Integrand size = 187, antiderivative size = 35 \[ \int \frac {-1576 x^4+36 x^6+e^8 \left (-300 x^2-80 x^3-4 x^4\right )+e^4 \left (1600 x^3+280 x^4\right )}{620944 x^4+370360 x^5+83593 x^6+8460 x^7+324 x^8+e^{16} \left (2500+2000 x+600 x^2+80 x^3+4 x^4\right )+e^{12} \left (-40000 x-30000 x^2-8400 x^3-1040 x^4-48 x^5\right )+e^8 \left (238800 x^2+167020 x^3+43552 x^4+5020 x^5+216 x^6\right )+e^4 \left (-630400 x^3-408640 x^4-99112 x^5-10680 x^6-432 x^7\right )} \, dx=\frac {x}{6+\frac {5 x}{2}-\left (-x+\frac {(5+x) \left (-e^4+4 x\right )}{x}\right )^2} \]
Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.40 \[ \int \frac {-1576 x^4+36 x^6+e^8 \left (-300 x^2-80 x^3-4 x^4\right )+e^4 \left (1600 x^3+280 x^4\right )}{620944 x^4+370360 x^5+83593 x^6+8460 x^7+324 x^8+e^{16} \left (2500+2000 x+600 x^2+80 x^3+4 x^4\right )+e^{12} \left (-40000 x-30000 x^2-8400 x^3-1040 x^4-48 x^5\right )+e^8 \left (238800 x^2+167020 x^3+43552 x^4+5020 x^5+216 x^6\right )+e^4 \left (-630400 x^3-408640 x^4-99112 x^5-10680 x^6-432 x^7\right )} \, dx=-\frac {4 x^3}{4 e^8 (5+x)^2-8 e^4 x \left (100+35 x+3 x^2\right )+2 x^2 \left (788+235 x+18 x^2\right )} \]
Integrate[(-1576*x^4 + 36*x^6 + E^8*(-300*x^2 - 80*x^3 - 4*x^4) + E^4*(160 0*x^3 + 280*x^4))/(620944*x^4 + 370360*x^5 + 83593*x^6 + 8460*x^7 + 324*x^ 8 + E^16*(2500 + 2000*x + 600*x^2 + 80*x^3 + 4*x^4) + E^12*(-40000*x - 300 00*x^2 - 8400*x^3 - 1040*x^4 - 48*x^5) + E^8*(238800*x^2 + 167020*x^3 + 43 552*x^4 + 5020*x^5 + 216*x^6) + E^4*(-630400*x^3 - 408640*x^4 - 99112*x^5 - 10680*x^6 - 432*x^7)),x]
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.
| \(\displaystyle \int \frac {36 x^6-1576 x^4+e^4 \left (280 x^4+1600 x^3\right )+e^8 \left (-4 x^4-80 x^3-300 x^2\right )}{324 x^8+8460 x^7+83593 x^6+370360 x^5+620944 x^4+e^{16} \left (4 x^4+80 x^3+600 x^2+2000 x+2500\right )+e^{12} \left (-48 x^5-1040 x^4-8400 x^3-30000 x^2-40000 x\right )+e^4 \left (-432 x^7-10680 x^6-99112 x^5-408640 x^4-630400 x^3\right )+e^8 \left (216 x^6+5020 x^5+43552 x^4+167020 x^3+238800 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {324 x^2-18 \left (235-12 e^4\right ) x+72 e^8-600 e^4+26857}{162 \left (18 x^4+\left (235-12 e^4\right ) x^3+2 \left (394-70 e^4+e^8\right ) x^2-20 e^4 \left (20-e^4\right ) x+50 e^8\right )}+\frac {-\left (\left (2978155-89676 e^4+4860 e^8-432 e^{12}\right ) x^3\right )-2 \left (10581658-1270390 e^4+44125 e^8-3480 e^{12}+72 e^{16}\right ) x^2+40 e^4 \left (268570-14141 e^4+750 e^8-36 e^{12}\right ) x-50 e^8 \left (26857-600 e^4+72 e^8\right )}{162 \left (18 x^4+\left (235-12 e^4\right ) x^3+2 \left (394-70 e^4+e^8\right ) x^2-20 e^4 \left (20-e^4\right ) x+50 e^8\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{162} \left (26857-600 e^4+72 e^8\right ) \int \frac {1}{-18 x^4-\left (235-12 e^4\right ) x^3-2 \left (394-70 e^4+e^8\right ) x^2+20 e^4 \left (20-e^4\right ) x-50 e^8}dx-\frac {5 e^4 \left (59563100+62585 e^4+78876 e^8-540 e^{12}+432 e^{16}\right ) \int \frac {1}{\left (18 x^4+\left (235-12 e^4\right ) x^3+2 \left (394-70 e^4+e^8\right ) x^2-20 e^4 \left (20-e^4\right ) x+50 e^8\right )^2}dx}{2916}+\frac {\left (1173393070-50432794 e^4+988795 e^8-60084 e^{12}+9180 e^{16}-432 e^{20}\right ) \int \frac {x}{\left (18 x^4+\left (235-12 e^4\right ) x^3+2 \left (394-70 e^4+e^8\right ) x^2-20 e^4 \left (20-e^4\right ) x+50 e^8\right )^2}dx}{2916}+\frac {\left (191946841+4167000 e^4+100212 e^8+7200 e^{12}+1728 e^{16}\right ) \int \frac {x^2}{\left (18 x^4+\left (235-12 e^4\right ) x^3+2 \left (394-70 e^4+e^8\right ) x^2-20 e^4 \left (20-e^4\right ) x+50 e^8\right )^2}dx}{3888}-\frac {1}{9} \left (235-12 e^4\right ) \int \frac {x}{18 x^4+\left (235-12 e^4\right ) x^3+2 \left (394-70 e^4+e^8\right ) x^2-20 e^4 \left (20-e^4\right ) x+50 e^8}dx+2 \int \frac {x^2}{18 x^4+\left (235-12 e^4\right ) x^3+2 \left (394-70 e^4+e^8\right ) x^2-20 e^4 \left (20-e^4\right ) x+50 e^8}dx+\frac {2978155-89676 e^4+4860 e^8-432 e^{12}}{11664 \left (18 x^4+\left (235-12 e^4\right ) x^3+2 \left (394-70 e^4+e^8\right ) x^2-20 e^4 \left (20-e^4\right ) x+50 e^8\right )}\) |
Int[(-1576*x^4 + 36*x^6 + E^8*(-300*x^2 - 80*x^3 - 4*x^4) + E^4*(1600*x^3 + 280*x^4))/(620944*x^4 + 370360*x^5 + 83593*x^6 + 8460*x^7 + 324*x^8 + E^ 16*(2500 + 2000*x + 600*x^2 + 80*x^3 + 4*x^4) + E^12*(-40000*x - 30000*x^2 - 8400*x^3 - 1040*x^4 - 48*x^5) + E^8*(238800*x^2 + 167020*x^3 + 43552*x^ 4 + 5020*x^5 + 216*x^6) + E^4*(-630400*x^3 - 408640*x^4 - 99112*x^5 - 1068 0*x^6 - 432*x^7)),x]
3.4.56.3.1 Defintions of rubi rules used
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]
Time = 3.46 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66
| method | result | size |
| risch | \(-\frac {x^{3}}{x^{2} {\mathrm e}^{8}-6 x^{3} {\mathrm e}^{4}+9 x^{4}+10 x \,{\mathrm e}^{8}-70 x^{2} {\mathrm e}^{4}+\frac {235 x^{3}}{2}+25 \,{\mathrm e}^{8}-200 x \,{\mathrm e}^{4}+394 x^{2}}\) | \(58\) |
| gosper | \(-\frac {2 x^{3}}{2 x^{2} {\mathrm e}^{8}-12 x^{3} {\mathrm e}^{4}+18 x^{4}+20 x \,{\mathrm e}^{8}-140 x^{2} {\mathrm e}^{4}+235 x^{3}+50 \,{\mathrm e}^{8}-400 x \,{\mathrm e}^{4}+788 x^{2}}\) | \(65\) |
| norman | \(-\frac {2 x^{3}}{2 x^{2} {\mathrm e}^{8}-12 x^{3} {\mathrm e}^{4}+18 x^{4}+20 x \,{\mathrm e}^{8}-140 x^{2} {\mathrm e}^{4}+235 x^{3}+50 \,{\mathrm e}^{8}-400 x \,{\mathrm e}^{4}+788 x^{2}}\) | \(65\) |
| parallelrisch | \(-\frac {2 x^{3}}{2 x^{2} {\mathrm e}^{8}-12 x^{3} {\mathrm e}^{4}+18 x^{4}+20 x \,{\mathrm e}^{8}-140 x^{2} {\mathrm e}^{4}+235 x^{3}+50 \,{\mathrm e}^{8}-400 x \,{\mathrm e}^{4}+788 x^{2}}\) | \(65\) |
| default | \(-2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (324 \textit {\_Z}^{8}+\left (-432 \,{\mathrm e}^{4}+8460\right ) \textit {\_Z}^{7}+\left (-10680 \,{\mathrm e}^{4}+216 \,{\mathrm e}^{8}+83593\right ) \textit {\_Z}^{6}+\left (-99112 \,{\mathrm e}^{4}-48 \,{\mathrm e}^{12}+5020 \,{\mathrm e}^{8}+370360\right ) \textit {\_Z}^{5}+\left (-408640 \,{\mathrm e}^{4}+4 \,{\mathrm e}^{16}-1040 \,{\mathrm e}^{12}+43552 \,{\mathrm e}^{8}+620944\right ) \textit {\_Z}^{4}+\left (-630400 \,{\mathrm e}^{4}+80 \,{\mathrm e}^{16}-8400 \,{\mathrm e}^{12}+167020 \,{\mathrm e}^{8}\right ) \textit {\_Z}^{3}+\left (600 \,{\mathrm e}^{16}-30000 \,{\mathrm e}^{12}+238800 \,{\mathrm e}^{8}\right ) \textit {\_Z}^{2}+\left (2000 \,{\mathrm e}^{16}-40000 \,{\mathrm e}^{12}\right ) \textit {\_Z} +2500 \,{\mathrm e}^{16}\right )}{\sum }\frac {\left (9 \textit {\_R}^{6}+\left (70 \,{\mathrm e}^{4}-{\mathrm e}^{8}-394\right ) \textit {\_R}^{4}+20 \left (20 \,{\mathrm e}^{4}-{\mathrm e}^{8}\right ) \textit {\_R}^{3}-75 \textit {\_R}^{2} {\mathrm e}^{8}\right ) \ln \left (x -\textit {\_R} \right )}{-600 \textit {\_R} \,{\mathrm e}^{16}+120 \,{\mathrm e}^{12} \textit {\_R}^{4}-8 \textit {\_R}^{3} {\mathrm e}^{16}+2080 \,{\mathrm e}^{12} \textit {\_R}^{3}+12600 \textit {\_R}^{2} {\mathrm e}^{12}-250530 \textit {\_R}^{2} {\mathrm e}^{8}+1512 \textit {\_R}^{6} {\mathrm e}^{4}-1000 \,{\mathrm e}^{16}+945600 \textit {\_R}^{2} {\mathrm e}^{4}+817280 \textit {\_R}^{3} {\mathrm e}^{4}+247780 \textit {\_R}^{4} {\mathrm e}^{4}+32040 \textit {\_R}^{5} {\mathrm e}^{4}-120 \textit {\_R}^{2} {\mathrm e}^{16}-29610 \textit {\_R}^{6}-1296 \textit {\_R}^{7}-1241888 \textit {\_R}^{3}-250779 \textit {\_R}^{5}-925900 \textit {\_R}^{4}-238800 \textit {\_R} \,{\mathrm e}^{8}-87104 \textit {\_R}^{3} {\mathrm e}^{8}-648 \textit {\_R}^{5} {\mathrm e}^{8}-12550 \textit {\_R}^{4} {\mathrm e}^{8}+20000 \,{\mathrm e}^{12}+30000 \textit {\_R} \,{\mathrm e}^{12}}\right )\) | \(327\) |
int(((-4*x^4-80*x^3-300*x^2)*exp(4)^2+(280*x^4+1600*x^3)*exp(4)+36*x^6-157 6*x^4)/((4*x^4+80*x^3+600*x^2+2000*x+2500)*exp(4)^4+(-48*x^5-1040*x^4-8400 *x^3-30000*x^2-40000*x)*exp(4)^3+(216*x^6+5020*x^5+43552*x^4+167020*x^3+23 8800*x^2)*exp(4)^2+(-432*x^7-10680*x^6-99112*x^5-408640*x^4-630400*x^3)*ex p(4)+324*x^8+8460*x^7+83593*x^6+370360*x^5+620944*x^4),x,method=_RETURNVER BOSE)
-x^3/(x^2*exp(8)-6*x^3*exp(4)+9*x^4+10*x*exp(8)-70*x^2*exp(4)+235/2*x^3+25 *exp(8)-200*x*exp(4)+394*x^2)
Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.51 \[ \int \frac {-1576 x^4+36 x^6+e^8 \left (-300 x^2-80 x^3-4 x^4\right )+e^4 \left (1600 x^3+280 x^4\right )}{620944 x^4+370360 x^5+83593 x^6+8460 x^7+324 x^8+e^{16} \left (2500+2000 x+600 x^2+80 x^3+4 x^4\right )+e^{12} \left (-40000 x-30000 x^2-8400 x^3-1040 x^4-48 x^5\right )+e^8 \left (238800 x^2+167020 x^3+43552 x^4+5020 x^5+216 x^6\right )+e^4 \left (-630400 x^3-408640 x^4-99112 x^5-10680 x^6-432 x^7\right )} \, dx=-\frac {2 \, x^{3}}{18 \, x^{4} + 235 \, x^{3} + 788 \, x^{2} + 2 \, {\left (x^{2} + 10 \, x + 25\right )} e^{8} - 4 \, {\left (3 \, x^{3} + 35 \, x^{2} + 100 \, x\right )} e^{4}} \]
integrate(((-4*x^4-80*x^3-300*x^2)*exp(4)^2+(280*x^4+1600*x^3)*exp(4)+36*x ^6-1576*x^4)/((4*x^4+80*x^3+600*x^2+2000*x+2500)*exp(4)^4+(-48*x^5-1040*x^ 4-8400*x^3-30000*x^2-40000*x)*exp(4)^3+(216*x^6+5020*x^5+43552*x^4+167020* x^3+238800*x^2)*exp(4)^2+(-432*x^7-10680*x^6-99112*x^5-408640*x^4-630400*x ^3)*exp(4)+324*x^8+8460*x^7+83593*x^6+370360*x^5+620944*x^4),x, algorithm= \
-2*x^3/(18*x^4 + 235*x^3 + 788*x^2 + 2*(x^2 + 10*x + 25)*e^8 - 4*(3*x^3 + 35*x^2 + 100*x)*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
Time = 4.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.51 \[ \int \frac {-1576 x^4+36 x^6+e^8 \left (-300 x^2-80 x^3-4 x^4\right )+e^4 \left (1600 x^3+280 x^4\right )}{620944 x^4+370360 x^5+83593 x^6+8460 x^7+324 x^8+e^{16} \left (2500+2000 x+600 x^2+80 x^3+4 x^4\right )+e^{12} \left (-40000 x-30000 x^2-8400 x^3-1040 x^4-48 x^5\right )+e^8 \left (238800 x^2+167020 x^3+43552 x^4+5020 x^5+216 x^6\right )+e^4 \left (-630400 x^3-408640 x^4-99112 x^5-10680 x^6-432 x^7\right )} \, dx=- \frac {2 x^{3}}{18 x^{4} + x^{3} \cdot \left (235 - 12 e^{4}\right ) + x^{2} \left (- 140 e^{4} + 788 + 2 e^{8}\right ) + x \left (- 400 e^{4} + 20 e^{8}\right ) + 50 e^{8}} \]
integrate(((-4*x**4-80*x**3-300*x**2)*exp(4)**2+(280*x**4+1600*x**3)*exp(4 )+36*x**6-1576*x**4)/((4*x**4+80*x**3+600*x**2+2000*x+2500)*exp(4)**4+(-48 *x**5-1040*x**4-8400*x**3-30000*x**2-40000*x)*exp(4)**3+(216*x**6+5020*x** 5+43552*x**4+167020*x**3+238800*x**2)*exp(4)**2+(-432*x**7-10680*x**6-9911 2*x**5-408640*x**4-630400*x**3)*exp(4)+324*x**8+8460*x**7+83593*x**6+37036 0*x**5+620944*x**4),x)
-2*x**3/(18*x**4 + x**3*(235 - 12*exp(4)) + x**2*(-140*exp(4) + 788 + 2*ex p(8)) + x*(-400*exp(4) + 20*exp(8)) + 50*exp(8))
Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.46 \[ \int \frac {-1576 x^4+36 x^6+e^8 \left (-300 x^2-80 x^3-4 x^4\right )+e^4 \left (1600 x^3+280 x^4\right )}{620944 x^4+370360 x^5+83593 x^6+8460 x^7+324 x^8+e^{16} \left (2500+2000 x+600 x^2+80 x^3+4 x^4\right )+e^{12} \left (-40000 x-30000 x^2-8400 x^3-1040 x^4-48 x^5\right )+e^8 \left (238800 x^2+167020 x^3+43552 x^4+5020 x^5+216 x^6\right )+e^4 \left (-630400 x^3-408640 x^4-99112 x^5-10680 x^6-432 x^7\right )} \, dx=-\frac {2 \, x^{3}}{18 \, x^{4} - x^{3} {\left (12 \, e^{4} - 235\right )} + 2 \, x^{2} {\left (e^{8} - 70 \, e^{4} + 394\right )} + 20 \, x {\left (e^{8} - 20 \, e^{4}\right )} + 50 \, e^{8}} \]
integrate(((-4*x^4-80*x^3-300*x^2)*exp(4)^2+(280*x^4+1600*x^3)*exp(4)+36*x ^6-1576*x^4)/((4*x^4+80*x^3+600*x^2+2000*x+2500)*exp(4)^4+(-48*x^5-1040*x^ 4-8400*x^3-30000*x^2-40000*x)*exp(4)^3+(216*x^6+5020*x^5+43552*x^4+167020* x^3+238800*x^2)*exp(4)^2+(-432*x^7-10680*x^6-99112*x^5-408640*x^4-630400*x ^3)*exp(4)+324*x^8+8460*x^7+83593*x^6+370360*x^5+620944*x^4),x, algorithm= \
-2*x^3/(18*x^4 - x^3*(12*e^4 - 235) + 2*x^2*(e^8 - 70*e^4 + 394) + 20*x*(e ^8 - 20*e^4) + 50*e^8)
Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.66 \[ \int \frac {-1576 x^4+36 x^6+e^8 \left (-300 x^2-80 x^3-4 x^4\right )+e^4 \left (1600 x^3+280 x^4\right )}{620944 x^4+370360 x^5+83593 x^6+8460 x^7+324 x^8+e^{16} \left (2500+2000 x+600 x^2+80 x^3+4 x^4\right )+e^{12} \left (-40000 x-30000 x^2-8400 x^3-1040 x^4-48 x^5\right )+e^8 \left (238800 x^2+167020 x^3+43552 x^4+5020 x^5+216 x^6\right )+e^4 \left (-630400 x^3-408640 x^4-99112 x^5-10680 x^6-432 x^7\right )} \, dx=-\frac {2 \, x^{3}}{18 \, x^{4} - 12 \, x^{3} e^{4} + 235 \, x^{3} + 2 \, x^{2} e^{8} - 140 \, x^{2} e^{4} + 788 \, x^{2} + 20 \, x e^{8} - 400 \, x e^{4} + 50 \, e^{8}} \]
integrate(((-4*x^4-80*x^3-300*x^2)*exp(4)^2+(280*x^4+1600*x^3)*exp(4)+36*x ^6-1576*x^4)/((4*x^4+80*x^3+600*x^2+2000*x+2500)*exp(4)^4+(-48*x^5-1040*x^ 4-8400*x^3-30000*x^2-40000*x)*exp(4)^3+(216*x^6+5020*x^5+43552*x^4+167020* x^3+238800*x^2)*exp(4)^2+(-432*x^7-10680*x^6-99112*x^5-408640*x^4-630400*x ^3)*exp(4)+324*x^8+8460*x^7+83593*x^6+370360*x^5+620944*x^4),x, algorithm= \
-2*x^3/(18*x^4 - 12*x^3*e^4 + 235*x^3 + 2*x^2*e^8 - 140*x^2*e^4 + 788*x^2 + 20*x*e^8 - 400*x*e^4 + 50*e^8)
Time = 8.87 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \frac {-1576 x^4+36 x^6+e^8 \left (-300 x^2-80 x^3-4 x^4\right )+e^4 \left (1600 x^3+280 x^4\right )}{620944 x^4+370360 x^5+83593 x^6+8460 x^7+324 x^8+e^{16} \left (2500+2000 x+600 x^2+80 x^3+4 x^4\right )+e^{12} \left (-40000 x-30000 x^2-8400 x^3-1040 x^4-48 x^5\right )+e^8 \left (238800 x^2+167020 x^3+43552 x^4+5020 x^5+216 x^6\right )+e^4 \left (-630400 x^3-408640 x^4-99112 x^5-10680 x^6-432 x^7\right )} \, dx=-\frac {2\,x^3}{18\,x^4+\left (235-12\,{\mathrm {e}}^4\right )\,x^3+\left (2\,{\mathrm {e}}^8-140\,{\mathrm {e}}^4+788\right )\,x^2+\left (20\,{\mathrm {e}}^8-400\,{\mathrm {e}}^4\right )\,x+50\,{\mathrm {e}}^8} \]
int((exp(4)*(1600*x^3 + 280*x^4) - exp(8)*(300*x^2 + 80*x^3 + 4*x^4) - 157 6*x^4 + 36*x^6)/(exp(8)*(238800*x^2 + 167020*x^3 + 43552*x^4 + 5020*x^5 + 216*x^6) - exp(4)*(630400*x^3 + 408640*x^4 + 99112*x^5 + 10680*x^6 + 432*x ^7) + exp(16)*(2000*x + 600*x^2 + 80*x^3 + 4*x^4 + 2500) - exp(12)*(40000* x + 30000*x^2 + 8400*x^3 + 1040*x^4 + 48*x^5) + 620944*x^4 + 370360*x^5 + 83593*x^6 + 8460*x^7 + 324*x^8),x)