Optimal. Leaf size=30 \[ \frac{1}{4} e \log (2 x+3)-\frac{2 d-3 e}{4 (2 x+3)} \]
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Rubi [A] time = 0.0168794, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {27, 43} \[ \frac{1}{4} e \log (2 x+3)-\frac{2 d-3 e}{4 (2 x+3)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{d+e x}{9+12 x+4 x^2} \, dx &=\int \frac{d+e x}{(3+2 x)^2} \, dx\\ &=\int \left (\frac{2 d-3 e}{2 (3+2 x)^2}+\frac{e}{2 (3+2 x)}\right ) \, dx\\ &=-\frac{2 d-3 e}{4 (3+2 x)}+\frac{1}{4} e \log (3+2 x)\\ \end{align*}
Mathematica [A] time = 0.0066698, size = 30, normalized size = 1. \[ \frac{3 e-2 d}{4 (2 x+3)}+\frac{1}{4} e \log (2 x+3) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 31, normalized size = 1. \begin{align*}{\frac{e\ln \left ( 3+2\,x \right ) }{4}}-{\frac{d}{6+4\,x}}+{\frac{3\,e}{12+8\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10739, size = 35, normalized size = 1.17 \begin{align*} \frac{1}{4} \, e \log \left (2 \, x + 3\right ) - \frac{2 \, d - 3 \, e}{4 \,{\left (2 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37845, size = 76, normalized size = 2.53 \begin{align*} \frac{{\left (2 \, e x + 3 \, e\right )} \log \left (2 \, x + 3\right ) - 2 \, d + 3 \, e}{4 \,{\left (2 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.335154, size = 20, normalized size = 0.67 \begin{align*} \frac{e \log{\left (2 x + 3 \right )}}{4} - \frac{2 d - 3 e}{8 x + 12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13272, size = 39, normalized size = 1.3 \begin{align*} \frac{1}{4} \, e \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac{2 \, d - 3 \, e}{4 \,{\left (2 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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