Integrand size = 100, antiderivative size = 26 \[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx=4 x^3 \left (x-\frac {4+x}{-e \left (6+\frac {3}{x^2}\right )+x}\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(26)=52\).
Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 3.42 \[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx=\frac {4 \left (x^5 \left (4+x-x^2\right )+648 e^4 \left (1+2 x^2\right )+e^2 \left (9+18 x^2-144 x^3\right )-216 e^3 \left (-2-4 x^2+x^3\right )+3 e x^3 \left (-1+x+2 x^3\right )\right )}{-x^3+e \left (3+6 x^2\right )} \]
Integrate[(-32*x^7 - 12*x^8 + 16*x^9 + E^2*(144*x^3 + 576*x^5 + 576*x^7) + E*(240*x^4 + 72*x^5 + 192*x^6 + 96*x^7 - 192*x^8))/(x^6 + E^2*(9 + 36*x^2 + 36*x^4) + E*(-6*x^3 - 12*x^5)),x]
(4*(x^5*(4 + x - x^2) + 648*E^4*(1 + 2*x^2) + E^2*(9 + 18*x^2 - 144*x^3) - 216*E^3*(-2 - 4*x^2 + x^3) + 3*E*x^3*(-1 + x + 2*x^3)))/(-x^3 + E*(3 + 6* x^2))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {16 x^9-12 x^8-32 x^7+e^2 \left (576 x^7+576 x^5+144 x^3\right )+e \left (-192 x^8+96 x^7+192 x^6+72 x^5+240 x^4\right )}{x^6+e \left (-12 x^5-6 x^3\right )+e^2 \left (36 x^4+36 x^2+9\right )} \, dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (16 x^3-12 x^2+\frac {48 e \left (-\left (\left (1+3 e+72 e^2+108 e^3\right ) x\right )-18 e \left (1+2 e+24 e^2+36 e^3\right )\right )}{-x^3+6 e x^2+3 e}+\frac {108 e^2 \left (\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2+4 \left (1+2 e+24 e^2+36 e^3\right ) x+4 e \left (8+15 e+144 e^2+216 e^3\right )\right )}{\left (-x^3+6 e x^2+3 e\right )^2}-16 (2+3 e) x-48 e (2+3 e)\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {4 x^3 \left (36 \left (2 e x^2+e\right )^2+\left (4 x^2-3 x-8\right ) x^4+6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 4 \int -\frac {x^3 \left (\left (-4 x^2+3 x+8\right ) x^4-6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x-36 e^2 \left (2 x^2+1\right )^2\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -4 \int \frac {x^3 \left (\left (-4 x^2+3 x+8\right ) x^4-6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x-36 e^2 \left (2 x^2+1\right )^2\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-4 x^3+3 x^2+4 (2+3 e) x+\frac {12 e \left (\left (1+3 e+72 e^2+108 e^3\right ) x+18 e \left (1+2 e+24 e^2+36 e^3\right )\right )}{-x^3+6 e x^2+3 e}+\frac {27 e^2 \left (-\left (\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2\right )-4 \left (1+2 e+24 e^2+36 e^3\right ) x-4 e \left (8+15 e+144 e^2+216 e^3\right )\right )}{\left (-x^3+6 e x^2+3 e\right )^2}+12 e (2+3 e)\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {x^3 \left (-\left (\left (4 x^2-3 x-8\right ) x^4\right )-6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x-36 \left (2 e x^2+e\right )^2\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-4 x^3+3 x^2+4 (2+3 e) x+\frac {12 e \left (\left (1+3 e+72 e^2+108 e^3\right ) x+18 e \left (1+2 e+24 e^2+36 e^3\right )\right )}{-x^3+6 e x^2+3 e}+\frac {27 e^2 \left (-\left (\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2\right )-4 \left (1+2 e+24 e^2+36 e^3\right ) x-4 e \left (8+15 e+144 e^2+216 e^3\right )\right )}{\left (-x^3+6 e x^2+3 e\right )^2}+12 e (2+3 e)\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {x^3 \left (-\left (\left (4 x^2-3 x-8\right ) x^4\right )-6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x-36 \left (2 e x^2+e\right )^2\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-4 x^3+3 x^2+4 (2+3 e) x+\frac {12 e \left (\left (1+3 e+72 e^2+108 e^3\right ) x+18 e \left (1+2 e+24 e^2+36 e^3\right )\right )}{-x^3+6 e x^2+3 e}+\frac {27 e^2 \left (-\left (\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2\right )-4 \left (1+2 e+24 e^2+36 e^3\right ) x-4 e \left (8+15 e+144 e^2+216 e^3\right )\right )}{\left (-x^3+6 e x^2+3 e\right )^2}+12 e (2+3 e)\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {x^3 \left (-\left (\left (4 x^2-3 x-8\right ) x^4\right )-6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x-36 \left (2 e x^2+e\right )^2\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-4 x^3+3 x^2+4 (2+3 e) x+\frac {12 e \left (\left (1+3 e+72 e^2+108 e^3\right ) x+18 e \left (1+2 e+24 e^2+36 e^3\right )\right )}{-x^3+6 e x^2+3 e}+\frac {27 e^2 \left (-\left (\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2\right )-4 \left (1+2 e+24 e^2+36 e^3\right ) x-4 e \left (8+15 e+144 e^2+216 e^3\right )\right )}{\left (-x^3+6 e x^2+3 e\right )^2}+12 e (2+3 e)\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {x^3 \left (-\left (\left (4 x^2-3 x-8\right ) x^4\right )-6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x-36 \left (2 e x^2+e\right )^2\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-4 x^3+3 x^2+4 (2+3 e) x+\frac {12 e \left (\left (1+3 e+72 e^2+108 e^3\right ) x+18 e \left (1+2 e+24 e^2+36 e^3\right )\right )}{-x^3+6 e x^2+3 e}+\frac {27 e^2 \left (-\left (\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2\right )-4 \left (1+2 e+24 e^2+36 e^3\right ) x-4 e \left (8+15 e+144 e^2+216 e^3\right )\right )}{\left (-x^3+6 e x^2+3 e\right )^2}+12 e (2+3 e)\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {x^3 \left (-\left (\left (4 x^2-3 x-8\right ) x^4\right )-6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x-36 \left (2 e x^2+e\right )^2\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-4 x^3+3 x^2+4 (2+3 e) x+\frac {12 e \left (\left (1+3 e+72 e^2+108 e^3\right ) x+18 e \left (1+2 e+24 e^2+36 e^3\right )\right )}{-x^3+6 e x^2+3 e}+\frac {27 e^2 \left (-\left (\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2\right )-4 \left (1+2 e+24 e^2+36 e^3\right ) x-4 e \left (8+15 e+144 e^2+216 e^3\right )\right )}{\left (-x^3+6 e x^2+3 e\right )^2}+12 e (2+3 e)\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {x^3 \left (-\left (\left (4 x^2-3 x-8\right ) x^4\right )-6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x-36 \left (2 e x^2+e\right )^2\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-4 x^3+3 x^2+4 (2+3 e) x+\frac {12 e \left (\left (1+3 e+72 e^2+108 e^3\right ) x+18 e \left (1+2 e+24 e^2+36 e^3\right )\right )}{-x^3+6 e x^2+3 e}+\frac {27 e^2 \left (-\left (\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2\right )-4 \left (1+2 e+24 e^2+36 e^3\right ) x-4 e \left (8+15 e+144 e^2+216 e^3\right )\right )}{\left (-x^3+6 e x^2+3 e\right )^2}+12 e (2+3 e)\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {x^3 \left (-\left (\left (4 x^2-3 x-8\right ) x^4\right )-6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x-36 \left (2 e x^2+e\right )^2\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-4 x^3+3 x^2+4 (2+3 e) x+\frac {12 e \left (\left (1+3 e+72 e^2+108 e^3\right ) x+18 e \left (1+2 e+24 e^2+36 e^3\right )\right )}{-x^3+6 e x^2+3 e}+\frac {27 e^2 \left (-\left (\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2\right )-4 \left (1+2 e+24 e^2+36 e^3\right ) x-4 e \left (8+15 e+144 e^2+216 e^3\right )\right )}{\left (-x^3+6 e x^2+3 e\right )^2}+12 e (2+3 e)\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {x^3 \left (-\left (\left (4 x^2-3 x-8\right ) x^4\right )-6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x-36 \left (2 e x^2+e\right )^2\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-4 x^3+3 x^2+4 (2+3 e) x+\frac {12 e \left (\left (1+3 e+72 e^2+108 e^3\right ) x+18 e \left (1+2 e+24 e^2+36 e^3\right )\right )}{-x^3+6 e x^2+3 e}+\frac {27 e^2 \left (-\left (\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2\right )-4 \left (1+2 e+24 e^2+36 e^3\right ) x-4 e \left (8+15 e+144 e^2+216 e^3\right )\right )}{\left (-x^3+6 e x^2+3 e\right )^2}+12 e (2+3 e)\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {x^3 \left (-\left (\left (4 x^2-3 x-8\right ) x^4\right )-6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x-36 \left (2 e x^2+e\right )^2\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-4 x^3+3 x^2+4 (2+3 e) x+\frac {12 e \left (\left (1+3 e+72 e^2+108 e^3\right ) x+18 e \left (1+2 e+24 e^2+36 e^3\right )\right )}{-x^3+6 e x^2+3 e}+\frac {27 e^2 \left (-\left (\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2\right )-4 \left (1+2 e+24 e^2+36 e^3\right ) x-4 e \left (8+15 e+144 e^2+216 e^3\right )\right )}{\left (-x^3+6 e x^2+3 e\right )^2}+12 e (2+3 e)\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {x^3 \left (-\left (\left (4 x^2-3 x-8\right ) x^4\right )-6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x-36 \left (2 e x^2+e\right )^2\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-4 x^3+3 x^2+4 (2+3 e) x+\frac {12 e \left (\left (1+3 e+72 e^2+108 e^3\right ) x+18 e \left (1+2 e+24 e^2+36 e^3\right )\right )}{-x^3+6 e x^2+3 e}+\frac {27 e^2 \left (-\left (\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2\right )-4 \left (1+2 e+24 e^2+36 e^3\right ) x-4 e \left (8+15 e+144 e^2+216 e^3\right )\right )}{\left (-x^3+6 e x^2+3 e\right )^2}+12 e (2+3 e)\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {x^3 \left (-\left (\left (4 x^2-3 x-8\right ) x^4\right )-6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x-36 \left (2 e x^2+e\right )^2\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-4 x^3+3 x^2+4 (2+3 e) x+\frac {12 e \left (\left (1+3 e+72 e^2+108 e^3\right ) x+18 e \left (1+2 e+24 e^2+36 e^3\right )\right )}{-x^3+6 e x^2+3 e}+\frac {27 e^2 \left (-\left (\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2\right )-4 \left (1+2 e+24 e^2+36 e^3\right ) x-4 e \left (8+15 e+144 e^2+216 e^3\right )\right )}{\left (-x^3+6 e x^2+3 e\right )^2}+12 e (2+3 e)\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {x^3 \left (-\left (\left (4 x^2-3 x-8\right ) x^4\right )-6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x-36 \left (2 e x^2+e\right )^2\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -4 \int \left (-4 x^3+3 x^2+4 (2+3 e) x+\frac {12 e \left (\left (1+3 e+72 e^2+108 e^3\right ) x+18 e \left (1+2 e+24 e^2+36 e^3\right )\right )}{-x^3+6 e x^2+3 e}+\frac {27 e^2 \left (-\left (\left (1+80 e+144 e^2+1152 e^3+1728 e^4\right ) x^2\right )-4 \left (1+2 e+24 e^2+36 e^3\right ) x-4 e \left (8+15 e+144 e^2+216 e^3\right )\right )}{\left (-x^3+6 e x^2+3 e\right )^2}+12 e (2+3 e)\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -4 \int \frac {x^3 \left (-\left (\left (4 x^2-3 x-8\right ) x^4\right )-6 e \left (-8 x^4+4 x^3+8 x^2+3 x+10\right ) x-36 \left (2 e x^2+e\right )^2\right )}{\left (-x^3+6 e x^2+3 e\right )^2}dx\) |
Int[(-32*x^7 - 12*x^8 + 16*x^9 + E^2*(144*x^3 + 576*x^5 + 576*x^7) + E*(24 0*x^4 + 72*x^5 + 192*x^6 + 96*x^7 - 192*x^8))/(x^6 + E^2*(9 + 36*x^2 + 36* x^4) + E*(-6*x^3 - 12*x^5)),x]
3.2.82.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85
| method | result | size |
| gosper | \(\frac {4 x^{4} \left (6 x^{2} {\mathrm e}-x^{3}+x^{2}+3 \,{\mathrm e}+4 x \right )}{6 x^{2} {\mathrm e}-x^{3}+3 \,{\mathrm e}}\) | \(48\) |
| norman | \(\frac {\left (4+24 \,{\mathrm e}\right ) x^{6}+16 x^{5}-4 x^{7}+12 x^{4} {\mathrm e}}{6 x^{2} {\mathrm e}-x^{3}+3 \,{\mathrm e}}\) | \(49\) |
| parallelrisch | \(\frac {24 x^{6} {\mathrm e}-4 x^{7}+4 x^{6}+12 x^{4} {\mathrm e}+16 x^{5}}{6 x^{2} {\mathrm e}-x^{3}+3 \,{\mathrm e}}\) | \(51\) |
| risch | \(-\frac {4 \left (-6 x^{6} {\mathrm e}+x^{7}-x^{6}+216 x^{3} {\mathrm e}^{3}+144 x^{3} {\mathrm e}^{2}-3 x^{4} {\mathrm e}-4 x^{5}-18 x^{2} {\mathrm e}^{2}+3 x^{3} {\mathrm e}-1296 x^{2} {\mathrm e}^{4}-864 x^{2} {\mathrm e}^{3}-9 \,{\mathrm e}^{2}-648 \,{\mathrm e}^{4}-432 \,{\mathrm e}^{3}\right )}{6 x^{2} {\mathrm e}-x^{3}+3 \,{\mathrm e}}\) | \(104\) |
| default | \(4 x^{4}-144 \,{\mathrm e}^{2} x -24 x^{2} {\mathrm e}-4 x^{3}-96 x \,{\mathrm e}-16 x^{2}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (36 \textit {\_Z}^{4} {\mathrm e}^{2}-12 \textit {\_Z}^{5} {\mathrm e}+\textit {\_Z}^{6}+36 \textit {\_Z}^{2} {\mathrm e}^{2}-6 \textit {\_Z}^{3} {\mathrm e}+9 \,{\mathrm e}^{2}\right )}{\sum }\frac {\left (4 \left (108 \,{\mathrm e}^{4}+72 \,{\mathrm e}^{3}+3 \,{\mathrm e}^{2}+{\mathrm e}\right ) \textit {\_R}^{4}+24 \left (3 \,{\mathrm e}^{3}+2 \,{\mathrm e}^{2}\right ) \textit {\_R}^{3}+9 \left (48 \,{\mathrm e}^{4}+32 \,{\mathrm e}^{3}+{\mathrm e}^{2}\right ) \textit {\_R}^{2}+12 \,{\mathrm e}^{2} \left (3 \,{\mathrm e}+2\right ) \textit {\_R} +108 \,{\mathrm e}^{4}+72 \,{\mathrm e} \,{\mathrm e}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{24 \textit {\_R}^{3} {\mathrm e}^{2}-10 \textit {\_R}^{4} {\mathrm e}+\textit {\_R}^{5}+12 \,{\mathrm e}^{2} \textit {\_R} -3 \textit {\_R}^{2} {\mathrm e}}\right )\) | \(196\) |
int(((576*x^7+576*x^5+144*x^3)*exp(1)^2+(-192*x^8+96*x^7+192*x^6+72*x^5+24 0*x^4)*exp(1)+16*x^9-12*x^8-32*x^7)/((36*x^4+36*x^2+9)*exp(1)^2+(-12*x^5-6 *x^3)*exp(1)+x^6),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.54 \[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx=\frac {4 \, {\left (x^{7} - x^{6} - 4 \, x^{5} - 648 \, {\left (2 \, x^{2} + 1\right )} e^{4} + 216 \, {\left (x^{3} - 4 \, x^{2} - 2\right )} e^{3} + 9 \, {\left (16 \, x^{3} - 2 \, x^{2} - 1\right )} e^{2} - 3 \, {\left (2 \, x^{6} + x^{4} - x^{3}\right )} e\right )}}{x^{3} - 3 \, {\left (2 \, x^{2} + 1\right )} e} \]
integrate(((576*x^7+576*x^5+144*x^3)*exp(1)^2+(-192*x^8+96*x^7+192*x^6+72* x^5+240*x^4)*exp(1)+16*x^9-12*x^8-32*x^7)/((36*x^4+36*x^2+9)*exp(1)^2+(-12 *x^5-6*x^3)*exp(1)+x^6),x, algorithm=\
4*(x^7 - x^6 - 4*x^5 - 648*(2*x^2 + 1)*e^4 + 216*(x^3 - 4*x^2 - 2)*e^3 + 9 *(16*x^3 - 2*x^2 - 1)*e^2 - 3*(2*x^6 + x^4 - x^3)*e)/(x^3 - 3*(2*x^2 + 1)* e)
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (22) = 44\).
Time = 1.31 (sec) , antiderivative size = 105, normalized size of antiderivative = 4.04 \[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx=4 x^{4} - 4 x^{3} + x^{2} \left (- 24 e - 16\right ) + x \left (- 144 e^{2} - 96 e\right ) + \frac {x^{2} \left (- 5184 e^{4} - 3456 e^{3} - 144 e^{2} - 48 e\right ) + x \left (- 432 e^{3} - 288 e^{2}\right ) - 2592 e^{4} - 1728 e^{3} - 36 e^{2}}{x^{3} - 6 e x^{2} - 3 e} \]
integrate(((576*x**7+576*x**5+144*x**3)*exp(1)**2+(-192*x**8+96*x**7+192*x **6+72*x**5+240*x**4)*exp(1)+16*x**9-12*x**8-32*x**7)/((36*x**4+36*x**2+9) *exp(1)**2+(-12*x**5-6*x**3)*exp(1)+x**6),x)
4*x**4 - 4*x**3 + x**2*(-24*E - 16) + x*(-144*exp(2) - 96*E) + (x**2*(-518 4*exp(4) - 3456*exp(3) - 144*exp(2) - 48*E) + x*(-432*exp(3) - 288*exp(2)) - 2592*exp(4) - 1728*exp(3) - 36*exp(2))/(x**3 - 6*E*x**2 - 3*E)
Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (26) = 52\).
Time = 0.18 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.77 \[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx=4 \, x^{4} - 4 \, x^{3} - 8 \, x^{2} {\left (3 \, e + 2\right )} - 48 \, x {\left (3 \, e^{2} + 2 \, e\right )} - \frac {12 \, {\left (4 \, x^{2} {\left (108 \, e^{4} + 72 \, e^{3} + 3 \, e^{2} + e\right )} + 12 \, x {\left (3 \, e^{3} + 2 \, e^{2}\right )} + 216 \, e^{4} + 144 \, e^{3} + 3 \, e^{2}\right )}}{x^{3} - 6 \, x^{2} e - 3 \, e} \]
integrate(((576*x^7+576*x^5+144*x^3)*exp(1)^2+(-192*x^8+96*x^7+192*x^6+72* x^5+240*x^4)*exp(1)+16*x^9-12*x^8-32*x^7)/((36*x^4+36*x^2+9)*exp(1)^2+(-12 *x^5-6*x^3)*exp(1)+x^6),x, algorithm=\
4*x^4 - 4*x^3 - 8*x^2*(3*e + 2) - 48*x*(3*e^2 + 2*e) - 12*(4*x^2*(108*e^4 + 72*e^3 + 3*e^2 + e) + 12*x*(3*e^3 + 2*e^2) + 216*e^4 + 144*e^3 + 3*e^2)/ (x^3 - 6*x^2*e - 3*e)
Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.96 \[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx=4 \, x^{4} - 4 \, x^{3} - 24 \, x^{2} e - 16 \, x^{2} - 144 \, x e^{2} - 96 \, x e - \frac {12 \, {\left (432 \, x^{2} e^{4} + 288 \, x^{2} e^{3} + 12 \, x^{2} e^{2} + 4 \, x^{2} e + 36 \, x e^{3} + 24 \, x e^{2} + 216 \, e^{4} + 144 \, e^{3} + 3 \, e^{2}\right )}}{x^{3} - 6 \, x^{2} e - 3 \, e} \]
integrate(((576*x^7+576*x^5+144*x^3)*exp(1)^2+(-192*x^8+96*x^7+192*x^6+72* x^5+240*x^4)*exp(1)+16*x^9-12*x^8-32*x^7)/((36*x^4+36*x^2+9)*exp(1)^2+(-12 *x^5-6*x^3)*exp(1)+x^6),x, algorithm=\
4*x^4 - 4*x^3 - 24*x^2*e - 16*x^2 - 144*x*e^2 - 96*x*e - 12*(432*x^2*e^4 + 288*x^2*e^3 + 12*x^2*e^2 + 4*x^2*e + 36*x*e^3 + 24*x*e^2 + 216*e^4 + 144* e^3 + 3*e^2)/(x^3 - 6*x^2*e - 3*e)
Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.15 \[ \int \frac {-32 x^7-12 x^8+16 x^9+e^2 \left (144 x^3+576 x^5+576 x^7\right )+e \left (240 x^4+72 x^5+192 x^6+96 x^7-192 x^8\right )}{x^6+e^2 \left (9+36 x^2+36 x^4\right )+e \left (-6 x^3-12 x^5\right )} \, dx=x\,\left (288\,\mathrm {e}+432\,{\mathrm {e}}^2-12\,\mathrm {e}\,\left (48\,\mathrm {e}+32\right )\right )-x^2\,\left (24\,\mathrm {e}+16\right )+\frac {\left (48\,\mathrm {e}+144\,{\mathrm {e}}^2+3456\,{\mathrm {e}}^3+5184\,{\mathrm {e}}^4\right )\,x^2+\left (288\,{\mathrm {e}}^2+432\,{\mathrm {e}}^3\right )\,x+36\,{\mathrm {e}}^2+1728\,{\mathrm {e}}^3+2592\,{\mathrm {e}}^4}{-x^3+6\,\mathrm {e}\,x^2+3\,\mathrm {e}}-4\,x^3+4\,x^4 \]
int((exp(1)*(240*x^4 + 72*x^5 + 192*x^6 + 96*x^7 - 192*x^8) + exp(2)*(144* x^3 + 576*x^5 + 576*x^7) - 32*x^7 - 12*x^8 + 16*x^9)/(exp(2)*(36*x^2 + 36* x^4 + 9) - exp(1)*(6*x^3 + 12*x^5) + x^6),x)