Optimal. Leaf size=64 \[ \frac{121 (69 x+61)}{7750 \left (5 x^2+3 x+2\right )^2}+\frac{11 (45710 x+17557)}{240250 \left (5 x^2+3 x+2\right )}+\frac{4330 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{961 \sqrt{31}} \]
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Rubi [A] time = 0.0516227, 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} \[ \frac{121 (69 x+61)}{7750 \left (5 x^2+3 x+2\right )^2}+\frac{11 (45710 x+17557)}{240250 \left (5 x^2+3 x+2\right )}+\frac{4330 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{961 \sqrt{31}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 12
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\left (3-x+2 x^2\right )^2}{\left (2+3 x+5 x^2\right )^3} \, dx &=\frac{121 (61+69 x)}{7750 \left (2+3 x+5 x^2\right )^2}+\frac{1}{62} \int \frac{\frac{48669}{125}-\frac{1984 x}{25}+\frac{248 x^2}{5}}{\left (2+3 x+5 x^2\right )^2} \, dx\\ &=\frac{121 (61+69 x)}{7750 \left (2+3 x+5 x^2\right )^2}+\frac{11 (17557+45710 x)}{240250 \left (2+3 x+5 x^2\right )}+\frac{\int \frac{4330}{2+3 x+5 x^2} \, dx}{1922}\\ &=\frac{121 (61+69 x)}{7750 \left (2+3 x+5 x^2\right )^2}+\frac{11 (17557+45710 x)}{240250 \left (2+3 x+5 x^2\right )}+\frac{2165}{961} \int \frac{1}{2+3 x+5 x^2} \, dx\\ &=\frac{121 (61+69 x)}{7750 \left (2+3 x+5 x^2\right )^2}+\frac{11 (17557+45710 x)}{240250 \left (2+3 x+5 x^2\right )}-\frac{4330}{961} \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )\\ &=\frac{121 (61+69 x)}{7750 \left (2+3 x+5 x^2\right )^2}+\frac{11 (17557+45710 x)}{240250 \left (2+3 x+5 x^2\right )}+\frac{4330 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{961 \sqrt{31}}\\ \end{align*}
Mathematica [A] time = 0.0233833, size = 53, normalized size = 0.83 \[ \frac{11 \left (45710 x^3+44983 x^2+33524 x+11183\right )}{48050 \left (5 x^2+3 x+2\right )^2}+\frac{4330 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{961 \sqrt{31}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 47, normalized size = 0.7 \begin{align*} 25\,{\frac{1}{ \left ( 5\,{x}^{2}+3\,x+2 \right ) ^{2}} \left ({\frac{50281\,{x}^{3}}{120125}}+{\frac{494813\,{x}^{2}}{1201250}}+{\frac{184382\,x}{600625}}+{\frac{123013}{1201250}} \right ) }+{\frac{4330\,\sqrt{31}}{29791}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4875, size = 76, normalized size = 1.19 \begin{align*} \frac{4330}{29791} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{11 \,{\left (45710 \, x^{3} + 44983 \, x^{2} + 33524 \, x + 11183\right )}}{48050 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.948494, size = 258, normalized size = 4.03 \begin{align*} \frac{15587110 \, x^{3} + 216500 \, \sqrt{31}{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + 15339203 \, x^{2} + 11431684 \, x + 3813403}{1489550 \,{\left (25 \, x^{4} + 30 \, x^{3} + 29 \, x^{2} + 12 \, x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.298147, size = 63, normalized size = 0.98 \begin{align*} \frac{502810 x^{3} + 494813 x^{2} + 368764 x + 123013}{1201250 x^{4} + 1441500 x^{3} + 1393450 x^{2} + 576600 x + 192200} + \frac{4330 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{29791} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23913, size = 62, normalized size = 0.97 \begin{align*} \frac{4330}{29791} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{11 \,{\left (45710 \, x^{3} + 44983 \, x^{2} + 33524 \, x + 11183\right )}}{48050 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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