Optimal. Leaf size=80 \[ -\frac {1}{2 x^2}+\frac {55 x^2}{2}+\frac {165 x^4}{4}+55 x^6+\frac {231 x^8}{4}+\frac {231 x^{10}}{5}+\frac {55 x^{12}}{2}+\frac {165 x^{14}}{14}+\frac {55 x^{16}}{16}+\frac {11 x^{18}}{18}+\frac {x^{20}}{20}+11 \log (x) \]
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Rubi [A]
time = 0.02, 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, 272, 45}
\begin {gather*} \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) \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 45
Rule 272
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} \text {Subst}\left (\int \frac {(1+x)^{11}}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 61, normalized size = 0.76
method | result | size |
default | \(-\frac {1}{2 x^{2}}+\frac {55 x^{2}}{2}+\frac {165 x^{4}}{4}+55 x^{6}+\frac {231 x^{8}}{4}+\frac {231 x^{10}}{5}+\frac {55 x^{12}}{2}+\frac {165 x^{14}}{14}+\frac {55 x^{16}}{16}+\frac {11 x^{18}}{18}+\frac {x^{20}}{20}+11 \ln \left (x \right )\) | \(61\) |
risch | \(-\frac {1}{2 x^{2}}+\frac {55 x^{2}}{2}+\frac {165 x^{4}}{4}+55 x^{6}+\frac {231 x^{8}}{4}+\frac {231 x^{10}}{5}+\frac {55 x^{12}}{2}+\frac {165 x^{14}}{14}+\frac {55 x^{16}}{16}+\frac {11 x^{18}}{18}+\frac {x^{20}}{20}+11 \ln \left (x \right )\) | \(61\) |
norman | \(\frac {-\frac {1}{2}+\frac {55}{2} x^{4}+\frac {165}{4} x^{6}+55 x^{8}+\frac {231}{4} x^{10}+\frac {231}{5} x^{12}+\frac {55}{2} x^{14}+\frac {165}{14} x^{16}+\frac {55}{16} x^{18}+\frac {11}{18} x^{20}+\frac {1}{20} x^{22}}{x^{2}}+11 \ln \left (x \right )\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 62, normalized size = 0.78 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 64, normalized size = 0.80 \begin {gather*} \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 \left (x\right ) - 2520}{5040 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 75, normalized size = 0.94 \begin {gather*} \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 {\left (x \right )} - \frac {1}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.06, size = 69, normalized size = 0.86 \begin {gather*} \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 ) \end {gather*}
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 \begin {gather*} 11\,\ln \left (x\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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