Integrand size = 124, antiderivative size = 26 \[ \int \frac {e^4 \left (320-1664 x+1536 x^2+4032 x^3-6720 x^4+1248 x^5+912 x^6-1008 x^7-516 x^8-80 x^9-4 x^{10}\right )}{-32-80 x-160 x^2-120 x^3-50 x^4+119 x^5+95 x^6+75 x^7-60 x^8-25 x^9-31 x^{10}+25 x^{11}+5 x^{13}-5 x^{14}+x^{15}} \, dx=\frac {e^4 \left (1+\frac {x (3+x)}{-2+x}\right )^4}{\left (1+x+x^2\right )^4} \] Output:
exp(1)^4/(x^2+x+1)^4*((3+x)/(-2+x)*x+1)^4
Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {e^4 \left (320-1664 x+1536 x^2+4032 x^3-6720 x^4+1248 x^5+912 x^6-1008 x^7-516 x^8-80 x^9-4 x^{10}\right )}{-32-80 x-160 x^2-120 x^3-50 x^4+119 x^5+95 x^6+75 x^7-60 x^8-25 x^9-31 x^{10}+25 x^{11}+5 x^{13}-5 x^{14}+x^{15}} \, dx=\frac {e^4 \left (-2+4 x+x^2\right )^4}{\left (2+x+x^2-x^3\right )^4} \] Input:
Integrate[(E^4*(320 - 1664*x + 1536*x^2 + 4032*x^3 - 6720*x^4 + 1248*x^5 + 912*x^6 - 1008*x^7 - 516*x^8 - 80*x^9 - 4*x^10))/(-32 - 80*x - 160*x^2 - 120*x^3 - 50*x^4 + 119*x^5 + 95*x^6 + 75*x^7 - 60*x^8 - 25*x^9 - 31*x^10 + 25*x^11 + 5*x^13 - 5*x^14 + x^15),x]
Output:
(E^4*(-2 + 4*x + x^2)^4)/(2 + x + x^2 - x^3)^4
Leaf count is larger than twice the leaf count of optimal. \(169\) vs. \(2(26)=52\).
Time = 1.00 (sec) , antiderivative size = 169, normalized size of antiderivative = 6.50, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {27, 27, 2462, 2009}
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.
\(\displaystyle \int \frac {e^4 \left (-4 x^{10}-80 x^9-516 x^8-1008 x^7+912 x^6+1248 x^5-6720 x^4+4032 x^3+1536 x^2-1664 x+320\right )}{x^{15}-5 x^{14}+5 x^{13}+25 x^{11}-31 x^{10}-25 x^9-60 x^8+75 x^7+95 x^6+119 x^5-50 x^4-120 x^3-160 x^2-80 x-32} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle e^4 \int -\frac {4 \left (-x^{10}-20 x^9-129 x^8-252 x^7+228 x^6+312 x^5-1680 x^4+1008 x^3+384 x^2-416 x+80\right )}{-x^{15}+5 x^{14}-5 x^{13}-25 x^{11}+31 x^{10}+25 x^9+60 x^8-75 x^7-95 x^6-119 x^5+50 x^4+120 x^3+160 x^2+80 x+32}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -4 e^4 \int \frac {-x^{10}-20 x^9-129 x^8-252 x^7+228 x^6+312 x^5-1680 x^4+1008 x^3+384 x^2-416 x+80}{-x^{15}+5 x^{14}-5 x^{13}-25 x^{11}+31 x^{10}+25 x^9+60 x^8-75 x^7-95 x^6-119 x^5+50 x^4+120 x^3+160 x^2+80 x+32}dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle -4 e^4 \int \left (\frac {720 (30 x+2099)}{823543 \left (x^2+x+1\right )^2}-\frac {123360}{823543 \left (x^2+x+1\right )}+\frac {123360}{823543 (x-2)^2}-\frac {91200}{117649 (x-2)^3}+\frac {243 (2843 x+2964)}{117649 \left (x^2+x+1\right )^3}+\frac {18000}{16807 (x-2)^4}-\frac {1458 (205 x+391)}{16807 \left (x^2+x+1\right )^4}+\frac {10000}{2401 (x-2)^5}+\frac {729 (71 x+94)}{2401 \left (x^2+x+1\right )^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 e^4 \left (-\frac {81 (2722-3085 x)}{235298 \left (x^2+x+1\right )^2}-\frac {62640 (2 x+1)}{117649 \left (x^2+x+1\right )}+\frac {240 (4168 x+2039)}{823543 \left (x^2+x+1\right )}-\frac {53595 (2 x+1)}{67228 \left (x^2+x+1\right )^2}+\frac {162 (19-577 x)}{16807 \left (x^2+x+1\right )^3}+\frac {3159 (2 x+1)}{1372 \left (x^2+x+1\right )^3}-\frac {729 (16-39 x)}{9604 \left (x^2+x+1\right )^4}+\frac {123360}{823543 (2-x)}+\frac {45600}{117649 (2-x)^2}+\frac {6000}{16807 (2-x)^3}-\frac {2500}{2401 (2-x)^4}\right )\) |
Input:
Int[(E^4*(320 - 1664*x + 1536*x^2 + 4032*x^3 - 6720*x^4 + 1248*x^5 + 912*x ^6 - 1008*x^7 - 516*x^8 - 80*x^9 - 4*x^10))/(-32 - 80*x - 160*x^2 - 120*x^ 3 - 50*x^4 + 119*x^5 + 95*x^6 + 75*x^7 - 60*x^8 - 25*x^9 - 31*x^10 + 25*x^ 11 + 5*x^13 - 5*x^14 + x^15),x]
Output:
-4*E^4*(-2500/(2401*(2 - x)^4) + 6000/(16807*(2 - x)^3) + 45600/(117649*(2 - x)^2) + 123360/(823543*(2 - x)) - (729*(16 - 39*x))/(9604*(1 + x + x^2) ^4) + (162*(19 - 577*x))/(16807*(1 + x + x^2)^3) + (3159*(1 + 2*x))/(1372* (1 + x + x^2)^3) - (81*(2722 - 3085*x))/(235298*(1 + x + x^2)^2) - (53595* (1 + 2*x))/(67228*(1 + x + x^2)^2) - (62640*(1 + 2*x))/(117649*(1 + x + x^ 2)) + (240*(2039 + 4168*x))/(823543*(1 + x + x^2)))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs. \(2(27)=54\).
Time = 0.14 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.12
method | result | size |
default | \(4 \,{\mathrm e}^{4} \left (\frac {2500}{2401 \left (-2+x \right )^{4}}+\frac {6000}{16807 \left (-2+x \right )^{3}}-\frac {45600}{117649 \left (-2+x \right )^{2}}+\frac {123360}{823543 \left (-2+x \right )}+\frac {-\frac {123360}{823543} x^{7}-\frac {420960}{823543} x^{6}-\frac {454320}{823543} x^{5}+\frac {4545543}{3294172} x^{4}+\frac {3863418}{823543} x^{3}+\frac {461016}{117649} x^{2}-\frac {678192}{823543} x +\frac {330522}{823543}}{\left (x^{2}+x +1\right )^{4}}\right )\) | \(81\) |
norman | \(\frac {160 x^{5} {\mathrm e}^{4}+88 \,{\mathrm e}^{4} x^{6}+16 \,{\mathrm e}^{4} x^{7}-104 x^{4} {\mathrm e}^{4}+x^{8} {\mathrm e}^{4}-320 x^{3} {\mathrm e}^{4}-128 x \,{\mathrm e}^{4}+352 x^{2} {\mathrm e}^{4}+16 \,{\mathrm e}^{4}}{\left (x^{3}-x^{2}-x -2\right )^{4}}\) | \(93\) |
risch | \(\frac {\left (x^{8}+16 x^{7}+88 x^{6}+160 x^{5}-104 x^{4}-320 x^{3}+352 x^{2}-128 x +16\right ) {\mathrm e}^{4}}{x^{12}-4 x^{11}+2 x^{10}+19 x^{8}-8 x^{7}-14 x^{6}-44 x^{5}+x^{4}+24 x^{3}+56 x^{2}+32 x +16}\) | \(95\) |
gosper | \(\frac {\left (x^{8}+16 x^{7}+88 x^{6}+160 x^{5}-104 x^{4}-320 x^{3}+352 x^{2}-128 x +16\right ) {\mathrm e}^{4}}{x^{12}-4 x^{11}+2 x^{10}+19 x^{8}-8 x^{7}-14 x^{6}-44 x^{5}+x^{4}+24 x^{3}+56 x^{2}+32 x +16}\) | \(97\) |
parallelrisch | \(\frac {\left (x^{8}+16 x^{7}+88 x^{6}+160 x^{5}-104 x^{4}-320 x^{3}+352 x^{2}-128 x +16\right ) {\mathrm e}^{4}}{x^{12}-4 x^{11}+2 x^{10}+19 x^{8}-8 x^{7}-14 x^{6}-44 x^{5}+x^{4}+24 x^{3}+56 x^{2}+32 x +16}\) | \(97\) |
orering | \(-\frac {\left (x^{2}+4 x -2\right ) \left (-2+x \right ) \left (x^{2}+x +1\right ) \left (-4 x^{10}-80 x^{9}-516 x^{8}-1008 x^{7}+912 x^{6}+1248 x^{5}-6720 x^{4}+4032 x^{3}+1536 x^{2}-1664 x +320\right ) {\mathrm e}^{4}}{4 \left (x^{4}+8 x^{3}-9 x^{2}+8 x +10\right ) \left (x^{15}-5 x^{14}+5 x^{13}+25 x^{11}-31 x^{10}-25 x^{9}-60 x^{8}+75 x^{7}+95 x^{6}+119 x^{5}-50 x^{4}-120 x^{3}-160 x^{2}-80 x -32\right )}\) | \(164\) |
Input:
int((-4*x^10-80*x^9-516*x^8-1008*x^7+912*x^6+1248*x^5-6720*x^4+4032*x^3+15 36*x^2-1664*x+320)*exp(1)^4/(x^15-5*x^14+5*x^13+25*x^11-31*x^10-25*x^9-60* x^8+75*x^7+95*x^6+119*x^5-50*x^4-120*x^3-160*x^2-80*x-32),x,method=_RETURN VERBOSE)
Output:
4*exp(1)^4*(2500/2401/(-2+x)^4+6000/16807/(-2+x)^3-45600/117649/(-2+x)^2+1 23360/823543/(-2+x)+3/823543*(-41120*x^7-140320*x^6-151440*x^5+1515181/4*x ^4+1287806*x^3+1075704*x^2-226064*x+110174)/(x^2+x+1)^4)
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (25) = 50\).
Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.62 \[ \int \frac {e^4 \left (320-1664 x+1536 x^2+4032 x^3-6720 x^4+1248 x^5+912 x^6-1008 x^7-516 x^8-80 x^9-4 x^{10}\right )}{-32-80 x-160 x^2-120 x^3-50 x^4+119 x^5+95 x^6+75 x^7-60 x^8-25 x^9-31 x^{10}+25 x^{11}+5 x^{13}-5 x^{14}+x^{15}} \, dx=\frac {{\left (x^{8} + 16 \, x^{7} + 88 \, x^{6} + 160 \, x^{5} - 104 \, x^{4} - 320 \, x^{3} + 352 \, x^{2} - 128 \, x + 16\right )} e^{4}}{x^{12} - 4 \, x^{11} + 2 \, x^{10} + 19 \, x^{8} - 8 \, x^{7} - 14 \, x^{6} - 44 \, x^{5} + x^{4} + 24 \, x^{3} + 56 \, x^{2} + 32 \, x + 16} \] Input:
integrate((-4*x^10-80*x^9-516*x^8-1008*x^7+912*x^6+1248*x^5-6720*x^4+4032* x^3+1536*x^2-1664*x+320)*exp(1)^4/(x^15-5*x^14+5*x^13+25*x^11-31*x^10-25*x ^9-60*x^8+75*x^7+95*x^6+119*x^5-50*x^4-120*x^3-160*x^2-80*x-32),x, algorit hm="fricas")
Output:
(x^8 + 16*x^7 + 88*x^6 + 160*x^5 - 104*x^4 - 320*x^3 + 352*x^2 - 128*x + 1 6)*e^4/(x^12 - 4*x^11 + 2*x^10 + 19*x^8 - 8*x^7 - 14*x^6 - 44*x^5 + x^4 + 24*x^3 + 56*x^2 + 32*x + 16)
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (22) = 44\).
Time = 0.56 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.69 \[ \int \frac {e^4 \left (320-1664 x+1536 x^2+4032 x^3-6720 x^4+1248 x^5+912 x^6-1008 x^7-516 x^8-80 x^9-4 x^{10}\right )}{-32-80 x-160 x^2-120 x^3-50 x^4+119 x^5+95 x^6+75 x^7-60 x^8-25 x^9-31 x^{10}+25 x^{11}+5 x^{13}-5 x^{14}+x^{15}} \, dx=- \frac {- x^{8} e^{4} - 16 x^{7} e^{4} - 88 x^{6} e^{4} - 160 x^{5} e^{4} + 104 x^{4} e^{4} + 320 x^{3} e^{4} - 352 x^{2} e^{4} + 128 x e^{4} - 16 e^{4}}{x^{12} - 4 x^{11} + 2 x^{10} + 19 x^{8} - 8 x^{7} - 14 x^{6} - 44 x^{5} + x^{4} + 24 x^{3} + 56 x^{2} + 32 x + 16} \] Input:
integrate((-4*x**10-80*x**9-516*x**8-1008*x**7+912*x**6+1248*x**5-6720*x** 4+4032*x**3+1536*x**2-1664*x+320)*exp(1)**4/(x**15-5*x**14+5*x**13+25*x**1 1-31*x**10-25*x**9-60*x**8+75*x**7+95*x**6+119*x**5-50*x**4-120*x**3-160*x **2-80*x-32),x)
Output:
-(-x**8*exp(4) - 16*x**7*exp(4) - 88*x**6*exp(4) - 160*x**5*exp(4) + 104*x **4*exp(4) + 320*x**3*exp(4) - 352*x**2*exp(4) + 128*x*exp(4) - 16*exp(4)) /(x**12 - 4*x**11 + 2*x**10 + 19*x**8 - 8*x**7 - 14*x**6 - 44*x**5 + x**4 + 24*x**3 + 56*x**2 + 32*x + 16)
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (25) = 50\).
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.62 \[ \int \frac {e^4 \left (320-1664 x+1536 x^2+4032 x^3-6720 x^4+1248 x^5+912 x^6-1008 x^7-516 x^8-80 x^9-4 x^{10}\right )}{-32-80 x-160 x^2-120 x^3-50 x^4+119 x^5+95 x^6+75 x^7-60 x^8-25 x^9-31 x^{10}+25 x^{11}+5 x^{13}-5 x^{14}+x^{15}} \, dx=\frac {{\left (x^{8} + 16 \, x^{7} + 88 \, x^{6} + 160 \, x^{5} - 104 \, x^{4} - 320 \, x^{3} + 352 \, x^{2} - 128 \, x + 16\right )} e^{4}}{x^{12} - 4 \, x^{11} + 2 \, x^{10} + 19 \, x^{8} - 8 \, x^{7} - 14 \, x^{6} - 44 \, x^{5} + x^{4} + 24 \, x^{3} + 56 \, x^{2} + 32 \, x + 16} \] Input:
integrate((-4*x^10-80*x^9-516*x^8-1008*x^7+912*x^6+1248*x^5-6720*x^4+4032* x^3+1536*x^2-1664*x+320)*exp(1)^4/(x^15-5*x^14+5*x^13+25*x^11-31*x^10-25*x ^9-60*x^8+75*x^7+95*x^6+119*x^5-50*x^4-120*x^3-160*x^2-80*x-32),x, algorit hm="maxima")
Output:
(x^8 + 16*x^7 + 88*x^6 + 160*x^5 - 104*x^4 - 320*x^3 + 352*x^2 - 128*x + 1 6)*e^4/(x^12 - 4*x^11 + 2*x^10 + 19*x^8 - 8*x^7 - 14*x^6 - 44*x^5 + x^4 + 24*x^3 + 56*x^2 + 32*x + 16)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (25) = 50\).
Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {e^4 \left (320-1664 x+1536 x^2+4032 x^3-6720 x^4+1248 x^5+912 x^6-1008 x^7-516 x^8-80 x^9-4 x^{10}\right )}{-32-80 x-160 x^2-120 x^3-50 x^4+119 x^5+95 x^6+75 x^7-60 x^8-25 x^9-31 x^{10}+25 x^{11}+5 x^{13}-5 x^{14}+x^{15}} \, dx=\frac {{\left (x^{8} + 16 \, x^{7} + 88 \, x^{6} + 160 \, x^{5} - 104 \, x^{4} - 320 \, x^{3} + 352 \, x^{2} - 128 \, x + 16\right )} e^{4}}{{\left (x^{3} - x^{2} - x - 2\right )}^{4}} \] Input:
integrate((-4*x^10-80*x^9-516*x^8-1008*x^7+912*x^6+1248*x^5-6720*x^4+4032* x^3+1536*x^2-1664*x+320)*exp(1)^4/(x^15-5*x^14+5*x^13+25*x^11-31*x^10-25*x ^9-60*x^8+75*x^7+95*x^6+119*x^5-50*x^4-120*x^3-160*x^2-80*x-32),x, algorit hm="giac")
Output:
(x^8 + 16*x^7 + 88*x^6 + 160*x^5 - 104*x^4 - 320*x^3 + 352*x^2 - 128*x + 1 6)*e^4/(x^3 - x^2 - x - 2)^4
Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 4.27 \[ \int \frac {e^4 \left (320-1664 x+1536 x^2+4032 x^3-6720 x^4+1248 x^5+912 x^6-1008 x^7-516 x^8-80 x^9-4 x^{10}\right )}{-32-80 x-160 x^2-120 x^3-50 x^4+119 x^5+95 x^6+75 x^7-60 x^8-25 x^9-31 x^{10}+25 x^{11}+5 x^{13}-5 x^{14}+x^{15}} \, dx=\frac {493440\,{\mathrm {e}}^4}{823543\,\left (x-2\right )}-\frac {182400\,{\mathrm {e}}^4}{117649\,{\left (x-2\right )}^2}+\frac {24000\,{\mathrm {e}}^4}{16807\,{\left (x-2\right )}^3}+\frac {10000\,{\mathrm {e}}^4}{2401\,{\left (x-2\right )}^4}-\frac {729\,{\mathrm {e}}^4\,\left (39\,x-16\right )}{2401\,{\left (x^2+x+1\right )}^4}-\frac {1920\,{\mathrm {e}}^4\,\left (257\,x+106\right )}{823543\,\left (x^2+x+1\right )}+\frac {81\,{\mathrm {e}}^4\,\left (794\,x-2063\right )}{16807\,{\left (x^2+x+1\right )}^3}+\frac {27\,{\mathrm {e}}^4\,\left (9280\,x+30227\right )}{117649\,{\left (x^2+x+1\right )}^2} \] Input:
int((exp(4)*(1664*x - 1536*x^2 - 4032*x^3 + 6720*x^4 - 1248*x^5 - 912*x^6 + 1008*x^7 + 516*x^8 + 80*x^9 + 4*x^10 - 320))/(80*x + 160*x^2 + 120*x^3 + 50*x^4 - 119*x^5 - 95*x^6 - 75*x^7 + 60*x^8 + 25*x^9 + 31*x^10 - 25*x^11 - 5*x^13 + 5*x^14 - x^15 + 32),x)
Output:
(493440*exp(4))/(823543*(x - 2)) - (182400*exp(4))/(117649*(x - 2)^2) + (2 4000*exp(4))/(16807*(x - 2)^3) + (10000*exp(4))/(2401*(x - 2)^4) - (729*ex p(4)*(39*x - 16))/(2401*(x + x^2 + 1)^4) - (1920*exp(4)*(257*x + 106))/(82 3543*(x + x^2 + 1)) + (81*exp(4)*(794*x - 2063))/(16807*(x + x^2 + 1)^3) + (27*exp(4)*(9280*x + 30227))/(117649*(x + x^2 + 1)^2)
Time = 0.16 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.65 \[ \int \frac {e^4 \left (320-1664 x+1536 x^2+4032 x^3-6720 x^4+1248 x^5+912 x^6-1008 x^7-516 x^8-80 x^9-4 x^{10}\right )}{-32-80 x-160 x^2-120 x^3-50 x^4+119 x^5+95 x^6+75 x^7-60 x^8-25 x^9-31 x^{10}+25 x^{11}+5 x^{13}-5 x^{14}+x^{15}} \, dx=\frac {e^{4} \left (x^{8}+16 x^{7}+88 x^{6}+160 x^{5}-104 x^{4}-320 x^{3}+352 x^{2}-128 x +16\right )}{x^{12}-4 x^{11}+2 x^{10}+19 x^{8}-8 x^{7}-14 x^{6}-44 x^{5}+x^{4}+24 x^{3}+56 x^{2}+32 x +16} \] Input:
int((-4*x^10-80*x^9-516*x^8-1008*x^7+912*x^6+1248*x^5-6720*x^4+4032*x^3+15 36*x^2-1664*x+320)*exp(1)^4/(x^15-5*x^14+5*x^13+25*x^11-31*x^10-25*x^9-60* x^8+75*x^7+95*x^6+119*x^5-50*x^4-120*x^3-160*x^2-80*x-32),x)
Output:
(e**4*(x**8 + 16*x**7 + 88*x**6 + 160*x**5 - 104*x**4 - 320*x**3 + 352*x** 2 - 128*x + 16))/(x**12 - 4*x**11 + 2*x**10 + 19*x**8 - 8*x**7 - 14*x**6 - 44*x**5 + x**4 + 24*x**3 + 56*x**2 + 32*x + 16)