Integrand size = 118, antiderivative size = 26 \[ \int \frac {e^{5 x/2} \left (66+5 x+e^2 \left (20-32 x+20 x^2\right )\right )}{5488+1176 x+84 x^2+2 x^3+e^2 \left (4704+672 x+4728 x^2+672 x^3+24 x^4\right )+e^4 \left (1344+96 x+2688 x^2+192 x^3+1344 x^4+96 x^5\right )+e^6 \left (128+384 x^2+384 x^4+128 x^6\right )} \, dx=\frac {e^{5 x/2}}{\left (-2+x+4 \left (4+e^2 \left (1+x^2\right )\right )\right )^2} \]
[Out]
Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {6820, 12, 2327} \[ \int \frac {e^{5 x/2} \left (66+5 x+e^2 \left (20-32 x+20 x^2\right )\right )}{5488+1176 x+84 x^2+2 x^3+e^2 \left (4704+672 x+4728 x^2+672 x^3+24 x^4\right )+e^4 \left (1344+96 x+2688 x^2+192 x^3+1344 x^4+96 x^5\right )+e^6 \left (128+384 x^2+384 x^4+128 x^6\right )} \, dx=\frac {e^{5 x/2}}{\left (4 e^2 x^2+x+2 \left (7+2 e^2\right )\right )^2} \]
[In]
[Out]
Rule 12
Rule 2327
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{5 x/2} \left (2 \left (33+10 e^2\right )+\left (5-32 e^2\right ) x+20 e^2 x^2\right )}{2 \left (2 \left (7+2 e^2\right )+x+4 e^2 x^2\right )^3} \, dx \\ & = \frac {1}{2} \int \frac {e^{5 x/2} \left (2 \left (33+10 e^2\right )+\left (5-32 e^2\right ) x+20 e^2 x^2\right )}{\left (2 \left (7+2 e^2\right )+x+4 e^2 x^2\right )^3} \, dx \\ & = \frac {e^{5 x/2}}{\left (2 \left (7+2 e^2\right )+x+4 e^2 x^2\right )^2} \\ \end{align*}
Time = 2.46 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {e^{5 x/2} \left (66+5 x+e^2 \left (20-32 x+20 x^2\right )\right )}{5488+1176 x+84 x^2+2 x^3+e^2 \left (4704+672 x+4728 x^2+672 x^3+24 x^4\right )+e^4 \left (1344+96 x+2688 x^2+192 x^3+1344 x^4+96 x^5\right )+e^6 \left (128+384 x^2+384 x^4+128 x^6\right )} \, dx=\frac {e^{5 x/2}}{\left (14+x+4 e^2 \left (1+x^2\right )\right )^2} \]
[In]
[Out]
Time = 0.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85
method | result | size |
risch | \(\frac {{\mathrm e}^{\frac {5 x}{2}}}{\left (4 x^{2} {\mathrm e}^{2}+4 \,{\mathrm e}^{2}+x +14\right )^{2}}\) | \(22\) |
norman | \(\frac {{\mathrm e}^{\frac {5 x}{2}}}{\left (4 x^{2} {\mathrm e}^{2}+4 \,{\mathrm e}^{2}+x +14\right )^{2}}\) | \(24\) |
gosper | \(\frac {{\mathrm e}^{\frac {5 x}{2}}}{16 x^{4} {\mathrm e}^{4}+32 x^{2} {\mathrm e}^{4}+8 x^{3} {\mathrm e}^{2}+112 x^{2} {\mathrm e}^{2}+16 \,{\mathrm e}^{4}+8 \,{\mathrm e}^{2} x +x^{2}+112 \,{\mathrm e}^{2}+28 x +196}\) | \(65\) |
parallelrisch | \(\frac {{\mathrm e}^{\frac {5 x}{2}}}{16 x^{4} {\mathrm e}^{4}+32 x^{2} {\mathrm e}^{4}+8 x^{3} {\mathrm e}^{2}+112 x^{2} {\mathrm e}^{2}+16 \,{\mathrm e}^{4}+8 \,{\mathrm e}^{2} x +x^{2}+112 \,{\mathrm e}^{2}+28 x +196}\) | \(65\) |
derivativedivides | \(\text {Expression too large to display}\) | \(6102\) |
default | \(\text {Expression too large to display}\) | \(6102\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {e^{5 x/2} \left (66+5 x+e^2 \left (20-32 x+20 x^2\right )\right )}{5488+1176 x+84 x^2+2 x^3+e^2 \left (4704+672 x+4728 x^2+672 x^3+24 x^4\right )+e^4 \left (1344+96 x+2688 x^2+192 x^3+1344 x^4+96 x^5\right )+e^6 \left (128+384 x^2+384 x^4+128 x^6\right )} \, dx=\frac {e^{\left (\frac {5}{2} \, x\right )}}{x^{2} + 16 \, {\left (x^{4} + 2 \, x^{2} + 1\right )} e^{4} + 8 \, {\left (x^{3} + 14 \, x^{2} + x + 14\right )} e^{2} + 28 \, x + 196} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (20) = 40\).
Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.50 \[ \int \frac {e^{5 x/2} \left (66+5 x+e^2 \left (20-32 x+20 x^2\right )\right )}{5488+1176 x+84 x^2+2 x^3+e^2 \left (4704+672 x+4728 x^2+672 x^3+24 x^4\right )+e^4 \left (1344+96 x+2688 x^2+192 x^3+1344 x^4+96 x^5\right )+e^6 \left (128+384 x^2+384 x^4+128 x^6\right )} \, dx=\frac {e^{\frac {5 x}{2}}}{16 x^{4} e^{4} + 8 x^{3} e^{2} + x^{2} + 112 x^{2} e^{2} + 32 x^{2} e^{4} + 28 x + 8 x e^{2} + 196 + 112 e^{2} + 16 e^{4}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (19) = 38\).
Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {e^{5 x/2} \left (66+5 x+e^2 \left (20-32 x+20 x^2\right )\right )}{5488+1176 x+84 x^2+2 x^3+e^2 \left (4704+672 x+4728 x^2+672 x^3+24 x^4\right )+e^4 \left (1344+96 x+2688 x^2+192 x^3+1344 x^4+96 x^5\right )+e^6 \left (128+384 x^2+384 x^4+128 x^6\right )} \, dx=\frac {e^{\left (\frac {5}{2} \, x\right )}}{16 \, x^{4} e^{4} + 8 \, x^{3} e^{2} + x^{2} {\left (32 \, e^{4} + 112 \, e^{2} + 1\right )} + 4 \, x {\left (2 \, e^{2} + 7\right )} + 16 \, e^{4} + 112 \, e^{2} + 196} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (19) = 38\).
Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19 \[ \int \frac {e^{5 x/2} \left (66+5 x+e^2 \left (20-32 x+20 x^2\right )\right )}{5488+1176 x+84 x^2+2 x^3+e^2 \left (4704+672 x+4728 x^2+672 x^3+24 x^4\right )+e^4 \left (1344+96 x+2688 x^2+192 x^3+1344 x^4+96 x^5\right )+e^6 \left (128+384 x^2+384 x^4+128 x^6\right )} \, dx=\frac {2 \, e^{\left (\frac {5}{2} \, x\right )}}{16 \, x^{4} e^{4} + 8 \, x^{3} e^{2} + 32 \, x^{2} e^{4} + 112 \, x^{2} e^{2} + x^{2} + 8 \, x e^{2} + 28 \, x + 16 \, e^{4} + 112 \, e^{2} + 196} \]
[In]
[Out]
Time = 15.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {e^{5 x/2} \left (66+5 x+e^2 \left (20-32 x+20 x^2\right )\right )}{5488+1176 x+84 x^2+2 x^3+e^2 \left (4704+672 x+4728 x^2+672 x^3+24 x^4\right )+e^4 \left (1344+96 x+2688 x^2+192 x^3+1344 x^4+96 x^5\right )+e^6 \left (128+384 x^2+384 x^4+128 x^6\right )} \, dx=\frac {{\mathrm {e}}^{\frac {5\,x}{2}-4}}{16\,\left (x^4+\frac {{\mathrm {e}}^{-2}\,x^3}{2}+\frac {{\mathrm {e}}^{-4}\,\left (112\,{\mathrm {e}}^2+32\,{\mathrm {e}}^4+1\right )\,x^2}{16}+\frac {{\mathrm {e}}^{-4}\,\left (8\,{\mathrm {e}}^2+28\right )\,x}{16}+\frac {{\mathrm {e}}^{-4}\,{\left (2\,{\mathrm {e}}^2+7\right )}^2}{4}\right )} \]
[In]
[Out]