Optimal. Leaf size=80 \[ \frac {x^{20}}{20}+\frac {11 x^{18}}{18}+\frac {55 x^{16}}{16}+\frac {165 x^{14}}{14}+\frac {55 x^{12}}{2}+\frac {231 x^{10}}{5}+\frac {231 x^8}{4}+55 x^6+\frac {165 x^4}{4}+\frac {55 x^2}{2}-\frac {1}{2 x^2}+11 \log (x) \]
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Rubi [A] time = 0.04, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {28, 266, 43} \[ \frac {x^{20}}{20}+\frac {11 x^{18}}{18}+\frac {55 x^{16}}{16}+\frac {165 x^{14}}{14}+\frac {55 x^{12}}{2}+\frac {231 x^{10}}{5}+\frac {231 x^8}{4}+55 x^6+\frac {165 x^4}{4}+\frac {55 x^2}{2}-\frac {1}{2 x^2}+11 \log (x) \]
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5}{x^3} \, dx &=\int \frac {\left (1+x^2\right )^{11}}{x^3} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(1+x)^{11}}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (55+\frac {1}{x^2}+\frac {11}{x}+165 x+330 x^2+462 x^3+462 x^4+330 x^5+165 x^6+55 x^7+11 x^8+x^9\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 x^2}+\frac {55 x^2}{2}+\frac {165 x^4}{4}+55 x^6+\frac {231 x^8}{4}+\frac {231 x^{10}}{5}+\frac {55 x^{12}}{2}+\frac {165 x^{14}}{14}+\frac {55 x^{16}}{16}+\frac {11 x^{18}}{18}+\frac {x^{20}}{20}+11 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 80, normalized size = 1.00 \[ \frac {x^{20}}{20}+\frac {11 x^{18}}{18}+\frac {55 x^{16}}{16}+\frac {165 x^{14}}{14}+\frac {55 x^{12}}{2}+\frac {231 x^{10}}{5}+\frac {231 x^8}{4}+55 x^6+\frac {165 x^4}{4}+\frac {55 x^2}{2}-\frac {1}{2 x^2}+11 \log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 64, normalized size = 0.80 \[ \frac {252 \, x^{22} + 3080 \, x^{20} + 17325 \, x^{18} + 59400 \, x^{16} + 138600 \, x^{14} + 232848 \, x^{12} + 291060 \, x^{10} + 277200 \, x^{8} + 207900 \, x^{6} + 138600 \, x^{4} + 55440 \, x^{2} \log \relax (x) - 2520}{5040 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 69, normalized size = 0.86 \[ \frac {1}{20} \, x^{20} + \frac {11}{18} \, x^{18} + \frac {55}{16} \, x^{16} + \frac {165}{14} \, x^{14} + \frac {55}{2} \, x^{12} + \frac {231}{5} \, x^{10} + \frac {231}{4} \, x^{8} + 55 \, x^{6} + \frac {165}{4} \, x^{4} + \frac {55}{2} \, x^{2} - \frac {11 \, x^{2} + 1}{2 \, x^{2}} + \frac {11}{2} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 61, normalized size = 0.76 \[ \frac {x^{20}}{20}+\frac {11 x^{18}}{18}+\frac {55 x^{16}}{16}+\frac {165 x^{14}}{14}+\frac {55 x^{12}}{2}+\frac {231 x^{10}}{5}+\frac {231 x^{8}}{4}+55 x^{6}+\frac {165 x^{4}}{4}+\frac {55 x^{2}}{2}+11 \ln \relax (x )-\frac {1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 62, normalized size = 0.78 \[ \frac {1}{20} \, x^{20} + \frac {11}{18} \, x^{18} + \frac {55}{16} \, x^{16} + \frac {165}{14} \, x^{14} + \frac {55}{2} \, x^{12} + \frac {231}{5} \, x^{10} + \frac {231}{4} \, x^{8} + 55 \, x^{6} + \frac {165}{4} \, x^{4} + \frac {55}{2} \, x^{2} - \frac {1}{2 \, x^{2}} + \frac {11}{2} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 60, normalized size = 0.75 \[ 11\,\ln \relax (x)-\frac {1}{2\,x^2}+\frac {55\,x^2}{2}+\frac {165\,x^4}{4}+55\,x^6+\frac {231\,x^8}{4}+\frac {231\,x^{10}}{5}+\frac {55\,x^{12}}{2}+\frac {165\,x^{14}}{14}+\frac {55\,x^{16}}{16}+\frac {11\,x^{18}}{18}+\frac {x^{20}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 75, normalized size = 0.94 \[ \frac {x^{20}}{20} + \frac {11 x^{18}}{18} + \frac {55 x^{16}}{16} + \frac {165 x^{14}}{14} + \frac {55 x^{12}}{2} + \frac {231 x^{10}}{5} + \frac {231 x^{8}}{4} + 55 x^{6} + \frac {165 x^{4}}{4} + \frac {55 x^{2}}{2} + 11 \log {\relax (x )} - \frac {1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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