Optimal. Leaf size=21 \[ -\frac{3 x+2}{2 \left (2 x^2+7 x+4\right )} \]
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Rubi [A] time = 0.0119356, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {1660, 8} \[ -\frac{3 x+2}{2 \left (2 x^2+7 x+4\right )} \]
Antiderivative was successfully verified.
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Rule 1660
Rule 8
Rubi steps
\begin{align*} \int \frac{1+4 x+3 x^2}{\left (4+7 x+2 x^2\right )^2} \, dx &=-\frac{2+3 x}{2 \left (4+7 x+2 x^2\right )}-\frac{\int 0 \, dx}{17}\\ &=-\frac{2+3 x}{2 \left (4+7 x+2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0074776, size = 21, normalized size = 1. \[ \frac{-3 x-2}{2 \left (2 x^2+7 x+4\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 17, normalized size = 0.8 \begin{align*}{ \left ( -{\frac{3\,x}{4}}-{\frac{1}{2}} \right ) \left ({x}^{2}+{\frac{7\,x}{2}}+2 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00437, size = 26, normalized size = 1.24 \begin{align*} -\frac{3 \, x + 2}{2 \,{\left (2 \, x^{2} + 7 \, x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6484, size = 46, normalized size = 2.19 \begin{align*} -\frac{3 \, x + 2}{2 \,{\left (2 \, x^{2} + 7 \, x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.112659, size = 15, normalized size = 0.71 \begin{align*} - \frac{3 x + 2}{4 x^{2} + 14 x + 8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22712, size = 26, normalized size = 1.24 \begin{align*} -\frac{3 \, x + 2}{2 \,{\left (2 \, x^{2} + 7 \, x + 4\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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