Optimal. Leaf size=52 \[ -\frac{2 d-3 e}{8 (2 x+3) \sqrt{4 x^2+12 x+9}}-\frac{e}{4 \sqrt{4 x^2+12 x+9}} \]
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Rubi [A] time = 0.0135592, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {640, 607} \[ -\frac{2 d-3 e}{8 (2 x+3) \sqrt{4 x^2+12 x+9}}-\frac{e}{4 \sqrt{4 x^2+12 x+9}} \]
Antiderivative was successfully verified.
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Rule 640
Rule 607
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx &=-\frac{e}{4 \sqrt{9+12 x+4 x^2}}+\frac{1}{2} (2 d-3 e) \int \frac{1}{\left (9+12 x+4 x^2\right )^{3/2}} \, dx\\ &=-\frac{e}{4 \sqrt{9+12 x+4 x^2}}-\frac{2 d-3 e}{8 (3+2 x) \sqrt{9+12 x+4 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0113364, size = 34, normalized size = 0.65 \[ \frac{-2 d-e (4 x+3)}{8 (2 x+3) \sqrt{(2 x+3)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 28, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 3+2\,x \right ) \left ( 4\,ex+2\,d+3\,e \right ) }{8} \left ( \left ( 3+2\,x \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56026, size = 49, normalized size = 0.94 \begin{align*} -\frac{e}{4 \, \sqrt{4 \, x^{2} + 12 \, x + 9}} - \frac{d}{4 \,{\left (2 \, x + 3\right )}^{2}} + \frac{3 \, e}{8 \,{\left (2 \, x + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54412, size = 61, normalized size = 1.17 \begin{align*} -\frac{4 \, e x + 2 \, d + 3 \, e}{8 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d + e x}{\left (\left (2 x + 3\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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