Optimal. Leaf size=72 \[ 11 x+\frac {55 x^2}{2}+55 x^3+\frac {165 x^4}{2}+\frac {462 x^5}{5}+77 x^6+\frac {330 x^7}{7}+\frac {165 x^8}{8}+\frac {55 x^9}{9}+\frac {11 x^{10}}{10}+\frac {x^{11}}{11}+\log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {27, 45}
\begin {gather*} \frac {x^{11}}{11}+\frac {11 x^{10}}{10}+\frac {55 x^9}{9}+\frac {165 x^8}{8}+\frac {330 x^7}{7}+77 x^6+\frac {462 x^5}{5}+\frac {165 x^4}{2}+55 x^3+\frac {55 x^2}{2}+11 x+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int \frac {(1+x) \left (1+2 x+x^2\right )^5}{x} \, dx &=\int \frac {(1+x)^{11}}{x} \, dx\\ &=\int \left (11+\frac {1}{x}+55 x+165 x^2+330 x^3+462 x^4+462 x^5+330 x^6+165 x^7+55 x^8+11 x^9+x^{10}\right ) \, dx\\ &=11 x+\frac {55 x^2}{2}+55 x^3+\frac {165 x^4}{2}+\frac {462 x^5}{5}+77 x^6+\frac {330 x^7}{7}+\frac {165 x^8}{8}+\frac {55 x^9}{9}+\frac {11 x^{10}}{10}+\frac {x^{11}}{11}+\log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 96, normalized size = 1.33 \begin {gather*} x+\frac {1}{2} (1+x)^2+\frac {1}{3} (1+x)^3+\frac {1}{4} (1+x)^4+\frac {1}{5} (1+x)^5+\frac {1}{6} (1+x)^6+\frac {1}{7} (1+x)^7+\frac {1}{8} (1+x)^8+\frac {1}{9} (1+x)^9+\frac {1}{10} (1+x)^{10}+\frac {1}{11} (1+x)^{11}+\log (-x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.63, size = 57, normalized size = 0.79
method | result | size |
default | \(11 x +\frac {55 x^{2}}{2}+55 x^{3}+\frac {165 x^{4}}{2}+\frac {462 x^{5}}{5}+77 x^{6}+\frac {330 x^{7}}{7}+\frac {165 x^{8}}{8}+\frac {55 x^{9}}{9}+\frac {11 x^{10}}{10}+\frac {x^{11}}{11}+\ln \left (x \right )\) | \(57\) |
norman | \(11 x +\frac {55 x^{2}}{2}+55 x^{3}+\frac {165 x^{4}}{2}+\frac {462 x^{5}}{5}+77 x^{6}+\frac {330 x^{7}}{7}+\frac {165 x^{8}}{8}+\frac {55 x^{9}}{9}+\frac {11 x^{10}}{10}+\frac {x^{11}}{11}+\ln \left (x \right )\) | \(57\) |
risch | \(11 x +\frac {55 x^{2}}{2}+55 x^{3}+\frac {165 x^{4}}{2}+\frac {462 x^{5}}{5}+77 x^{6}+\frac {330 x^{7}}{7}+\frac {165 x^{8}}{8}+\frac {55 x^{9}}{9}+\frac {11 x^{10}}{10}+\frac {x^{11}}{11}+\ln \left (x \right )\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 56, normalized size = 0.78 \begin {gather*} \frac {1}{11} \, x^{11} + \frac {11}{10} \, x^{10} + \frac {55}{9} \, x^{9} + \frac {165}{8} \, x^{8} + \frac {330}{7} \, x^{7} + 77 \, x^{6} + \frac {462}{5} \, x^{5} + \frac {165}{2} \, x^{4} + 55 \, x^{3} + \frac {55}{2} \, x^{2} + 11 \, x + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.76, size = 56, normalized size = 0.78 \begin {gather*} \frac {1}{11} \, x^{11} + \frac {11}{10} \, x^{10} + \frac {55}{9} \, x^{9} + \frac {165}{8} \, x^{8} + \frac {330}{7} \, x^{7} + 77 \, x^{6} + \frac {462}{5} \, x^{5} + \frac {165}{2} \, x^{4} + 55 \, x^{3} + \frac {55}{2} \, x^{2} + 11 \, x + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 68, normalized size = 0.94 \begin {gather*} \frac {x^{11}}{11} + \frac {11 x^{10}}{10} + \frac {55 x^{9}}{9} + \frac {165 x^{8}}{8} + \frac {330 x^{7}}{7} + 77 x^{6} + \frac {462 x^{5}}{5} + \frac {165 x^{4}}{2} + 55 x^{3} + \frac {55 x^{2}}{2} + 11 x + \log {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.16, size = 57, normalized size = 0.79 \begin {gather*} \frac {1}{11} \, x^{11} + \frac {11}{10} \, x^{10} + \frac {55}{9} \, x^{9} + \frac {165}{8} \, x^{8} + \frac {330}{7} \, x^{7} + 77 \, x^{6} + \frac {462}{5} \, x^{5} + \frac {165}{2} \, x^{4} + 55 \, x^{3} + \frac {55}{2} \, x^{2} + 11 \, x + \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 56, normalized size = 0.78 \begin {gather*} 11\,x+\ln \left (x\right )+\frac {55\,x^2}{2}+55\,x^3+\frac {165\,x^4}{2}+\frac {462\,x^5}{5}+77\,x^6+\frac {330\,x^7}{7}+\frac {165\,x^8}{8}+\frac {55\,x^9}{9}+\frac {11\,x^{10}}{10}+\frac {x^{11}}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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