Optimal. Leaf size=43 \[ -\frac{19 x+39}{28 \left (3 x^2+2 x+5\right )}-\frac{19 \tan ^{-1}\left (\frac{3 x+1}{\sqrt{14}}\right )}{28 \sqrt{14}} \]
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Rubi [A] time = 0.0206964, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {638, 618, 204} \[ -\frac{19 x+39}{28 \left (3 x^2+2 x+5\right )}-\frac{19 \tan ^{-1}\left (\frac{3 x+1}{\sqrt{14}}\right )}{28 \sqrt{14}} \]
Antiderivative was successfully verified.
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Rule 638
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{-4+7 x}{\left (5+2 x+3 x^2\right )^2} \, dx &=-\frac{39+19 x}{28 \left (5+2 x+3 x^2\right )}-\frac{19}{28} \int \frac{1}{5+2 x+3 x^2} \, dx\\ &=-\frac{39+19 x}{28 \left (5+2 x+3 x^2\right )}+\frac{19}{14} \operatorname{Subst}\left (\int \frac{1}{-56-x^2} \, dx,x,2+6 x\right )\\ &=-\frac{39+19 x}{28 \left (5+2 x+3 x^2\right )}-\frac{19 \tan ^{-1}\left (\frac{1+3 x}{\sqrt{14}}\right )}{28 \sqrt{14}}\\ \end{align*}
Mathematica [A] time = 0.033955, size = 43, normalized size = 1. \[ \frac{-19 x-39}{28 \left (3 x^2+2 x+5\right )}-\frac{19 \tan ^{-1}\left (\frac{3 x+1}{\sqrt{14}}\right )}{28 \sqrt{14}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 37, normalized size = 0.9 \begin{align*}{\frac{-38\,x-78}{168\,{x}^{2}+112\,x+280}}-{\frac{19\,\sqrt{14}}{392}\arctan \left ({\frac{ \left ( 6\,x+2 \right ) \sqrt{14}}{28}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40944, size = 49, normalized size = 1.14 \begin{align*} -\frac{19}{392} \, \sqrt{14} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (3 \, x + 1\right )}\right ) - \frac{19 \, x + 39}{28 \,{\left (3 \, x^{2} + 2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96407, size = 140, normalized size = 3.26 \begin{align*} -\frac{19 \, \sqrt{14}{\left (3 \, x^{2} + 2 \, x + 5\right )} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (3 \, x + 1\right )}\right ) + 266 \, x + 546}{392 \,{\left (3 \, x^{2} + 2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.128665, size = 42, normalized size = 0.98 \begin{align*} - \frac{19 x + 39}{84 x^{2} + 56 x + 140} - \frac{19 \sqrt{14} \operatorname{atan}{\left (\frac{3 \sqrt{14} x}{14} + \frac{\sqrt{14}}{14} \right )}}{392} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06648, size = 49, normalized size = 1.14 \begin{align*} -\frac{19}{392} \, \sqrt{14} \arctan \left (\frac{1}{14} \, \sqrt{14}{\left (3 \, x + 1\right )}\right ) - \frac{19 \, x + 39}{28 \,{\left (3 \, x^{2} + 2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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