Integrand size = 24, antiderivative size = 20 \[ \int e^{4 x} \left (10 x+17 x^2-4 x^3\right ) \log ^2(3) \, dx=1+e^{4 x} (5-x) x^2 \log ^2(3) \]
[Out]
Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45, number of steps used = 13, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {12, 1608, 2227, 2207, 2225} \[ \int e^{4 x} \left (10 x+17 x^2-4 x^3\right ) \log ^2(3) \, dx=5 e^{4 x} x^2 \log ^2(3)-e^{4 x} x^3 \log ^2(3) \]
[In]
[Out]
Rule 12
Rule 1608
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \log ^2(3) \int e^{4 x} \left (10 x+17 x^2-4 x^3\right ) \, dx \\ & = \log ^2(3) \int e^{4 x} x \left (10+17 x-4 x^2\right ) \, dx \\ & = \log ^2(3) \int \left (10 e^{4 x} x+17 e^{4 x} x^2-4 e^{4 x} x^3\right ) \, dx \\ & = -\left (\left (4 \log ^2(3)\right ) \int e^{4 x} x^3 \, dx\right )+\left (10 \log ^2(3)\right ) \int e^{4 x} x \, dx+\left (17 \log ^2(3)\right ) \int e^{4 x} x^2 \, dx \\ & = \frac {5}{2} e^{4 x} x \log ^2(3)+\frac {17}{4} e^{4 x} x^2 \log ^2(3)-e^{4 x} x^3 \log ^2(3)-\frac {1}{2} \left (5 \log ^2(3)\right ) \int e^{4 x} \, dx+\left (3 \log ^2(3)\right ) \int e^{4 x} x^2 \, dx-\frac {1}{2} \left (17 \log ^2(3)\right ) \int e^{4 x} x \, dx \\ & = -\frac {5}{8} e^{4 x} \log ^2(3)+\frac {3}{8} e^{4 x} x \log ^2(3)+5 e^{4 x} x^2 \log ^2(3)-e^{4 x} x^3 \log ^2(3)-\frac {1}{2} \left (3 \log ^2(3)\right ) \int e^{4 x} x \, dx+\frac {1}{8} \left (17 \log ^2(3)\right ) \int e^{4 x} \, dx \\ & = -\frac {3}{32} e^{4 x} \log ^2(3)+5 e^{4 x} x^2 \log ^2(3)-e^{4 x} x^3 \log ^2(3)+\frac {1}{8} \left (3 \log ^2(3)\right ) \int e^{4 x} \, dx \\ & = 5 e^{4 x} x^2 \log ^2(3)-e^{4 x} x^3 \log ^2(3) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int e^{4 x} \left (10 x+17 x^2-4 x^3\right ) \log ^2(3) \, dx=-e^{4 x} (-5+x) x^2 \log ^2(3) \]
[In]
[Out]
Time = 0.37 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85
method | result | size |
gosper | \(-{\mathrm e}^{4 x} \ln \left (3\right )^{2} x^{2} \left (-5+x \right )\) | \(17\) |
risch | \(\ln \left (3\right )^{2} \left (-x^{3}+5 x^{2}\right ) {\mathrm e}^{4 x}\) | \(21\) |
default | \(\ln \left (3\right )^{2} \left (5 x^{2} {\mathrm e}^{4 x}-x^{3} {\mathrm e}^{4 x}\right )\) | \(25\) |
parallelrisch | \(\ln \left (3\right )^{2} \left (5 x^{2} {\mathrm e}^{4 x}-x^{3} {\mathrm e}^{4 x}\right )\) | \(25\) |
norman | \(5 x^{2} \ln \left (3\right )^{2} {\mathrm e}^{4 x}-x^{3} \ln \left (3\right )^{2} {\mathrm e}^{4 x}\) | \(28\) |
meijerg | \(-\frac {\ln \left (3\right )^{2} \left (6-\frac {\left (-256 x^{3}+192 x^{2}-96 x +24\right ) {\mathrm e}^{4 x}}{4}\right )}{64}-\frac {17 \ln \left (3\right )^{2} \left (2-\frac {\left (48 x^{2}-24 x +6\right ) {\mathrm e}^{4 x}}{3}\right )}{64}+\frac {5 \ln \left (3\right )^{2} \left (1-\frac {\left (-8 x +2\right ) {\mathrm e}^{4 x}}{2}\right )}{8}\) | \(74\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int e^{4 x} \left (10 x+17 x^2-4 x^3\right ) \log ^2(3) \, dx=-{\left (x^{3} - 5 \, x^{2}\right )} e^{\left (4 \, x\right )} \log \left (3\right )^{2} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int e^{4 x} \left (10 x+17 x^2-4 x^3\right ) \log ^2(3) \, dx=\left (- x^{3} \log {\left (3 \right )}^{2} + 5 x^{2} \log {\left (3 \right )}^{2}\right ) e^{4 x} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (18) = 36\).
Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.70 \[ \int e^{4 x} \left (10 x+17 x^2-4 x^3\right ) \log ^2(3) \, dx=-\frac {1}{32} \, {\left ({\left (32 \, x^{3} - 24 \, x^{2} + 12 \, x - 3\right )} e^{\left (4 \, x\right )} - 17 \, {\left (8 \, x^{2} - 4 \, x + 1\right )} e^{\left (4 \, x\right )} - 20 \, {\left (4 \, x - 1\right )} e^{\left (4 \, x\right )}\right )} \log \left (3\right )^{2} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int e^{4 x} \left (10 x+17 x^2-4 x^3\right ) \log ^2(3) \, dx=-{\left (x^{3} - 5 \, x^{2}\right )} e^{\left (4 \, x\right )} \log \left (3\right )^{2} \]
[In]
[Out]
Time = 9.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int e^{4 x} \left (10 x+17 x^2-4 x^3\right ) \log ^2(3) \, dx=-x^2\,{\mathrm {e}}^{4\,x}\,{\ln \left (3\right )}^2\,\left (x-5\right ) \]
[In]
[Out]