Optimal. Leaf size=31 \[ -\frac{2 d-3 e}{20 (2 x+3)^5}-\frac{e}{16 (2 x+3)^4} \]
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Rubi [A] time = 0.0165174, 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} \[ -\frac{2 d-3 e}{20 (2 x+3)^5}-\frac{e}{16 (2 x+3)^4} \]
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 )^3} \, dx &=\int \frac{d+e x}{(3+2 x)^6} \, dx\\ &=\int \left (\frac{2 d-3 e}{2 (3+2 x)^6}+\frac{e}{2 (3+2 x)^5}\right ) \, dx\\ &=-\frac{2 d-3 e}{20 (3+2 x)^5}-\frac{e}{16 (3+2 x)^4}\\ \end{align*}
Mathematica [A] time = 0.0074101, size = 22, normalized size = 0.71 \[ -\frac{8 d+e (10 x+3)}{80 (2 x+3)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 28, normalized size = 0.9 \begin{align*} -{\frac{e}{16\, \left ( 3+2\,x \right ) ^{4}}}-{\frac{1}{5\, \left ( 3+2\,x \right ) ^{5}} \left ({\frac{d}{2}}-{\frac{3\,e}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0152, size = 54, normalized size = 1.74 \begin{align*} -\frac{10 \, e x + 8 \, d + 3 \, e}{80 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67236, size = 111, normalized size = 3.58 \begin{align*} -\frac{10 \, e x + 8 \, d + 3 \, e}{80 \,{\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.496349, size = 37, normalized size = 1.19 \begin{align*} - \frac{8 d + 10 e x + 3 e}{2560 x^{5} + 19200 x^{4} + 57600 x^{3} + 86400 x^{2} + 64800 x + 19440} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13009, size = 30, normalized size = 0.97 \begin{align*} -\frac{10 \, x e + 8 \, d + 3 \, e}{80 \,{\left (2 \, x + 3\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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