Optimal. Leaf size=68 \[ \frac {1}{3} e^{3 x} x^5-\frac {5}{9} e^{3 x} x^4+\frac {38}{27} e^{3 x} x^3-\frac {38}{27} e^{3 x} x^2+\frac {76}{81} e^{3 x} x-\frac {724 e^{3 x}}{243} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2196, 2194, 2176} \[ \frac {1}{3} e^{3 x} x^5-\frac {5}{9} e^{3 x} x^4+\frac {38}{27} e^{3 x} x^3-\frac {38}{27} e^{3 x} x^2+\frac {76}{81} e^{3 x} x-\frac {724 e^{3 x}}{243} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2176
Rule 2194
Rule 2196
Rubi steps
\begin {align*} \int e^{3 x} \left (-8+2 x^3+x^5\right ) \, dx &=\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*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 34, normalized size = 0.50 \[ \frac {1}{243} e^{3 x} \left (81 x^5-135 x^4+342 x^3-342 x^2+228 x-724\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.38, size = 31, normalized size = 0.46 \[ \frac {1}{243} \, {\left (81 \, x^{5} - 135 \, x^{4} + 342 \, x^{3} - 342 \, x^{2} + 228 \, x - 724\right )} e^{\left (3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 31, normalized size = 0.46 \[ \frac {1}{243} \, {\left (81 \, x^{5} - 135 \, x^{4} + 342 \, x^{3} - 342 \, x^{2} + 228 \, x - 724\right )} e^{\left (3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 32, normalized size = 0.47 \[ \frac {\left (81 x^{5}-135 x^{4}+342 x^{3}-342 x^{2}+228 x -724\right ) {\mathrm e}^{3 x}}{243} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.86, size = 59, normalized size = 0.87 \[ \frac {1}{243} \, {\left (81 \, x^{5} - 135 \, x^{4} + 180 \, x^{3} - 180 \, x^{2} + 120 \, x - 40\right )} e^{\left (3 \, x\right )} + \frac {2}{27} \, {\left (9 \, x^{3} - 9 \, x^{2} + 6 \, x - 2\right )} e^{\left (3 \, x\right )} - \frac {8}{3} \, e^{\left (3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.53, size = 31, normalized size = 0.46 \[ \frac {{\mathrm {e}}^{3\,x}\,\left (81\,x^5-135\,x^4+342\,x^3-342\,x^2+228\,x-724\right )}{243} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.10, size = 31, normalized size = 0.46 \[ \frac {\left (81 x^{5} - 135 x^{4} + 342 x^{3} - 342 x^{2} + 228 x - 724\right ) e^{3 x}}{243} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________