Optimal. Leaf size=97 \[ \frac {5 x}{4}-\frac {3 x^2}{4}+\frac {x^3}{3}+\frac {x^4}{4}+\frac {1}{24} \sqrt {\frac {5}{3}} \tan ^{-1}\left (\frac {1-4 x}{\sqrt {15}}\right )-\frac {10 \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} \log \left (1+x+x^2\right )-\frac {13}{48} \log \left (2-x+2 x^2\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2100, 648,
632, 210, 642} \begin {gather*} \frac {1}{24} \sqrt {\frac {5}{3}} \text {ArcTan}\left (\frac {1-4 x}{\sqrt {15}}\right )-\frac {10 \text {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {x^4}{4}+\frac {x^3}{3}-\frac {3 x^2}{4}+\frac {1}{3} \log \left (x^2+x+1\right )-\frac {13}{48} \log \left (2 x^2-x+2\right )+\frac {5 x}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 2100
Rubi steps
\begin {align*} \int \frac {x^4 \left (5+x+3 x^2+2 x^3\right )}{2+x+3 x^2+x^3+2 x^4} \, dx &=\int \left (\frac {5}{4}-\frac {3 x}{2}+x^2+x^3+\frac {2 (-2+x)}{3 \left (1+x+x^2\right )}+\frac {2-13 x}{12 \left (2-x+2 x^2\right )}\right ) \, dx\\ &=\frac {5 x}{4}-\frac {3 x^2}{4}+\frac {x^3}{3}+\frac {x^4}{4}+\frac {1}{12} \int \frac {2-13 x}{2-x+2 x^2} \, dx+\frac {2}{3} \int \frac {-2+x}{1+x+x^2} \, dx\\ &=\frac {5 x}{4}-\frac {3 x^2}{4}+\frac {x^3}{3}+\frac {x^4}{4}-\frac {5}{48} \int \frac {1}{2-x+2 x^2} \, dx-\frac {13}{48} \int \frac {-1+4 x}{2-x+2 x^2} \, dx+\frac {1}{3} \int \frac {1+2 x}{1+x+x^2} \, dx-\frac {5}{3} \int \frac {1}{1+x+x^2} \, dx\\ &=\frac {5 x}{4}-\frac {3 x^2}{4}+\frac {x^3}{3}+\frac {x^4}{4}+\frac {1}{3} \log \left (1+x+x^2\right )-\frac {13}{48} \log \left (2-x+2 x^2\right )+\frac {5}{24} \text {Subst}\left (\int \frac {1}{-15-x^2} \, dx,x,-1+4 x\right )+\frac {10}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\frac {5 x}{4}-\frac {3 x^2}{4}+\frac {x^3}{3}+\frac {x^4}{4}+\frac {1}{24} \sqrt {\frac {5}{3}} \tan ^{-1}\left (\frac {1-4 x}{\sqrt {15}}\right )-\frac {10 \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{3} \log \left (1+x+x^2\right )-\frac {13}{48} \log \left (2-x+2 x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 83, normalized size = 0.86 \begin {gather*} \frac {1}{144} \left (180 x-108 x^2+48 x^3+36 x^4-160 \sqrt {3} \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )-2 \sqrt {15} \tan ^{-1}\left (\frac {-1+4 x}{\sqrt {15}}\right )+48 \log \left (1+x+x^2\right )-39 \log \left (2-x+2 x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 74, normalized size = 0.76
method | result | size |
default | \(\frac {x^{4}}{4}+\frac {x^{3}}{3}-\frac {3 x^{2}}{4}+\frac {5 x}{4}+\frac {\ln \left (x^{2}+x +1\right )}{3}-\frac {10 \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{9}-\frac {13 \ln \left (2 x^{2}-x +2\right )}{48}-\frac {\sqrt {15}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {15}}{15}\right )}{72}\) | \(74\) |
risch | \(\frac {x^{4}}{4}+\frac {x^{3}}{3}-\frac {3 x^{2}}{4}+\frac {5 x}{4}-\frac {13 \ln \left (16 x^{2}-8 x +16\right )}{48}-\frac {\sqrt {15}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {15}}{15}\right )}{72}+\frac {\ln \left (4 x^{2}+4 x +4\right )}{3}-\frac {10 \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{9}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 73, normalized size = 0.75 \begin {gather*} \frac {1}{4} \, x^{4} + \frac {1}{3} \, x^{3} - \frac {3}{4} \, x^{2} - \frac {1}{72} \, \sqrt {15} \arctan \left (\frac {1}{15} \, \sqrt {15} {\left (4 \, x - 1\right )}\right ) - \frac {10}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {5}{4} \, x - \frac {13}{48} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac {1}{3} \, \log \left (x^{2} + x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 79, normalized size = 0.81 \begin {gather*} \frac {1}{4} \, x^{4} + \frac {1}{3} \, x^{3} - \frac {3}{4} \, x^{2} - \frac {1}{72} \, \sqrt {5} \sqrt {3} \arctan \left (\frac {1}{15} \, \sqrt {5} \sqrt {3} {\left (4 \, x - 1\right )}\right ) - \frac {10}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {5}{4} \, x - \frac {13}{48} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac {1}{3} \, \log \left (x^{2} + x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 97, normalized size = 1.00 \begin {gather*} \frac {x^{4}}{4} + \frac {x^{3}}{3} - \frac {3 x^{2}}{4} + \frac {5 x}{4} - \frac {13 \log {\left (x^{2} - \frac {x}{2} + 1 \right )}}{48} + \frac {\log {\left (x^{2} + x + 1 \right )}}{3} - \frac {\sqrt {15} \operatorname {atan}{\left (\frac {4 \sqrt {15} x}{15} - \frac {\sqrt {15}}{15} \right )}}{72} - \frac {10 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.66, size = 73, normalized size = 0.75 \begin {gather*} \frac {1}{4} \, x^{4} + \frac {1}{3} \, x^{3} - \frac {3}{4} \, x^{2} - \frac {1}{72} \, \sqrt {15} \arctan \left (\frac {1}{15} \, \sqrt {15} {\left (4 \, x - 1\right )}\right ) - \frac {10}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) + \frac {5}{4} \, x - \frac {13}{48} \, \log \left (2 \, x^{2} - x + 2\right ) + \frac {1}{3} \, \log \left (x^{2} + x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 97, normalized size = 1.00 \begin {gather*} \frac {5\,x}{4}+\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{3}+\frac {\sqrt {3}\,5{}\mathrm {i}}{9}\right )-\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{3}+\frac {\sqrt {3}\,5{}\mathrm {i}}{9}\right )+\ln \left (x-\frac {1}{4}-\frac {\sqrt {15}\,1{}\mathrm {i}}{4}\right )\,\left (-\frac {13}{48}+\frac {\sqrt {15}\,1{}\mathrm {i}}{144}\right )-\ln \left (x-\frac {1}{4}+\frac {\sqrt {15}\,1{}\mathrm {i}}{4}\right )\,\left (\frac {13}{48}+\frac {\sqrt {15}\,1{}\mathrm {i}}{144}\right )-\frac {3\,x^2}{4}+\frac {x^3}{3}+\frac {x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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