Integrand size = 71, antiderivative size = 24 \[ \int \frac {-25 e^2 x^2+e^4 \left (-200 x+50 x^2\right )}{4 x^2-4 x^3+x^4+e^4 \left (64-64 x+16 x^2\right )+e^2 \left (32 x-32 x^2+8 x^3\right )} \, dx=\frac {25 x^2}{2 (2-x) \left (-4-\frac {x}{e^2}\right )} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1694, 12, 1828, 8} \[ \int \frac {-25 e^2 x^2+e^4 \left (-200 x+50 x^2\right )}{4 x^2-4 x^3+x^4+e^4 \left (64-64 x+16 x^2\right )+e^2 \left (32 x-32 x^2+8 x^3\right )} \, dx=-\frac {25 e^2 \left (\left (1-2 e^2\right ) x+4 e^2\right )}{-x^2+2 \left (1-2 e^2\right ) x+8 e^2} \]
[In]
[Out]
Rule 8
Rule 12
Rule 1694
Rule 1828
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {25 \left (-e^2 \left (1-2 e^2\right ) \left (1+2 e^2\right )^2-2 e^2 \left (1+4 e^4\right ) x-e^2 \left (1-2 e^2\right ) x^2\right )}{\left (1+4 e^2+4 e^4-x^2\right )^2} \, dx,x,\frac {1}{4} \left (-4+8 e^2\right )+x\right ) \\ & = 25 \text {Subst}\left (\int \frac {-e^2 \left (1-2 e^2\right ) \left (1+2 e^2\right )^2-2 e^2 \left (1+4 e^4\right ) x-e^2 \left (1-2 e^2\right ) x^2}{\left (1+4 e^2+4 e^4-x^2\right )^2} \, dx,x,\frac {1}{4} \left (-4+8 e^2\right )+x\right ) \\ & = -\frac {25 e^2 \left (4 e^2+\left (1-2 e^2\right ) x\right )}{8 e^2+2 \left (1-2 e^2\right ) x-x^2}-\frac {25 \text {Subst}\left (\int 0 \, dx,x,\frac {1}{4} \left (-4+8 e^2\right )+x\right )}{2 \left (1+2 e^2\right )^2} \\ & = -\frac {25 e^2 \left (4 e^2+\left (1-2 e^2\right ) x\right )}{8 e^2+2 \left (1-2 e^2\right ) x-x^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {-25 e^2 x^2+e^4 \left (-200 x+50 x^2\right )}{4 x^2-4 x^3+x^4+e^4 \left (64-64 x+16 x^2\right )+e^2 \left (32 x-32 x^2+8 x^3\right )} \, dx=\frac {25 e^2 \left (-2 e^2 (-2+x)+x\right )}{(-2+x) \left (4 e^2+x\right )} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46
method | result | size |
norman | \(\frac {\left (-50 \,{\mathrm e}^{4}+25 \,{\mathrm e}^{2}\right ) x +100 \,{\mathrm e}^{4}}{\left (-2+x \right ) \left (x +4 \,{\mathrm e}^{2}\right )}\) | \(35\) |
gosper | \(-\frac {25 \left (2 \,{\mathrm e}^{2} x -4 \,{\mathrm e}^{2}-x \right ) {\mathrm e}^{2}}{4 \,{\mathrm e}^{2} x +x^{2}-8 \,{\mathrm e}^{2}-2 x}\) | \(36\) |
risch | \(\frac {\left (-\frac {25 \,{\mathrm e}^{4}}{2}+\frac {25 \,{\mathrm e}^{2}}{4}\right ) x +25 \,{\mathrm e}^{4}}{{\mathrm e}^{2} x +\frac {x^{2}}{4}-2 \,{\mathrm e}^{2}-\frac {x}{2}}\) | \(37\) |
parallelrisch | \(-\frac {50 x \,{\mathrm e}^{4}-100 \,{\mathrm e}^{4}-25 \,{\mathrm e}^{2} x}{4 \,{\mathrm e}^{2} x +x^{2}-8 \,{\mathrm e}^{2}-2 x}\) | \(40\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {-25 e^2 x^2+e^4 \left (-200 x+50 x^2\right )}{4 x^2-4 x^3+x^4+e^4 \left (64-64 x+16 x^2\right )+e^2 \left (32 x-32 x^2+8 x^3\right )} \, dx=-\frac {25 \, {\left (2 \, {\left (x - 2\right )} e^{4} - x e^{2}\right )}}{x^{2} + 4 \, {\left (x - 2\right )} e^{2} - 2 \, x} \]
[In]
[Out]
Time = 0.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {-25 e^2 x^2+e^4 \left (-200 x+50 x^2\right )}{4 x^2-4 x^3+x^4+e^4 \left (64-64 x+16 x^2\right )+e^2 \left (32 x-32 x^2+8 x^3\right )} \, dx=- \frac {x \left (- 25 e^{2} + 50 e^{4}\right ) - 100 e^{4}}{x^{2} + x \left (-2 + 4 e^{2}\right ) - 8 e^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (18) = 36\).
Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {-25 e^2 x^2+e^4 \left (-200 x+50 x^2\right )}{4 x^2-4 x^3+x^4+e^4 \left (64-64 x+16 x^2\right )+e^2 \left (32 x-32 x^2+8 x^3\right )} \, dx=-\frac {25 \, {\left (x {\left (2 \, e^{4} - e^{2}\right )} - 4 \, e^{4}\right )}}{x^{2} + 2 \, x {\left (2 \, e^{2} - 1\right )} - 8 \, e^{2}} \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {-25 e^2 x^2+e^4 \left (-200 x+50 x^2\right )}{4 x^2-4 x^3+x^4+e^4 \left (64-64 x+16 x^2\right )+e^2 \left (32 x-32 x^2+8 x^3\right )} \, dx=-\frac {25 \, {\left (2 \, x e^{4} - x e^{2} - 4 \, e^{4}\right )}}{x^{2} + 4 \, x e^{2} - 2 \, x - 8 \, e^{2}} \]
[In]
[Out]
Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.17 \[ \int \frac {-25 e^2 x^2+e^4 \left (-200 x+50 x^2\right )}{4 x^2-4 x^3+x^4+e^4 \left (64-64 x+16 x^2\right )+e^2 \left (32 x-32 x^2+8 x^3\right )} \, dx=\frac {25\,{\mathrm {e}}^2\,\left (x+4\,{\mathrm {e}}^2-2\,x\,{\mathrm {e}}^2\right )}{\left (x+4\,{\mathrm {e}}^2\right )\,\left (x-2\right )} \]
[In]
[Out]