Integrand size = 65, antiderivative size = 22 \[ \int \frac {-2592 x^2+144 x^3+576 x^5+e^4 \left (-18+4 x^3\right )+e^2 \left (432 x-96 x^4\right )}{3 e^4 x^3-72 e^2 x^4+432 x^5} \, dx=\frac {4}{e^2-12 x}+\frac {3}{x^2}+\frac {4 x}{3} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1608, 27, 12, 1634} \[ \int \frac {-2592 x^2+144 x^3+576 x^5+e^4 \left (-18+4 x^3\right )+e^2 \left (432 x-96 x^4\right )}{3 e^4 x^3-72 e^2 x^4+432 x^5} \, dx=\frac {3}{x^2}+\frac {4 x}{3}+\frac {4}{e^2-12 x} \]
[In]
[Out]
Rule 12
Rule 27
Rule 1608
Rule 1634
Rubi steps \begin{align*} \text {integral}& = \int \frac {-2592 x^2+144 x^3+576 x^5+e^4 \left (-18+4 x^3\right )+e^2 \left (432 x-96 x^4\right )}{x^3 \left (3 e^4-72 e^2 x+432 x^2\right )} \, dx \\ & = \int \frac {-2592 x^2+144 x^3+576 x^5+e^4 \left (-18+4 x^3\right )+e^2 \left (432 x-96 x^4\right )}{3 \left (e^2-12 x\right )^2 x^3} \, dx \\ & = \frac {1}{3} \int \frac {-2592 x^2+144 x^3+576 x^5+e^4 \left (-18+4 x^3\right )+e^2 \left (432 x-96 x^4\right )}{\left (e^2-12 x\right )^2 x^3} \, dx \\ & = \frac {1}{3} \int \left (4+\frac {144}{\left (e^2-12 x\right )^2}-\frac {18}{x^3}\right ) \, dx \\ & = \frac {4}{e^2-12 x}+\frac {3}{x^2}+\frac {4 x}{3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {-2592 x^2+144 x^3+576 x^5+e^4 \left (-18+4 x^3\right )+e^2 \left (432 x-96 x^4\right )}{3 e^4 x^3-72 e^2 x^4+432 x^5} \, dx=\frac {2}{3} \left (\frac {9}{2 x^2}+2 x-\frac {6}{-e^2+12 x}\right ) \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36
method | result | size |
risch | \(\frac {4 x}{3}+\frac {4 x^{2}+3 \,{\mathrm e}^{2}-36 x}{x^{2} \left (-12 x +{\mathrm e}^{2}\right )}\) | \(30\) |
norman | \(\frac {\left (\frac {{\mathrm e}^{4}}{9}+4\right ) x^{2}-36 x -16 x^{4}+3 \,{\mathrm e}^{2}}{x^{2} \left (-12 x +{\mathrm e}^{2}\right )}\) | \(38\) |
gosper | \(\frac {x^{2} {\mathrm e}^{4}-144 x^{4}+36 x^{2}+27 \,{\mathrm e}^{2}-324 x}{9 x^{2} \left (-12 x +{\mathrm e}^{2}\right )}\) | \(40\) |
parallelrisch | \(\frac {4 x^{2} {\mathrm e}^{4}-576 x^{4}+144 x^{2}+108 \,{\mathrm e}^{2}-1296 x}{36 x^{2} \left (-12 x +{\mathrm e}^{2}\right )}\) | \(41\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {-2592 x^2+144 x^3+576 x^5+e^4 \left (-18+4 x^3\right )+e^2 \left (432 x-96 x^4\right )}{3 e^4 x^3-72 e^2 x^4+432 x^5} \, dx=\frac {48 \, x^{4} - 12 \, x^{2} - {\left (4 \, x^{3} + 9\right )} e^{2} + 108 \, x}{3 \, {\left (12 \, x^{3} - x^{2} e^{2}\right )}} \]
[In]
[Out]
Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {-2592 x^2+144 x^3+576 x^5+e^4 \left (-18+4 x^3\right )+e^2 \left (432 x-96 x^4\right )}{3 e^4 x^3-72 e^2 x^4+432 x^5} \, dx=\frac {4 x}{3} + \frac {- 4 x^{2} + 36 x - 3 e^{2}}{12 x^{3} - x^{2} e^{2}} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {-2592 x^2+144 x^3+576 x^5+e^4 \left (-18+4 x^3\right )+e^2 \left (432 x-96 x^4\right )}{3 e^4 x^3-72 e^2 x^4+432 x^5} \, dx=\frac {4}{3} \, x - \frac {4 \, x^{2} - 36 \, x + 3 \, e^{2}}{12 \, x^{3} - x^{2} e^{2}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-2592 x^2+144 x^3+576 x^5+e^4 \left (-18+4 x^3\right )+e^2 \left (432 x-96 x^4\right )}{3 e^4 x^3-72 e^2 x^4+432 x^5} \, dx=\frac {4}{3} \, x - \frac {4}{12 \, x - e^{2}} + \frac {3}{x^{2}} \]
[In]
[Out]
Time = 0.14 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {-2592 x^2+144 x^3+576 x^5+e^4 \left (-18+4 x^3\right )+e^2 \left (432 x-96 x^4\right )}{3 e^4 x^3-72 e^2 x^4+432 x^5} \, dx=\frac {4\,x}{3}-\frac {4\,x^2-36\,x+3\,{\mathrm {e}}^2}{x^2\,\left (12\,x-{\mathrm {e}}^2\right )} \]
[In]
[Out]