Optimal. Leaf size=30 \[ -\frac {2 d-3 e}{4 (3+2 x)}+\frac {1}{4} e \log (3+2 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 30, 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 {1}{4} e \log (2 x+3)-\frac {2 d-3 e}{4 (2 x+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 45
Rubi steps
\begin {align*} \int \frac {d+e x}{9+12 x+4 x^2} \, dx &=\int \frac {d+e x}{(3+2 x)^2} \, dx\\ &=\int \left (\frac {2 d-3 e}{2 (3+2 x)^2}+\frac {e}{2 (3+2 x)}\right ) \, dx\\ &=-\frac {2 d-3 e}{4 (3+2 x)}+\frac {1}{4} e \log (3+2 x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 30, normalized size = 1.00 \begin {gather*} \frac {-2 d+3 e}{4 (3+2 x)}+\frac {1}{4} e \log (3+2 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.57, size = 27, normalized size = 0.90
method | result | size |
norman | \(\frac {-\frac {d}{2}+\frac {3 e}{4}}{2 x +3}+\frac {e \ln \left (2 x +3\right )}{4}\) | \(26\) |
default | \(-\frac {\frac {d}{2}-\frac {3 e}{4}}{2 x +3}+\frac {e \ln \left (2 x +3\right )}{4}\) | \(27\) |
risch | \(-\frac {d}{4 \left (x +\frac {3}{2}\right )}+\frac {3 e}{8 \left (x +\frac {3}{2}\right )}+\frac {e \ln \left (2 x +3\right )}{4}\) | \(27\) |
meijerg | \(\frac {e \left (-\frac {2 x}{3 \left (1+\frac {2 x}{3}\right )}+\ln \left (1+\frac {2 x}{3}\right )\right )}{4}+\frac {d x}{9+6 x}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 28, normalized size = 0.93 \begin {gather*} \frac {1}{4} \, e \log \left (2 \, x + 3\right ) - \frac {2 \, d - 3 \, e}{4 \, {\left (2 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.92, size = 31, normalized size = 1.03 \begin {gather*} \frac {{\left (2 \, x + 3\right )} e \log \left (2 \, x + 3\right ) - 2 \, d + 3 \, e}{4 \, {\left (2 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 20, normalized size = 0.67 \begin {gather*} \frac {e \log {\left (2 x + 3 \right )}}{4} + \frac {- 2 d + 3 e}{8 x + 12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.75, size = 29, normalized size = 0.97 \begin {gather*} \frac {1}{4} \, e \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {2 \, d - 3 \, e}{4 \, {\left (2 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 24, normalized size = 0.80 \begin {gather*} \frac {e\,\ln \left (x+\frac {3}{2}\right )}{4}-\frac {\frac {d}{2}-\frac {3\,e}{4}}{2\,x+3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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