Integrand size = 16, antiderivative size = 68 \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=-\frac {724 e^{3 x}}{243}+\frac {76}{81} e^{3 x} x-\frac {38}{27} e^{3 x} x^2+\frac {38}{27} e^{3 x} x^3-\frac {5}{9} e^{3 x} x^4+\frac {1}{3} e^{3 x} x^5 \]
[Out]
Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2227, 2225, 2207} \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=\frac {1}{3} e^{3 x} x^5-\frac {5}{9} e^{3 x} x^4+\frac {38}{27} e^{3 x} x^3-\frac {38}{27} e^{3 x} x^2+\frac {76}{81} e^{3 x} x-\frac {724 e^{3 x}}{243} \]
[In]
[Out]
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \int \left (-8 e^{3 x}+2 e^{3 x} x^3+e^{3 x} x^5\right ) \, dx \\ & = 2 \int e^{3 x} x^3 \, dx-8 \int e^{3 x} \, dx+\int e^{3 x} x^5 \, dx \\ & = -\frac {8 e^{3 x}}{3}+\frac {2}{3} e^{3 x} x^3+\frac {1}{3} e^{3 x} x^5-\frac {5}{3} \int e^{3 x} x^4 \, dx-2 \int e^{3 x} x^2 \, dx \\ & = -\frac {8 e^{3 x}}{3}-\frac {2}{3} e^{3 x} x^2+\frac {2}{3} e^{3 x} x^3-\frac {5}{9} e^{3 x} x^4+\frac {1}{3} e^{3 x} x^5+\frac {4}{3} \int e^{3 x} x \, dx+\frac {20}{9} \int e^{3 x} x^3 \, dx \\ & = -\frac {8 e^{3 x}}{3}+\frac {4}{9} e^{3 x} x-\frac {2}{3} e^{3 x} x^2+\frac {38}{27} e^{3 x} x^3-\frac {5}{9} e^{3 x} x^4+\frac {1}{3} e^{3 x} x^5-\frac {4}{9} \int e^{3 x} \, dx-\frac {20}{9} \int e^{3 x} x^2 \, dx \\ & = -\frac {76 e^{3 x}}{27}+\frac {4}{9} e^{3 x} x-\frac {38}{27} e^{3 x} x^2+\frac {38}{27} e^{3 x} x^3-\frac {5}{9} e^{3 x} x^4+\frac {1}{3} e^{3 x} x^5+\frac {40}{27} \int e^{3 x} x \, dx \\ & = -\frac {76 e^{3 x}}{27}+\frac {76}{81} e^{3 x} x-\frac {38}{27} e^{3 x} x^2+\frac {38}{27} e^{3 x} x^3-\frac {5}{9} e^{3 x} x^4+\frac {1}{3} e^{3 x} x^5-\frac {40}{81} \int e^{3 x} \, dx \\ & = -\frac {724 e^{3 x}}{243}+\frac {76}{81} e^{3 x} x-\frac {38}{27} e^{3 x} x^2+\frac {38}{27} e^{3 x} x^3-\frac {5}{9} e^{3 x} x^4+\frac {1}{3} e^{3 x} x^5 \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.50 \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=\frac {1}{243} e^{3 x} \left (-724+228 x-342 x^2+342 x^3-135 x^4+81 x^5\right ) \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.46
method | result | size |
risch | \(\left (\frac {1}{3} x^{5}-\frac {5}{9} x^{4}+\frac {38}{27} x^{3}-\frac {38}{27} x^{2}+\frac {76}{81} x -\frac {724}{243}\right ) {\mathrm e}^{3 x}\) | \(31\) |
gosper | \(\frac {{\mathrm e}^{3 x} \left (81 x^{5}-135 x^{4}+342 x^{3}-342 x^{2}+228 x -724\right )}{243}\) | \(32\) |
derivativedivides | \(-\frac {724 \,{\mathrm e}^{3 x}}{243}+\frac {76 \,{\mathrm e}^{3 x} x}{81}-\frac {38 \,{\mathrm e}^{3 x} x^{2}}{27}+\frac {38 \,{\mathrm e}^{3 x} x^{3}}{27}-\frac {5 \,{\mathrm e}^{3 x} x^{4}}{9}+\frac {{\mathrm e}^{3 x} x^{5}}{3}\) | \(51\) |
default | \(-\frac {724 \,{\mathrm e}^{3 x}}{243}+\frac {76 \,{\mathrm e}^{3 x} x}{81}-\frac {38 \,{\mathrm e}^{3 x} x^{2}}{27}+\frac {38 \,{\mathrm e}^{3 x} x^{3}}{27}-\frac {5 \,{\mathrm e}^{3 x} x^{4}}{9}+\frac {{\mathrm e}^{3 x} x^{5}}{3}\) | \(51\) |
norman | \(-\frac {724 \,{\mathrm e}^{3 x}}{243}+\frac {76 \,{\mathrm e}^{3 x} x}{81}-\frac {38 \,{\mathrm e}^{3 x} x^{2}}{27}+\frac {38 \,{\mathrm e}^{3 x} x^{3}}{27}-\frac {5 \,{\mathrm e}^{3 x} x^{4}}{9}+\frac {{\mathrm e}^{3 x} x^{5}}{3}\) | \(51\) |
parallelrisch | \(-\frac {724 \,{\mathrm e}^{3 x}}{243}+\frac {76 \,{\mathrm e}^{3 x} x}{81}-\frac {38 \,{\mathrm e}^{3 x} x^{2}}{27}+\frac {38 \,{\mathrm e}^{3 x} x^{3}}{27}-\frac {5 \,{\mathrm e}^{3 x} x^{4}}{9}+\frac {{\mathrm e}^{3 x} x^{5}}{3}\) | \(51\) |
parts | \(-\frac {724 \,{\mathrm e}^{3 x}}{243}+\frac {76 \,{\mathrm e}^{3 x} x}{81}-\frac {38 \,{\mathrm e}^{3 x} x^{2}}{27}+\frac {38 \,{\mathrm e}^{3 x} x^{3}}{27}-\frac {5 \,{\mathrm e}^{3 x} x^{4}}{9}+\frac {{\mathrm e}^{3 x} x^{5}}{3}\) | \(51\) |
meijerg | \(\frac {724}{243}-\frac {\left (-1458 x^{5}+2430 x^{4}-3240 x^{3}+3240 x^{2}-2160 x +720\right ) {\mathrm e}^{3 x}}{4374}-\frac {\left (-108 x^{3}+108 x^{2}-72 x +24\right ) {\mathrm e}^{3 x}}{162}-\frac {8 \,{\mathrm e}^{3 x}}{3}\) | \(61\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.46 \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=\frac {1}{243} \, {\left (81 \, x^{5} - 135 \, x^{4} + 342 \, x^{3} - 342 \, x^{2} + 228 \, x - 724\right )} e^{\left (3 \, x\right )} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.46 \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=\frac {\left (81 x^{5} - 135 x^{4} + 342 x^{3} - 342 x^{2} + 228 x - 724\right ) e^{3 x}}{243} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.87 \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=\frac {1}{243} \, {\left (81 \, x^{5} - 135 \, x^{4} + 180 \, x^{3} - 180 \, x^{2} + 120 \, x - 40\right )} e^{\left (3 \, x\right )} + \frac {2}{27} \, {\left (9 \, x^{3} - 9 \, x^{2} + 6 \, x - 2\right )} e^{\left (3 \, x\right )} - \frac {8}{3} \, e^{\left (3 \, x\right )} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.46 \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=\frac {1}{243} \, {\left (81 \, x^{5} - 135 \, x^{4} + 342 \, x^{3} - 342 \, x^{2} + 228 \, x - 724\right )} e^{\left (3 \, x\right )} \]
[In]
[Out]
Time = 0.23 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.46 \[ \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx=\frac {{\mathrm {e}}^{3\,x}\,\left (81\,x^5-135\,x^4+342\,x^3-342\,x^2+228\,x-724\right )}{243} \]
[In]
[Out]