Integrand size = 32, antiderivative size = 22 \[ \int \frac {1}{6} \left (24 x+36 x^2+e^{3+2 x} \left (3 x^2+2 x^3\right )\right ) \, dx=2 x \left (x+x^2+\frac {1}{12} e^{3+2 x} x^2\right ) \]
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Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {12, 1607, 2227, 2207, 2225} \[ \int \frac {1}{6} \left (24 x+36 x^2+e^{3+2 x} \left (3 x^2+2 x^3\right )\right ) \, dx=\frac {1}{6} e^{2 x+3} x^3+2 x^3+2 x^2 \]
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Rule 12
Rule 1607
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} \int \left (24 x+36 x^2+e^{3+2 x} \left (3 x^2+2 x^3\right )\right ) \, dx \\ & = 2 x^2+2 x^3+\frac {1}{6} \int e^{3+2 x} \left (3 x^2+2 x^3\right ) \, dx \\ & = 2 x^2+2 x^3+\frac {1}{6} \int e^{3+2 x} x^2 (3+2 x) \, dx \\ & = 2 x^2+2 x^3+\frac {1}{6} \int \left (3 e^{3+2 x} x^2+2 e^{3+2 x} x^3\right ) \, dx \\ & = 2 x^2+2 x^3+\frac {1}{3} \int e^{3+2 x} x^3 \, dx+\frac {1}{2} \int e^{3+2 x} x^2 \, dx \\ & = 2 x^2+\frac {1}{4} e^{3+2 x} x^2+2 x^3+\frac {1}{6} e^{3+2 x} x^3-\frac {1}{2} \int e^{3+2 x} x \, dx-\frac {1}{2} \int e^{3+2 x} x^2 \, dx \\ & = -\frac {1}{4} e^{3+2 x} x+2 x^2+2 x^3+\frac {1}{6} e^{3+2 x} x^3+\frac {1}{4} \int e^{3+2 x} \, dx+\frac {1}{2} \int e^{3+2 x} x \, dx \\ & = \frac {1}{8} e^{3+2 x}+2 x^2+2 x^3+\frac {1}{6} e^{3+2 x} x^3-\frac {1}{4} \int e^{3+2 x} \, dx \\ & = 2 x^2+2 x^3+\frac {1}{6} e^{3+2 x} x^3 \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {1}{6} \left (24 x+36 x^2+e^{3+2 x} \left (3 x^2+2 x^3\right )\right ) \, dx=\frac {1}{6} x^2 \left (12+\left (12+e^{3+2 x}\right ) x\right ) \]
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Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05
method | result | size |
norman | \(2 x^{2}+2 x^{3}+\frac {x^{3} {\mathrm e}^{3} {\mathrm e}^{2 x}}{6}\) | \(23\) |
risch | \(2 x^{2}+2 x^{3}+\frac {x^{3} {\mathrm e}^{3+2 x}}{6}\) | \(23\) |
parallelrisch | \(2 x^{2}+2 x^{3}+\frac {{\mathrm e}^{3+x} {\mathrm e}^{x} x^{3}}{6}\) | \(23\) |
default | \(\frac {{\mathrm e}^{3} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )}{2}+\frac {{\mathrm e}^{3} \left (\frac {{\mathrm e}^{2 x} x^{3}}{2}-\frac {3 \,{\mathrm e}^{2 x} x^{2}}{4}+\frac {3 x \,{\mathrm e}^{2 x}}{4}-\frac {3 \,{\mathrm e}^{2 x}}{8}\right )}{3}+2 x^{2}+2 x^{3}\) | \(75\) |
parts | \(\frac {{\mathrm e}^{3} \left (\frac {{\mathrm e}^{2 x} x^{2}}{2}-\frac {x \,{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{2 x}}{4}\right )}{2}+\frac {{\mathrm e}^{3} \left (\frac {{\mathrm e}^{2 x} x^{3}}{2}-\frac {3 \,{\mathrm e}^{2 x} x^{2}}{4}+\frac {3 x \,{\mathrm e}^{2 x}}{4}-\frac {3 \,{\mathrm e}^{2 x}}{8}\right )}{3}+2 x^{2}+2 x^{3}\) | \(75\) |
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Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{6} \left (24 x+36 x^2+e^{3+2 x} \left (3 x^2+2 x^3\right )\right ) \, dx=\frac {1}{6} \, x^{3} e^{\left (2 \, x + 3\right )} + 2 \, x^{3} + 2 \, x^{2} \]
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Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{6} \left (24 x+36 x^2+e^{3+2 x} \left (3 x^2+2 x^3\right )\right ) \, dx=\frac {x^{3} e^{3} e^{2 x}}{6} + 2 x^{3} + 2 x^{2} \]
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Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{6} \left (24 x+36 x^2+e^{3+2 x} \left (3 x^2+2 x^3\right )\right ) \, dx=\frac {1}{6} \, x^{3} e^{\left (2 \, x + 3\right )} + 2 \, x^{3} + 2 \, x^{2} \]
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Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{6} \left (24 x+36 x^2+e^{3+2 x} \left (3 x^2+2 x^3\right )\right ) \, dx=\frac {1}{6} \, x^{3} e^{\left (2 \, x + 3\right )} + 2 \, x^{3} + 2 \, x^{2} \]
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Time = 10.37 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1}{6} \left (24 x+36 x^2+e^{3+2 x} \left (3 x^2+2 x^3\right )\right ) \, dx=\frac {x^2\,\left (12\,x+x\,{\mathrm {e}}^{2\,x+3}+12\right )}{6} \]
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