Optimal. Leaf size=31 \[ -\frac{2 d-3 e}{12 (2 x+3)^3}-\frac{e}{8 (2 x+3)^2} \]
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Rubi [A] time = 0.0162679, antiderivative size = 31, 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} \[ -\frac{2 d-3 e}{12 (2 x+3)^3}-\frac{e}{8 (2 x+3)^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (9+12 x+4 x^2\right )^2} \, dx &=\int \frac{d+e x}{(3+2 x)^4} \, dx\\ &=\int \left (\frac{2 d-3 e}{2 (3+2 x)^4}+\frac{e}{2 (3+2 x)^3}\right ) \, dx\\ &=-\frac{2 d-3 e}{12 (3+2 x)^3}-\frac{e}{8 (3+2 x)^2}\\ \end{align*}
Mathematica [A] time = 0.0077466, size = 22, normalized size = 0.71 \[ -\frac{4 d+6 e x+3 e}{24 (2 x+3)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 28, normalized size = 0.9 \begin{align*} -{\frac{1}{3\, \left ( 3+2\,x \right ) ^{3}} \left ({\frac{d}{2}}-{\frac{3\,e}{4}} \right ) }-{\frac{e}{8\, \left ( 3+2\,x \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15613, size = 41, normalized size = 1.32 \begin{align*} -\frac{6 \, e x + 4 \, d + 3 \, e}{24 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39814, size = 76, normalized size = 2.45 \begin{align*} -\frac{6 \, e x + 4 \, d + 3 \, e}{24 \,{\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.374043, size = 27, normalized size = 0.87 \begin{align*} - \frac{4 d + 6 e x + 3 e}{192 x^{3} + 864 x^{2} + 1296 x + 648} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19541, size = 30, normalized size = 0.97 \begin{align*} -\frac{6 \, x e + 4 \, d + 3 \, e}{24 \,{\left (2 \, x + 3\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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