Optimal. Leaf size=91 \[ \frac {125 x^5}{4}+\frac {2125 x^4}{16}+\frac {9775 x^3}{48}-\frac {1185 x^2}{8}-\frac {14641 (79 x+101)}{2944 \left (2 x^2-x+3\right )}-\frac {30613}{128} \log \left (2 x^2-x+3\right )-\frac {89359 x}{64}-\frac {13292697 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1472 \sqrt {23}} \]
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Rubi [A] time = 0.09, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1660, 1657, 634, 618, 204, 628} \[ \frac {125 x^5}{4}+\frac {2125 x^4}{16}+\frac {9775 x^3}{48}-\frac {1185 x^2}{8}-\frac {14641 (79 x+101)}{2944 \left (2 x^2-x+3\right )}-\frac {30613}{128} \log \left (2 x^2-x+3\right )-\frac {89359 x}{64}-\frac {13292697 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1472 \sqrt {23}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 1657
Rule 1660
Rubi steps
\begin {align*} \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^2} \, dx &=-\frac {14641 (101+79 x)}{2944 \left (3-x+2 x^2\right )}+\frac {1}{23} \int \frac {\frac {832627}{64}-\frac {661181 x}{64}-\frac {488267 x^2}{32}+\frac {143635 x^3}{16}+\frac {213325 x^4}{8}+\frac {83375 x^5}{4}+\frac {14375 x^6}{2}}{3-x+2 x^2} \, dx\\ &=-\frac {14641 (101+79 x)}{2944 \left (3-x+2 x^2\right )}+\frac {1}{23} \int \left (-\frac {2055257}{64}-\frac {27255 x}{4}+\frac {224825 x^2}{16}+\frac {48875 x^3}{4}+\frac {14375 x^4}{4}+\frac {1331 (2629-529 x)}{32 \left (3-x+2 x^2\right )}\right ) \, dx\\ &=-\frac {89359 x}{64}-\frac {1185 x^2}{8}+\frac {9775 x^3}{48}+\frac {2125 x^4}{16}+\frac {125 x^5}{4}-\frac {14641 (101+79 x)}{2944 \left (3-x+2 x^2\right )}+\frac {1331}{736} \int \frac {2629-529 x}{3-x+2 x^2} \, dx\\ &=-\frac {89359 x}{64}-\frac {1185 x^2}{8}+\frac {9775 x^3}{48}+\frac {2125 x^4}{16}+\frac {125 x^5}{4}-\frac {14641 (101+79 x)}{2944 \left (3-x+2 x^2\right )}-\frac {30613}{128} \int \frac {-1+4 x}{3-x+2 x^2} \, dx+\frac {13292697 \int \frac {1}{3-x+2 x^2} \, dx}{2944}\\ &=-\frac {89359 x}{64}-\frac {1185 x^2}{8}+\frac {9775 x^3}{48}+\frac {2125 x^4}{16}+\frac {125 x^5}{4}-\frac {14641 (101+79 x)}{2944 \left (3-x+2 x^2\right )}-\frac {30613}{128} \log \left (3-x+2 x^2\right )-\frac {13292697 \operatorname {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )}{1472}\\ &=-\frac {89359 x}{64}-\frac {1185 x^2}{8}+\frac {9775 x^3}{48}+\frac {2125 x^4}{16}+\frac {125 x^5}{4}-\frac {14641 (101+79 x)}{2944 \left (3-x+2 x^2\right )}-\frac {13292697 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{1472 \sqrt {23}}-\frac {30613}{128} \log \left (3-x+2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 91, normalized size = 1.00 \[ \frac {125 x^5}{4}+\frac {2125 x^4}{16}+\frac {9775 x^3}{48}-\frac {1185 x^2}{8}-\frac {14641 (79 x+101)}{2944 \left (2 x^2-x+3\right )}-\frac {30613}{128} \log \left (2 x^2-x+3\right )-\frac {89359 x}{64}+\frac {13292697 \tan ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{1472 \sqrt {23}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 98, normalized size = 1.08 \[ \frac {12696000 \, x^{7} + 47610000 \, x^{6} + 74800600 \, x^{5} - 20609840 \, x^{4} - 413058012 \, x^{3} + 79756182 \, \sqrt {23} {\left (2 \, x^{2} - x + 3\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 193356906 \, x^{2} - 48582831 \, {\left (2 \, x^{2} - x + 3\right )} \log \left (2 \, x^{2} - x + 3\right ) - 930684489 \, x - 102033129}{203136 \, {\left (2 \, x^{2} - x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 72, normalized size = 0.79 \[ \frac {125}{4} \, x^{5} + \frac {2125}{16} \, x^{4} + \frac {9775}{48} \, x^{3} - \frac {1185}{8} \, x^{2} + \frac {13292697}{33856} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {89359}{64} \, x - \frac {14641 \, {\left (79 \, x + 101\right )}}{2944 \, {\left (2 \, x^{2} - x + 3\right )}} - \frac {30613}{128} \, \log \left (2 \, x^{2} - x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 71, normalized size = 0.78 \[ \frac {125 x^{5}}{4}+\frac {2125 x^{4}}{16}+\frac {9775 x^{3}}{48}-\frac {1185 x^{2}}{8}-\frac {89359 x}{64}+\frac {13292697 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{33856}-\frac {30613 \ln \left (2 x^{2}-x +3\right )}{128}-\frac {1331 \left (\frac {869 x}{92}+\frac {1111}{92}\right )}{64 \left (x^{2}-\frac {1}{2} x +\frac {3}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 72, normalized size = 0.79 \[ \frac {125}{4} \, x^{5} + \frac {2125}{16} \, x^{4} + \frac {9775}{48} \, x^{3} - \frac {1185}{8} \, x^{2} + \frac {13292697}{33856} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {89359}{64} \, x - \frac {14641 \, {\left (79 \, x + 101\right )}}{2944 \, {\left (2 \, x^{2} - x + 3\right )}} - \frac {30613}{128} \, \log \left (2 \, x^{2} - x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.46, size = 72, normalized size = 0.79 \[ \frac {13292697\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{33856}-\frac {30613\,\ln \left (2\,x^2-x+3\right )}{128}-\frac {\frac {1156639\,x}{5888}+\frac {1478741}{5888}}{x^2-\frac {x}{2}+\frac {3}{2}}-\frac {89359\,x}{64}-\frac {1185\,x^2}{8}+\frac {9775\,x^3}{48}+\frac {2125\,x^4}{16}+\frac {125\,x^5}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 90, normalized size = 0.99 \[ \frac {125 x^{5}}{4} + \frac {2125 x^{4}}{16} + \frac {9775 x^{3}}{48} - \frac {1185 x^{2}}{8} - \frac {89359 x}{64} + \frac {- 1156639 x - 1478741}{5888 x^{2} - 2944 x + 8832} - \frac {30613 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{128} + \frac {13292697 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{33856} \]
Verification of antiderivative is not currently implemented for this CAS.
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