Integrand size = 33, antiderivative size = 25 \[ \int \left (3 e^{4+3 x} \left (5+4 e^2+2 e^4\right )+e^{2 x} (1+2 x)\right ) \, dx=e^{2 x} \left (e^{4+x} \left (3+2 \left (1+e^2\right )^2\right )+x\right ) \]
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Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2225, 2207} \[ \int \left (3 e^{4+3 x} \left (5+4 e^2+2 e^4\right )+e^{2 x} (1+2 x)\right ) \, dx=\frac {1}{2} e^{2 x} (2 x+1)-\frac {e^{2 x}}{2}+\left (5+4 e^2+2 e^4\right ) e^{3 x+4} \]
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Rule 2207
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \left (3 \left (5+4 e^2+2 e^4\right )\right ) \int e^{4+3 x} \, dx+\int e^{2 x} (1+2 x) \, dx \\ & = e^{4+3 x} \left (5+4 e^2+2 e^4\right )+\frac {1}{2} e^{2 x} (1+2 x)-\int e^{2 x} \, dx \\ & = -\frac {e^{2 x}}{2}+e^{4+3 x} \left (5+4 e^2+2 e^4\right )+\frac {1}{2} e^{2 x} (1+2 x) \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \left (3 e^{4+3 x} \left (5+4 e^2+2 e^4\right )+e^{2 x} (1+2 x)\right ) \, dx=e^{4+3 x} \left (5+4 e^2+2 e^4\right )+e^{2 x} x \]
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Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20
method | result | size |
parallelrisch | \(x \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x} {\mathrm e}^{\ln \left (2 \,{\mathrm e}^{4}+4 \,{\mathrm e}^{2}+5\right )+4+x}\) | \(30\) |
norman | \(x \,{\mathrm e}^{2 x}+\left (2 \left ({\mathrm e}^{4}\right )^{2}+4 \,{\mathrm e}^{2} {\mathrm e}^{4}+5 \,{\mathrm e}^{4}\right ) {\mathrm e}^{3 x}\) | \(32\) |
risch | \(5 \,{\mathrm e}^{4+3 x}+2 \,{\mathrm e}^{8+3 x}+4 \,{\mathrm e}^{6+3 x}+x \,{\mathrm e}^{2 x}\) | \(32\) |
default | \(x \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{3 x} \left ({\mathrm e}^{4}\right )^{2}+4 \,{\mathrm e}^{3 x} {\mathrm e}^{4} {\mathrm e}^{2}+5 \,{\mathrm e}^{3 x} {\mathrm e}^{4}\) | \(38\) |
meijerg | \(-{\mathrm e}^{\ln \left (2 \,{\mathrm e}^{4}+4 \,{\mathrm e}^{2}+5\right )+4} \left (1-{\mathrm e}^{3 x}\right )+\frac {{\mathrm e}^{2 x}}{2}-\frac {\left (-4 x +2\right ) {\mathrm e}^{2 x}}{4}\) | \(43\) |
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Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \left (3 e^{4+3 x} \left (5+4 e^2+2 e^4\right )+e^{2 x} (1+2 x)\right ) \, dx={\left (2 \, e^{8} + 4 \, e^{6} + 5 \, e^{4}\right )} e^{\left (3 \, x\right )} + x e^{\left (2 \, x\right )} \]
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Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \left (3 e^{4+3 x} \left (5+4 e^2+2 e^4\right )+e^{2 x} (1+2 x)\right ) \, dx=x e^{2 x} + \left (5 e^{4} + 4 e^{6} + 2 e^{8}\right ) \left (e^{2 x}\right )^{\frac {3}{2}} \]
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Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \left (3 e^{4+3 x} \left (5+4 e^2+2 e^4\right )+e^{2 x} (1+2 x)\right ) \, dx=\frac {1}{2} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + {\left (2 \, e^{4} + 4 \, e^{2} + 5\right )} e^{\left (3 \, x + 4\right )} + \frac {1}{2} \, e^{\left (2 \, x\right )} \]
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Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \left (3 e^{4+3 x} \left (5+4 e^2+2 e^4\right )+e^{2 x} (1+2 x)\right ) \, dx=x e^{\left (2 \, x\right )} + e^{\left (3 \, x + \log \left (2 \, e^{4} + 4 \, e^{2} + 5\right ) + 4\right )} \]
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Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \left (3 e^{4+3 x} \left (5+4 e^2+2 e^4\right )+e^{2 x} (1+2 x)\right ) \, dx=x\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x+4}\,\left (4\,{\mathrm {e}}^2+2\,{\mathrm {e}}^4+5\right ) \]
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