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 = 0.0529804, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, 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 1660
Rule 12
Rule 618
Rule 204
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*}
Mathematica [A] time = 0.0290323, size = 51, normalized size = 0.8 \[ \frac{4330 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{529 \sqrt{23}}-\frac{11 \left (1660 x^3+4045 x^2+938 x+4909\right )}{4232 \left (-2 x^2+x-3\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 47, normalized size = 0.7 \begin{align*} 4\,{\frac{1}{ \left ( 2\,{x}^{2}-x+3 \right ) ^{2}} \left ( -{\frac{4565\,{x}^{3}}{4232}}-{\frac{44495\,{x}^{2}}{16928}}-{\frac{5159\,x}{8464}}-{\frac{53999}{16928}} \right ) }+{\frac{4330\,\sqrt{23}}{12167}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{23}}{23}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43885, size = 76, normalized size = 1.19 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.00681, size = 239, normalized size = 3.73 \begin{align*} -\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.275541, size = 61, normalized size = 0.95 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29135, size = 62, normalized size = 0.97 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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