Optimal. Leaf size=52 \[ -\frac{2 d-3 e}{16 (2 x+3) \left (4 x^2+12 x+9\right )^{3/2}}-\frac{e}{12 \left (4 x^2+12 x+9\right )^{3/2}} \]
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Rubi [A] time = 0.0126159, 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} \[ -\frac{2 d-3 e}{16 (2 x+3) \left (4 x^2+12 x+9\right )^{3/2}}-\frac{e}{12 \left (4 x^2+12 x+9\right )^{3/2}} \]
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 )^{5/2}} \, dx &=-\frac{e}{12 \left (9+12 x+4 x^2\right )^{3/2}}+\frac{1}{2} (2 d-3 e) \int \frac{1}{\left (9+12 x+4 x^2\right )^{5/2}} \, dx\\ &=-\frac{e}{12 \left (9+12 x+4 x^2\right )^{3/2}}-\frac{2 d-3 e}{16 (3+2 x) \left (9+12 x+4 x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0140349, size = 34, normalized size = 0.65 \[ \frac{-6 d-e (8 x+3)}{48 (2 x+3)^3 \sqrt{(2 x+3)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.078, size = 28, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 3+2\,x \right ) \left ( 8\,ex+6\,d+3\,e \right ) }{48} \left ( \left ( 3+2\,x \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60159, size = 49, normalized size = 0.94 \begin{align*} -\frac{e}{12 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}}} - \frac{d}{8 \,{\left (2 \, x + 3\right )}^{4}} + \frac{3 \, e}{16 \,{\left (2 \, x + 3\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51083, size = 92, normalized size = 1.77 \begin{align*} -\frac{8 \, e x + 6 \, d + 3 \, e}{48 \,{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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{5}{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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