Optimal. Leaf size=31 \[ -\frac {2 d-3 e}{12 (2 x+3)^3}-\frac {e}{8 (2 x+3)^2} \]
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Rubi [A] time = 0.02, 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, 43} \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 43
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+6 e x+3 e}{24 (2 x+3)^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}{\left (9+12 x+4 x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 30, normalized size = 0.97 \begin {gather*} -\frac {6 \, e x + 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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giac [A] time = 0.19, 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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maple [A] time = 0.04, size = 28, normalized size = 0.90 \begin {gather*} -\frac {e}{8 \left (2 x +3\right )^{2}}-\frac {\frac {d}{2}-\frac {3 e}{4}}{3 \left (2 x +3\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.03, size = 30, normalized size = 0.97 \begin {gather*} -\frac {6 \, e x + 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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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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sympy [A] time = 0.22, 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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