7}{4}+\frac {154\,x^6}{3}+\frac {231\,x^5}{5}+30\,x^4+\frac {55\,x^3}{4}+\frac {55\,x^2}{13}+\frac {11\,x}{14}+\frac {1}{15}}{x^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 61, normalized size = 1.24 \begin {gather*} \frac {- 1365 x^{11} - 12012 x^{10} - 50050 x^{9} - 128700 x^{8} - 225225 x^{7} - 280280 x^{6} - 252252 x^{5} - 163800 x^{4} - 75075 x^{3} - 23100 x^{2} - 4290 x - 364}{5460 x^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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