Optimal. Leaf size=30 \[ \frac {1}{4} e \log (2 x+3)-\frac {2 d-3 e}{4 (2 x+3)} \]
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Rubi [A] time = 0.02, 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, 43} \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 43
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 {3 e-2 d}{4 (2 x+3)}+\frac {1}{4} e \log (2 x+3) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{9+12 x+4 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 31, normalized size = 1.03 \begin {gather*} \frac {{\left (2 \, e x + 3 \, e\right )} \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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giac [A] time = 0.19, 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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maple [A] time = 0.05, size = 31, normalized size = 1.03 \begin {gather*} \frac {e \ln \left (2 x +3\right )}{4}-\frac {d}{2 \left (2 x +3\right )}+\frac {3 e}{4 \left (2 x +3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 26, normalized size = 0.87 \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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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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sympy [A] time = 0.14, 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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