Optimal. Leaf size=31 \[ -\frac {2 d-3 e}{20 (3+2 x)^5}-\frac {e}{16 (3+2 x)^4} \]
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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}{20 (2 x+3)^5}-\frac {e}{16 (2 x+3)^4} \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 )^3} \, dx &=\int \frac {d+e x}{(3+2 x)^6} \, dx\\ &=\int \left (\frac {2 d-3 e}{2 (3+2 x)^6}+\frac {e}{2 (3+2 x)^5}\right ) \, dx\\ &=-\frac {2 d-3 e}{20 (3+2 x)^5}-\frac {e}{16 (3+2 x)^4}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.71 \begin {gather*} -\frac {8 d+e (3+10 x)}{80 (3+2 x)^5} \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 | \(\frac {-\frac {1}{8} e x -\frac {3}{80} e -\frac {1}{10} d}{\left (2 x +3\right )^{5}}\) | \(20\) |
risch | \(\frac {-\frac {1}{8} e x -\frac {3}{80} e -\frac {1}{10} d}{\left (2 x +3\right )^{5}}\) | \(21\) |
default | \(-\frac {\frac {d}{2}-\frac {3 e}{4}}{5 \left (2 x +3\right )^{5}}-\frac {e}{16 \left (2 x +3\right )^{4}}\) | \(28\) |
gosper | \(-\frac {10 e x +8 d +3 e}{80 \left (2 x +3\right ) \left (4 x^{2}+12 x +9\right )^{2}}\) | \(33\) |
meijerg | \(\frac {e \,x^{2} \left (\frac {8}{27} x^{3}+\frac {20}{9} x^{2}+\frac {20}{3} x +10\right )}{14580 \left (1+\frac {2 x}{3}\right )^{5}}+\frac {d x \left (\frac {16}{81} x^{4}+\frac {40}{27} x^{3}+\frac {40}{9} x^{2}+\frac {20}{3} x +5\right )}{3645 \left (1+\frac {2 x}{3}\right )^{5}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 42, normalized size = 1.35 \begin {gather*} -\frac {10 \, x e + 8 \, d + 3 \, e}{80 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.71, size = 41, normalized size = 1.32 \begin {gather*} -\frac {{\left (10 \, x + 3\right )} e + 8 \, d}{80 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 37, normalized size = 1.19 \begin {gather*} \frac {- 8 d - 10 e x - 3 e}{2560 x^{5} + 19200 x^{4} + 57600 x^{3} + 86400 x^{2} + 64800 x + 19440} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.18, size = 22, normalized size = 0.71 \begin {gather*} -\frac {10 \, x e + 8 \, d + 3 \, e}{80 \, {\left (2 \, x + 3\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 20, normalized size = 0.65 \begin {gather*} -\frac {8\,d+3\,e+10\,e\,x}{80\,{\left (2\,x+3\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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