Optimal. Leaf size=77 \[ \frac {125 x^3}{12}+\frac {175 x^2}{4}-\frac {1331 (17-45 x)}{736 \left (2 x^2-x+3\right )}-\frac {2057}{32} \log \left (2 x^2-x+3\right )+\frac {915 x}{16}+\frac {223971 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{368 \sqrt {23}} \]
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Rubi [A] time = 0.07, antiderivative size = 77, 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} \begin {gather*} \frac {125 x^3}{12}+\frac {175 x^2}{4}-\frac {1331 (17-45 x)}{736 \left (2 x^2-x+3\right )}-\frac {2057}{32} \log \left (2 x^2-x+3\right )+\frac {915 x}{16}+\frac {223971 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{368 \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 )^3}{\left (3-x+2 x^2\right )^2} \, dx &=-\frac {1331 (17-45 x)}{736 \left (3-x+2 x^2\right )}+\frac {1}{23} \int \frac {-\frac {25195}{16}-\frac {19067 x}{16}+\frac {22195 x^2}{8}+\frac {13225 x^3}{4}+\frac {2875 x^4}{2}}{3-x+2 x^2} \, dx\\ &=-\frac {1331 (17-45 x)}{736 \left (3-x+2 x^2\right )}+\frac {1}{23} \int \left (\frac {21045}{16}+\frac {4025 x}{2}+\frac {2875 x^2}{4}-\frac {121 (365+391 x)}{8 \left (3-x+2 x^2\right )}\right ) \, dx\\ &=\frac {915 x}{16}+\frac {175 x^2}{4}+\frac {125 x^3}{12}-\frac {1331 (17-45 x)}{736 \left (3-x+2 x^2\right )}-\frac {121}{184} \int \frac {365+391 x}{3-x+2 x^2} \, dx\\ &=\frac {915 x}{16}+\frac {175 x^2}{4}+\frac {125 x^3}{12}-\frac {1331 (17-45 x)}{736 \left (3-x+2 x^2\right )}-\frac {2057}{32} \int \frac {-1+4 x}{3-x+2 x^2} \, dx-\frac {223971}{736} \int \frac {1}{3-x+2 x^2} \, dx\\ &=\frac {915 x}{16}+\frac {175 x^2}{4}+\frac {125 x^3}{12}-\frac {1331 (17-45 x)}{736 \left (3-x+2 x^2\right )}-\frac {2057}{32} \log \left (3-x+2 x^2\right )+\frac {223971}{368} \operatorname {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )\\ &=\frac {915 x}{16}+\frac {175 x^2}{4}+\frac {125 x^3}{12}-\frac {1331 (17-45 x)}{736 \left (3-x+2 x^2\right )}+\frac {223971 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{368 \sqrt {23}}-\frac {2057}{32} \log \left (3-x+2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 77, normalized size = 1.00 \begin {gather*} \frac {125 x^3}{12}+\frac {175 x^2}{4}+\frac {1331 (45 x-17)}{736 \left (2 x^2-x+3\right )}-\frac {2057}{32} \log \left (2 x^2-x+3\right )+\frac {915 x}{16}-\frac {223971 \tan ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{368 \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 )^3}{\left (3-x+2 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.43, size = 88, normalized size = 1.14 \begin {gather*} \frac {1058000 \, x^{5} + 3914600 \, x^{4} + 5173620 \, x^{3} - 1343826 \, \sqrt {23} {\left (2 \, x^{2} - x + 3\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 3761190 \, x^{2} - 3264459 \, {\left (2 \, x^{2} - x + 3\right )} \log \left (2 \, x^{2} - x + 3\right ) + 12845385 \, x - 1561263}{50784 \, {\left (2 \, x^{2} - x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 62, normalized size = 0.81 \begin {gather*} \frac {125}{12} \, x^{3} + \frac {175}{4} \, x^{2} - \frac {223971}{8464} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {915}{16} \, x + \frac {1331 \, {\left (45 \, x - 17\right )}}{736 \, {\left (2 \, x^{2} - x + 3\right )}} - \frac {2057}{32} \, \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 = 61, normalized size = 0.79 \begin {gather*} \frac {125 x^{3}}{12}+\frac {175 x^{2}}{4}+\frac {915 x}{16}-\frac {223971 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{8464}-\frac {2057 \ln \left (2 x^{2}-x +3\right )}{32}-\frac {121 \left (-\frac {495 x}{92}+\frac {187}{92}\right )}{16 \left (x^{2}-\frac {1}{2} x +\frac {3}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 62, normalized size = 0.81 \begin {gather*} \frac {125}{12} \, x^{3} + \frac {175}{4} \, x^{2} - \frac {223971}{8464} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + \frac {915}{16} \, x + \frac {1331 \, {\left (45 \, x - 17\right )}}{736 \, {\left (2 \, x^{2} - x + 3\right )}} - \frac {2057}{32} \, \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 = 3.42, size = 61, normalized size = 0.79 \begin {gather*} \frac {915\,x}{16}-\frac {2057\,\ln \left (2\,x^2-x+3\right )}{32}+\frac {\frac {59895\,x}{1472}-\frac {22627}{1472}}{x^2-\frac {x}{2}+\frac {3}{2}}-\frac {223971\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{8464}+\frac {175\,x^2}{4}+\frac {125\,x^3}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 75, normalized size = 0.97 \begin {gather*} \frac {125 x^{3}}{12} + \frac {175 x^{2}}{4} + \frac {915 x}{16} + \frac {59895 x - 22627}{1472 x^{2} - 736 x + 2208} - \frac {2057 \log {\left (x^{2} - \frac {x}{2} + \frac {3}{2} \right )}}{32} - \frac {223971 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{8464} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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