Optimal. Leaf size=43 \[ \frac{11 (13 x+7)}{155 \left (5 x^2+3 x+2\right )}+\frac{82 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{31 \sqrt{31}} \]
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Rubi [A] time = 0.0267706, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {1660, 12, 618, 204} \[ \frac{11 (13 x+7)}{155 \left (5 x^2+3 x+2\right )}+\frac{82 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{31 \sqrt{31}} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 12
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{3-x+2 x^2}{\left (2+3 x+5 x^2\right )^2} \, dx &=\frac{11 (7+13 x)}{155 \left (2+3 x+5 x^2\right )}+\frac{1}{31} \int \frac{41}{2+3 x+5 x^2} \, dx\\ &=\frac{11 (7+13 x)}{155 \left (2+3 x+5 x^2\right )}+\frac{41}{31} \int \frac{1}{2+3 x+5 x^2} \, dx\\ &=\frac{11 (7+13 x)}{155 \left (2+3 x+5 x^2\right )}-\frac{82}{31} \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )\\ &=\frac{11 (7+13 x)}{155 \left (2+3 x+5 x^2\right )}+\frac{82 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{31 \sqrt{31}}\\ \end{align*}
Mathematica [A] time = 0.0150616, size = 43, normalized size = 1. \[ \frac{11 (13 x+7)}{155 \left (5 x^2+3 x+2\right )}+\frac{82 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{31 \sqrt{31}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 34, normalized size = 0.8 \begin{align*}{ \left ({\frac{143\,x}{775}}+{\frac{77}{775}} \right ) \left ({x}^{2}+{\frac{3\,x}{5}}+{\frac{2}{5}} \right ) ^{-1}}+{\frac{82\,\sqrt{31}}{961}\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 = 1.46585, size = 49, normalized size = 1.14 \begin{align*} \frac{82}{961} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{11 \,{\left (13 \, x + 7\right )}}{155 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.85103, size = 146, normalized size = 3.4 \begin{align*} \frac{410 \, \sqrt{31}{\left (5 \, x^{2} + 3 \, x + 2\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + 4433 \, x + 2387}{4805 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.213093, size = 42, normalized size = 0.98 \begin{align*} \frac{143 x + 77}{775 x^{2} + 465 x + 310} + \frac{82 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{961} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22253, size = 49, normalized size = 1.14 \begin{align*} \frac{82}{961} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{11 \,{\left (13 \, x + 7\right )}}{155 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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