Optimal. Leaf size=98 \[ \frac {625 x^3}{24}+\frac {4875 x^2}{32}+\frac {1331 (76420 x+5229)}{135424 \left (2 x^2-x+3\right )}-\frac {14641 (79 x+101)}{5888 \left (2 x^2-x+3\right )^2}-\frac {13915}{64} \log \left (2 x^2-x+3\right )+\frac {2725 x}{8}+\frac {63799791 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{16928 \sqrt {23}} \]
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Rubi [A] time = 0.11, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \begin {gather*} \frac {625 x^3}{24}+\frac {4875 x^2}{32}+\frac {1331 (76420 x+5229)}{135424 \left (2 x^2-x+3\right )}-\frac {14641 (79 x+101)}{5888 \left (2 x^2-x+3\right )^2}-\frac {13915}{64} \log \left (2 x^2-x+3\right )+\frac {2725 x}{8}+\frac {63799791 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{16928 \sqrt {23}} \end {gather*}
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 )^3} \, dx &=-\frac {14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac {1}{46} \int \frac {\frac {2173869}{128}-\frac {661181 x}{32}-\frac {488267 x^2}{16}+\frac {143635 x^3}{8}+\frac {213325 x^4}{4}+\frac {83375 x^5}{2}+14375 x^6}{\left (3-x+2 x^2\right )^2} \, dx\\ &=-\frac {14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac {1331 (5229+76420 x)}{135424 \left (3-x+2 x^2\right )}+\frac {\int \frac {-\frac {5460539}{8}-\frac {626865 x}{2}+\frac {5170975 x^2}{8}+\frac {1124125 x^3}{2}+\frac {330625 x^4}{2}}{3-x+2 x^2} \, dx}{1058}\\ &=-\frac {14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac {1331 (5229+76420 x)}{135424 \left (3-x+2 x^2\right )}+\frac {\int \left (\frac {1441525}{4}+\frac {2578875 x}{8}+\frac {330625 x^2}{4}-\frac {121 (116609+60835 x)}{8 \left (3-x+2 x^2\right )}\right ) \, dx}{1058}\\ &=\frac {2725 x}{8}+\frac {4875 x^2}{32}+\frac {625 x^3}{24}-\frac {14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac {1331 (5229+76420 x)}{135424 \left (3-x+2 x^2\right )}-\frac {121 \int \frac {116609+60835 x}{3-x+2 x^2} \, dx}{8464}\\ &=\frac {2725 x}{8}+\frac {4875 x^2}{32}+\frac {625 x^3}{24}-\frac {14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac {1331 (5229+76420 x)}{135424 \left (3-x+2 x^2\right )}-\frac {13915}{64} \int \frac {-1+4 x}{3-x+2 x^2} \, dx-\frac {63799791 \int \frac {1}{3-x+2 x^2} \, dx}{33856}\\ &=\frac {2725 x}{8}+\frac {4875 x^2}{32}+\frac {625 x^3}{24}-\frac {14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac {1331 (5229+76420 x)}{135424 \left (3-x+2 x^2\right )}-\frac {13915}{64} \log \left (3-x+2 x^2\right )+\frac {63799791 \operatorname {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )}{16928}\\ &=\frac {2725 x}{8}+\frac {4875 x^2}{32}+\frac {625 x^3}{24}-\frac {14641 (101+79 x)}{5888 \left (3-x+2 x^2\right )^2}+\frac {1331 (5229+76420 x)}{135424 \left (3-x+2 x^2\right )}+\frac {63799791 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{16928 \sqrt {23}}-\frac {13915}{64} \log \left (3-x+2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 98, normalized size = 1.00 \begin {gather*} \frac {625 x^3}{24}+\frac {4875 x^2}{32}+\frac {1331 (76420 x+5229)}{135424 \left (2 x^2-x+3\right )}-\frac {14641 (79 x+101)}{5888 \left (2 x^2-x+3\right )^2}-\frac {13915}{64} \log \left (2 x^2-x+3\right )+\frac {2725 x}{8}-\frac {63799791 \tan ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{16928 \sqrt {23}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 128, normalized size = 1.31 \begin {gather*} \frac {486680000 \, x^{7} + 2360398000 \, x^{6} + 5100406400 \, x^{5} + 2157209100 \, x^{4} + 24531516180 \, x^{3} - 765597492 \, \sqrt {23} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - 6171678159 \, x^{2} - 1015822830 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x^{2} - x + 3\right ) + 23692590858 \, x - 453041787}{4672128 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 72, normalized size = 0.73 \begin {gather*} \frac {625}{24} \, x^{3} + \frac {4875}{32} \, x^{2} - \frac {63799791}{389344} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {2725}{8} \, x + \frac {1331 \, {\left (76420 \, x^{3} - 32981 \, x^{2} + 102022 \, x - 4933\right )}}{67712 \, {\left (2 \, x^{2} - x + 3\right )}^{2}} - \frac {13915}{64} \, \log \left (2 \, x^{2} - x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 73, normalized size = 0.74 \begin {gather*} \frac {625 x^{3}}{24}+\frac {4875 x^{2}}{32}+\frac {2725 x}{8}-\frac {63799791 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{389344}-\frac {13915 \ln \left (2 x^{2}-x +3\right )}{64}-\frac {121 \left (-\frac {210155}{4232} x^{3}+\frac {362791}{16928} x^{2}-\frac {561121}{8464} x +\frac {54263}{16928}\right )}{4 \left (2 x^{2}-x +3\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 82, normalized size = 0.84 \begin {gather*} \frac {625}{24} \, x^{3} + \frac {4875}{32} \, x^{2} - \frac {63799791}{389344} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {2725}{8} \, x + \frac {1331 \, {\left (76420 \, x^{3} - 32981 \, x^{2} + 102022 \, x - 4933\right )}}{67712 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} - \frac {13915}{64} \, \log \left (2 \, x^{2} - x + 3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 81, normalized size = 0.83 \begin {gather*} \frac {2725\,x}{8}-\frac {13915\,\ln \left (2\,x^2-x+3\right )}{64}-\frac {63799791\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{389344}+\frac {4875\,x^2}{32}+\frac {625\,x^3}{24}+\frac {\frac {25428755\,x^3}{67712}-\frac {43897711\,x^2}{270848}+\frac {67895641\,x}{135424}-\frac {6565823}{270848}}{x^4-x^3+\frac {13\,x^2}{4}-\frac {3\,x}{2}+\frac {9}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 95, normalized size = 0.97 \begin {gather*} \frac {625 x^{3}}{24} + \frac {4875 x^{2}}{32} + \frac {2725 x}{8} + \frac {101715020 x^{3} - 43897711 x^{2} + 135791282 x - 6565823}{270848 x^{4} - 270848 x^{3} + 880256 x^{2} - 406272 x + 609408} - \frac {13915 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{64} - \frac {63799791 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{389344} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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