Optimal. Leaf size=31 \[ -\frac {2 d-3 e}{12 (3+2 x)^3}-\frac {e}{8 (3+2 x)^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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, 45}
\begin {gather*} -\frac {2 d-3 e}{12 (2 x+3)^3}-\frac {e}{8 (2 x+3)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
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*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.71 \begin {gather*} -\frac {4 d+3 e+6 e x}{24 (3+2 x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.55, size = 28, normalized size = 0.90
method | result | size |
norman | \(\frac {-\frac {1}{4} e x -\frac {1}{8} e -\frac {1}{6} d}{\left (2 x +3\right )^{3}}\) | \(20\) |
risch | \(\frac {-\frac {1}{4} e x -\frac {1}{8} e -\frac {1}{6} d}{\left (2 x +3\right )^{3}}\) | \(21\) |
default | \(-\frac {\frac {d}{2}-\frac {3 e}{4}}{3 \left (2 x +3\right )^{3}}-\frac {e}{8 \left (2 x +3\right )^{2}}\) | \(28\) |
gosper | \(-\frac {6 e x +4 d +3 e}{24 \left (2 x +3\right ) \left (4 x^{2}+12 x +9\right )}\) | \(33\) |
meijerg | \(\frac {e \,x^{2} \left (3+\frac {2 x}{3}\right )}{486 \left (1+\frac {2 x}{3}\right )^{3}}+\frac {d x \left (\frac {4}{9} x^{2}+2 x +3\right )}{243 \left (1+\frac {2 x}{3}\right )^{3}}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 32, normalized size = 1.03 \begin {gather*} -\frac {6 \, x e + 4 \, d + 3 \, e}{24 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.79, size = 32, normalized size = 1.03 \begin {gather*} -\frac {3 \, {\left (2 \, x + 1\right )} e + 4 \, d}{24 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 27, normalized size = 0.87 \begin {gather*} \frac {- 4 d - 6 e x - 3 e}{192 x^{3} + 864 x^{2} + 1296 x + 648} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.41, size = 22, normalized size = 0.71 \begin {gather*} -\frac {6 \, x e + 4 \, d + 3 \, e}{24 \, {\left (2 \, x + 3\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 20, normalized size = 0.65 \begin {gather*} -\frac {4\,d+3\,e+6\,e\,x}{24\,{\left (2\,x+3\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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