Optimal. Leaf size=23 \[ \frac {3}{2 \left (3+x+\frac {12 e^{-1-x}}{2+x}\right )^2} \]
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Rubi [F]
time = 4.91, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {e^{2+2 x} \left (216+180 x+36 x^2\right )+e^{3+3 x} \left (-24-36 x-18 x^2-3 x^3\right )}{1728+e^{1+x} \left (2592+2160 x+432 x^2\right )+e^{2+2 x} \left (1296+2160 x+1332 x^2+360 x^3+36 x^4\right )+e^{3+3 x} \left (216+540 x+558 x^2+305 x^3+93 x^4+15 x^5+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 e^{2+2 x} (2+x) \left (-e^{1+x} (2+x)^2+12 (3+x)\right )}{\left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^3} \, dx\\ &=3 \int \frac {e^{2+2 x} (2+x) \left (-e^{1+x} (2+x)^2+12 (3+x)\right )}{\left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^3} \, dx\\ &=3 \int \left (-\frac {e^{2+2 x} (2+x)^2}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2}+\frac {12 e^{2+2 x} \left (22+25 x+9 x^2+x^3\right )}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3}\right ) \, dx\\ &=-\left (3 \int \frac {e^{2+2 x} (2+x)^2}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2} \, dx\right )+36 \int \frac {e^{2+2 x} \left (22+25 x+9 x^2+x^3\right )}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3} \, dx\\ &=-\left (3 \int \frac {e^{2+2 x} (2+x)^2}{(3+x) \left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^2} \, dx\right )+36 \int \frac {e^{2+2 x} \left (22+25 x+9 x^2+x^3\right )}{(3+x) \left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^3} \, dx\\ &=-\left (3 \int \left (\frac {e^{2+2 x}}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2}+\frac {e^{2+2 x} x}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2}+\frac {e^{2+2 x}}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2}\right ) \, dx\right )+36 \int \left (\frac {7 e^{2+2 x}}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3}+\frac {6 e^{2+2 x} x}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3}+\frac {e^{2+2 x} x^2}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3}+\frac {e^{2+2 x}}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3}\right ) \, dx\\ &=-\left (3 \int \frac {e^{2+2 x}}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2} \, dx\right )-3 \int \frac {e^{2+2 x} x}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2} \, dx-3 \int \frac {e^{2+2 x}}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^2} \, dx+36 \int \frac {e^{2+2 x} x^2}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3} \, dx+36 \int \frac {e^{2+2 x}}{(3+x) \left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3} \, dx+216 \int \frac {e^{2+2 x} x}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3} \, dx+252 \int \frac {e^{2+2 x}}{\left (12+6 e^{1+x}+5 e^{1+x} x+e^{1+x} x^2\right )^3} \, dx\\ &=-\left (3 \int \frac {e^{2+2 x}}{\left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^2} \, dx\right )-3 \int \frac {e^{2+2 x} x}{\left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^2} \, dx-3 \int \frac {e^{2+2 x}}{(3+x) \left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^2} \, dx+36 \int \frac {e^{2+2 x} x^2}{\left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^3} \, dx+36 \int \frac {e^{2+2 x}}{(3+x) \left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^3} \, dx+216 \int \frac {e^{2+2 x} x}{\left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^3} \, dx+252 \int \frac {e^{2+2 x}}{\left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 8.86, size = 34, normalized size = 1.48 \begin {gather*} \frac {3 e^{2+2 x} (2+x)^2}{2 \left (12+e^{1+x} \left (6+5 x+x^2\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs.
\(2(20)=40\).
time = 0.77, size = 56, normalized size = 2.43
method | result | size |
norman | \(\frac {6 \,{\mathrm e}^{2 x +2}+6 x \,{\mathrm e}^{2 x +2}+\frac {3 x^{2} {\mathrm e}^{2 x +2}}{2}}{\left (x^{2} {\mathrm e}^{x +1}+5 x \,{\mathrm e}^{x +1}+6 \,{\mathrm e}^{x +1}+12\right )^{2}}\) | \(56\) |
risch | \(\frac {3}{2 \left (x^{2}+6 x +9\right )}-\frac {36 \left (x^{2} {\mathrm e}^{x +1}+5 x \,{\mathrm e}^{x +1}+6 \,{\mathrm e}^{x +1}+6\right )}{\left (x^{2}+6 x +9\right ) \left (x^{2} {\mathrm e}^{x +1}+5 x \,{\mathrm e}^{x +1}+6 \,{\mathrm e}^{x +1}+12\right )^{2}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs.
\(2 (20) = 40\).
time = 0.33, size = 81, normalized size = 3.52 \begin {gather*} \frac {3 \, {\left (x^{2} e^{2} + 4 \, x e^{2} + 4 \, e^{2}\right )} e^{\left (2 \, x\right )}}{2 \, {\left ({\left (x^{4} e^{2} + 10 \, x^{3} e^{2} + 37 \, x^{2} e^{2} + 60 \, x e^{2} + 36 \, e^{2}\right )} e^{\left (2 \, x\right )} + 24 \, {\left (x^{2} e + 5 \, x e + 6 \, e\right )} e^{x} + 144\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 59 vs.
\(2 (20) = 40\).
time = 0.37, size = 59, normalized size = 2.57 \begin {gather*} \frac {3 \, {\left (x^{2} + 4 \, x + 4\right )} e^{\left (2 \, x + 2\right )}}{2 \, {\left ({\left (x^{4} + 10 \, x^{3} + 37 \, x^{2} + 60 \, x + 36\right )} e^{\left (2 \, x + 2\right )} + 24 \, {\left (x^{2} + 5 \, x + 6\right )} e^{\left (x + 1\right )} + 144\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 100 vs.
\(2 (19) = 38\).
time = 0.23, size = 100, normalized size = 4.35 \begin {gather*} \frac {\left (- 36 x^{2} - 180 x - 216\right ) e^{x + 1} - 216}{144 x^{2} + 864 x + \left (24 x^{4} + 264 x^{3} + 1080 x^{2} + 1944 x + 1296\right ) e^{x + 1} + \left (x^{6} + 16 x^{5} + 106 x^{4} + 372 x^{3} + 729 x^{2} + 756 x + 324\right ) e^{2 x + 2} + 1296} + \frac {3}{2 x^{2} + 12 x + 18} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 105 vs.
\(2 (20) = 40\).
time = 0.43, size = 105, normalized size = 4.57 \begin {gather*} \frac {3 \, {\left (x^{2} e^{\left (2 \, x + 2\right )} + 4 \, x e^{\left (2 \, x + 2\right )} + 4 \, e^{\left (2 \, x + 2\right )}\right )}}{2 \, {\left (x^{4} e^{\left (2 \, x + 2\right )} + 10 \, x^{3} e^{\left (2 \, x + 2\right )} + 37 \, x^{2} e^{\left (2 \, x + 2\right )} + 24 \, x^{2} e^{\left (x + 1\right )} + 60 \, x e^{\left (2 \, x + 2\right )} + 120 \, x e^{\left (x + 1\right )} + 36 \, e^{\left (2 \, x + 2\right )} + 144 \, e^{\left (x + 1\right )} + 144\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{2\,x+2}\,\left (36\,x^2+180\,x+216\right )-{\mathrm {e}}^{3\,x+3}\,\left (3\,x^3+18\,x^2+36\,x+24\right )}{{\mathrm {e}}^{x+1}\,\left (432\,x^2+2160\,x+2592\right )+{\mathrm {e}}^{2\,x+2}\,\left (36\,x^4+360\,x^3+1332\,x^2+2160\,x+1296\right )+{\mathrm {e}}^{3\,x+3}\,\left (x^6+15\,x^5+93\,x^4+305\,x^3+558\,x^2+540\,x+216\right )+1728} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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