Optimal. Leaf size=32 \[ \frac{1}{3} e^{3 x} x^2-\frac{5}{9} e^{3 x} x+\frac{5 e^{3 x}}{27} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0506393, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {1593, 2196, 2176, 2194} \[ \frac{1}{3} e^{3 x} x^2-\frac{5}{9} e^{3 x} x+\frac{5 e^{3 x}}{27} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1593
Rule 2196
Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{3 x} \left (-x+x^2\right ) \, dx &=\int e^{3 x} (-1+x) x \, dx\\ &=\int \left (-e^{3 x} x+e^{3 x} x^2\right ) \, dx\\ &=-\int e^{3 x} x \, dx+\int e^{3 x} x^2 \, dx\\ &=-\frac{1}{3} e^{3 x} x+\frac{1}{3} e^{3 x} x^2+\frac{1}{3} \int e^{3 x} \, dx-\frac{2}{3} \int e^{3 x} x \, dx\\ &=\frac{e^{3 x}}{9}-\frac{5}{9} e^{3 x} x+\frac{1}{3} e^{3 x} x^2+\frac{2}{9} \int e^{3 x} \, dx\\ &=\frac{5 e^{3 x}}{27}-\frac{5}{9} e^{3 x} x+\frac{1}{3} e^{3 x} x^2\\ \end{align*}
Mathematica [A] time = 0.0284713, size = 19, normalized size = 0.59 \[ \frac{1}{27} e^{3 x} \left (9 x^2-15 x+5\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.022, size = 17, normalized size = 0.5 \begin{align*}{\frac{{{\rm e}^{3\,x}} \left ( 9\,{x}^{2}-15\,x+5 \right ) }{27}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.975857, size = 38, normalized size = 1.19 \begin{align*} \frac{1}{27} \,{\left (9 \, x^{2} - 6 \, x + 2\right )} e^{\left (3 \, x\right )} - \frac{1}{9} \,{\left (3 \, x - 1\right )} e^{\left (3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.808133, size = 45, normalized size = 1.41 \begin{align*} \frac{1}{27} \,{\left (9 \, x^{2} - 15 \, x + 5\right )} e^{\left (3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.096386, size = 15, normalized size = 0.47 \begin{align*} \frac{\left (9 x^{2} - 15 x + 5\right ) e^{3 x}}{27} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.2162, size = 22, normalized size = 0.69 \begin{align*} \frac{1}{27} \,{\left (9 \, x^{2} - 15 \, x + 5\right )} e^{\left (3 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]