Optimal. Leaf size=64 \[ \frac {121 (19-7 x)}{368 \left (2 x^2-x+3\right )^2}-\frac {55 (332 x+975)}{8464 \left (2 x^2-x+3\right )}-\frac {4330 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{529 \sqrt {23}} \]
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Rubi [A] time = 0.05, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1660, 12, 618, 204} \[ \frac {121 (19-7 x)}{368 \left (2 x^2-x+3\right )^2}-\frac {55 (332 x+975)}{8464 \left (2 x^2-x+3\right )}-\frac {4330 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{529 \sqrt {23}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 618
Rule 1660
Rubi steps
\begin {align*} \int \frac {\left (2+3 x+5 x^2\right )^2}{\left (3-x+2 x^2\right )^3} \, dx &=\frac {121 (19-7 x)}{368 \left (3-x+2 x^2\right )^2}+\frac {1}{46} \int \frac {-\frac {195}{8}+\frac {1955 x}{2}+575 x^2}{\left (3-x+2 x^2\right )^2} \, dx\\ &=\frac {121 (19-7 x)}{368 \left (3-x+2 x^2\right )^2}-\frac {55 (975+332 x)}{8464 \left (3-x+2 x^2\right )}+\frac {\int \frac {4330}{3-x+2 x^2} \, dx}{1058}\\ &=\frac {121 (19-7 x)}{368 \left (3-x+2 x^2\right )^2}-\frac {55 (975+332 x)}{8464 \left (3-x+2 x^2\right )}+\frac {2165}{529} \int \frac {1}{3-x+2 x^2} \, dx\\ &=\frac {121 (19-7 x)}{368 \left (3-x+2 x^2\right )^2}-\frac {55 (975+332 x)}{8464 \left (3-x+2 x^2\right )}-\frac {4330}{529} \operatorname {Subst}\left (\int \frac {1}{-23-x^2} \, dx,x,-1+4 x\right )\\ &=\frac {121 (19-7 x)}{368 \left (3-x+2 x^2\right )^2}-\frac {55 (975+332 x)}{8464 \left (3-x+2 x^2\right )}-\frac {4330 \tan ^{-1}\left (\frac {1-4 x}{\sqrt {23}}\right )}{529 \sqrt {23}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 51, normalized size = 0.80 \[ \frac {4330 \tan ^{-1}\left (\frac {4 x-1}{\sqrt {23}}\right )}{529 \sqrt {23}}-\frac {11 \left (1660 x^3+4045 x^2+938 x+4909\right )}{4232 \left (-2 x^2+x-3\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 75, normalized size = 1.17 \[ -\frac {419980 \, x^{3} - 34640 \, \sqrt {23} {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) + 1023385 \, x^{2} + 237314 \, x + 1241977}{97336 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 46, normalized size = 0.72 \[ \frac {4330}{12167} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {11 \, {\left (1660 \, x^{3} + 4045 \, x^{2} + 938 \, x + 4909\right )}}{4232 \, {\left (2 \, x^{2} - x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.73 \[ \frac {4330 \sqrt {23}\, \arctan \left (\frac {\left (4 x -1\right ) \sqrt {23}}{23}\right )}{12167}+\frac {-\frac {4565}{1058} x^{3}-\frac {44495}{4232} x^{2}-\frac {5159}{2116} x -\frac {53999}{4232}}{\left (2 x^{2}-x +3\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 56, normalized size = 0.88 \[ \frac {4330}{12167} \, \sqrt {23} \arctan \left (\frac {1}{23} \, \sqrt {23} {\left (4 \, x - 1\right )}\right ) - \frac {11 \, {\left (1660 \, x^{3} + 4045 \, x^{2} + 938 \, x + 4909\right )}}{4232 \, {\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.47, size = 56, normalized size = 0.88 \[ \frac {4330\,\sqrt {23}\,\mathrm {atan}\left (\frac {4\,\sqrt {23}\,x}{23}-\frac {\sqrt {23}}{23}\right )}{12167}-\frac {\frac {4565\,x^3}{4232}+\frac {44495\,x^2}{16928}+\frac {5159\,x}{8464}+\frac {53999}{16928}}{x^4-x^3+\frac {13\,x^2}{4}-\frac {3\,x}{2}+\frac {9}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 63, normalized size = 0.98 \[ \frac {- 18260 x^{3} - 44495 x^{2} - 10318 x - 53999}{16928 x^{4} - 16928 x^{3} + 55016 x^{2} - 25392 x + 38088} + \frac {4330 \sqrt {23} \operatorname {atan}{\left (\frac {4 \sqrt {23} x}{23} - \frac {\sqrt {23}}{23} \right )}}{12167} \]
Verification of antiderivative is not currently implemented for this CAS.
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