3.46.43 \(\int \frac {e^{10} (5+e^4 (-3 x+4 x^2))+e^5 (-5 x^2+e^4 (4 x^2-5 x^3))}{3125 x^2-6250 x^3+3125 x^4+e^4 (3125 x^3-6250 x^4+3125 x^5)+e^8 (1250 x^4-2500 x^5+1250 x^6)+e^{12} (250 x^5-500 x^6+250 x^7)+e^{16} (25 x^6-50 x^7+25 x^8)+e^{20} (x^7-2 x^8+x^9)+e^{10} (3125-6250 x+3125 x^2+e^4 (3125 x-6250 x^2+3125 x^3)+e^8 (1250 x^2-2500 x^3+1250 x^4)+e^{12} (250 x^3-500 x^4+250 x^5)+e^{16} (25 x^4-50 x^5+25 x^6)+e^{20} (x^5-2 x^6+x^7))+e^5 (-6250 x+12500 x^2-6250 x^3+e^4 (-6250 x^2+12500 x^3-6250 x^4)+e^8 (-2500 x^3+5000 x^4-2500 x^5)+e^{12} (-500 x^4+1000 x^5-500 x^6)+e^{16} (-50 x^5+100 x^6-50 x^7)+e^{20} (-2 x^6+4 x^7-2 x^8))} \, dx\) [4543]

Optimal. Leaf size=30 \[ \frac {x}{\left (5+e^4 x\right )^4 \left (1-x-\frac {x-x^2}{e^5}\right )} \]

[Out]

x/(5+x*exp(4))^4/(1-x-(-x^2+x)/exp(5))

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(226\) vs. \(2(30)=60\).
time = 1.45, antiderivative size = 226, normalized size of antiderivative = 7.53, number of steps used = 2, number of rules used = 1, integrand size = 386, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.003, Rules used = {2099} \begin {gather*} \frac {e^{10}}{\left (1-e^5\right ) \left (5+e^9\right )^4 \left (e^5-x\right )}-\frac {e^9 \left (625-150 e^{13}-20 e^{17}-e^{21}-20 e^{22}-e^{26}-e^{31}\right )}{\left (5+e^4\right )^4 \left (5+e^9\right )^4 \left (e^4 x+5\right )}-\frac {e^9 \left (125-15 e^{13}-e^{17}-e^{22}\right )}{\left (5+e^4\right )^3 \left (5+e^9\right )^3 \left (e^4 x+5\right )^2}-\frac {e^9 \left (25-e^{13}\right )}{\left (5+e^4\right )^2 \left (5+e^9\right )^2 \left (e^4 x+5\right )^3}-\frac {5 e^9}{\left (5+e^4\right ) \left (5+e^9\right ) \left (e^4 x+5\right )^4}-\frac {e^5}{\left (5+e^4\right )^4 \left (1-e^5\right ) (1-x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^10*(5 + E^4*(-3*x + 4*x^2)) + E^5*(-5*x^2 + E^4*(4*x^2 - 5*x^3)))/(3125*x^2 - 6250*x^3 + 3125*x^4 + E^4
*(3125*x^3 - 6250*x^4 + 3125*x^5) + E^8*(1250*x^4 - 2500*x^5 + 1250*x^6) + E^12*(250*x^5 - 500*x^6 + 250*x^7)
+ E^16*(25*x^6 - 50*x^7 + 25*x^8) + E^20*(x^7 - 2*x^8 + x^9) + E^10*(3125 - 6250*x + 3125*x^2 + E^4*(3125*x -
6250*x^2 + 3125*x^3) + E^8*(1250*x^2 - 2500*x^3 + 1250*x^4) + E^12*(250*x^3 - 500*x^4 + 250*x^5) + E^16*(25*x^
4 - 50*x^5 + 25*x^6) + E^20*(x^5 - 2*x^6 + x^7)) + E^5*(-6250*x + 12500*x^2 - 6250*x^3 + E^4*(-6250*x^2 + 1250
0*x^3 - 6250*x^4) + E^8*(-2500*x^3 + 5000*x^4 - 2500*x^5) + E^12*(-500*x^4 + 1000*x^5 - 500*x^6) + E^16*(-50*x
^5 + 100*x^6 - 50*x^7) + E^20*(-2*x^6 + 4*x^7 - 2*x^8))),x]

[Out]

-(E^5/((5 + E^4)^4*(1 - E^5)*(1 - x))) + E^10/((1 - E^5)*(5 + E^9)^4*(E^5 - x)) - (5*E^9)/((5 + E^4)*(5 + E^9)
*(5 + E^4*x)^4) - (E^9*(25 - E^13))/((5 + E^4)^2*(5 + E^9)^2*(5 + E^4*x)^3) - (E^9*(125 - 15*E^13 - E^17 - E^2
2))/((5 + E^4)^3*(5 + E^9)^3*(5 + E^4*x)^2) - (E^9*(625 - 150*E^13 - 20*E^17 - E^21 - 20*E^22 - E^26 - E^31))/
((5 + E^4)^4*(5 + E^9)^4*(5 + E^4*x))

Rule 2099

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {e^{10}}{\left (-1+e^5\right ) \left (5+e^9\right )^4 \left (e^5-x\right )^2}+\frac {e^5}{\left (5+e^4\right )^4 \left (-1+e^5\right ) (-1+x)^2}+\frac {20 e^{13}}{\left (5+e^4\right ) \left (5+e^9\right ) \left (5+e^4 x\right )^5}+\frac {3 e^{13} \left (25-e^{13}\right )}{\left (5+e^4\right )^2 \left (5+e^9\right )^2 \left (5+e^4 x\right )^4}+\frac {2 e^{13} \left (125-15 e^{13}-e^{17}-e^{22}\right )}{\left (5+e^4\right )^3 \left (5+e^9\right )^3 \left (5+e^4 x\right )^3}+\frac {e^{13} \left (625-150 e^{13}-20 e^{17}-e^{21}-20 e^{22}-e^{26}-e^{31}\right )}{\left (5+e^4\right )^4 \left (5+e^9\right )^4 \left (5+e^4 x\right )^2}\right ) \, dx\\ &=-\frac {e^5}{\left (5+e^4\right )^4 \left (1-e^5\right ) (1-x)}+\frac {e^{10}}{\left (1-e^5\right ) \left (5+e^9\right )^4 \left (e^5-x\right )}-\frac {5 e^9}{\left (5+e^4\right ) \left (5+e^9\right ) \left (5+e^4 x\right )^4}-\frac {e^9 \left (25-e^{13}\right )}{\left (5+e^4\right )^2 \left (5+e^9\right )^2 \left (5+e^4 x\right )^3}-\frac {e^9 \left (125-15 e^{13}-e^{17}-e^{22}\right )}{\left (5+e^4\right )^3 \left (5+e^9\right )^3 \left (5+e^4 x\right )^2}-\frac {e^9 \left (625-150 e^{13}-20 e^{17}-e^{21}-20 e^{22}-e^{26}-e^{31}\right )}{\left (5+e^4\right )^4 \left (5+e^9\right )^4 \left (5+e^4 x\right )}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.19, size = 29, normalized size = 0.97 \begin {gather*} -\frac {e^5 x}{\left (e^5-x\right ) (-1+x) \left (5+e^4 x\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^10*(5 + E^4*(-3*x + 4*x^2)) + E^5*(-5*x^2 + E^4*(4*x^2 - 5*x^3)))/(3125*x^2 - 6250*x^3 + 3125*x^4
+ E^4*(3125*x^3 - 6250*x^4 + 3125*x^5) + E^8*(1250*x^4 - 2500*x^5 + 1250*x^6) + E^12*(250*x^5 - 500*x^6 + 250
*x^7) + E^16*(25*x^6 - 50*x^7 + 25*x^8) + E^20*(x^7 - 2*x^8 + x^9) + E^10*(3125 - 6250*x + 3125*x^2 + E^4*(312
5*x - 6250*x^2 + 3125*x^3) + E^8*(1250*x^2 - 2500*x^3 + 1250*x^4) + E^12*(250*x^3 - 500*x^4 + 250*x^5) + E^16*
(25*x^4 - 50*x^5 + 25*x^6) + E^20*(x^5 - 2*x^6 + x^7)) + E^5*(-6250*x + 12500*x^2 - 6250*x^3 + E^4*(-6250*x^2
+ 12500*x^3 - 6250*x^4) + E^8*(-2500*x^3 + 5000*x^4 - 2500*x^5) + E^12*(-500*x^4 + 1000*x^5 - 500*x^6) + E^16*
(-50*x^5 + 100*x^6 - 50*x^7) + E^20*(-2*x^6 + 4*x^7 - 2*x^8))),x]

[Out]

-((E^5*x)/((E^5 - x)*(-1 + x)*(5 + E^4*x)^4))

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 3.02, size = 1549, normalized size = 51.63

method	result	size
norman	\(-\frac {x \,{\mathrm e}^{5}}{\left (x -1\right ) \left (5+x \,{\mathrm e}^{4}\right )^{4} \left ({\mathrm e}^{5}-x \right )}\)	\(27\)
risch	\(-\frac {x \,{\mathrm e}^{5}}{x^{5} {\mathrm e}^{21}-x^{4} {\mathrm e}^{21}-{\mathrm e}^{16} x^{6}+20 x^{4} {\mathrm e}^{17}+{\mathrm e}^{16} x^{5}-20 x^{3} {\mathrm e}^{17}-20 \,{\mathrm e}^{12} x^{5}+150 x^{3} {\mathrm e}^{13}+20 \,{\mathrm e}^{12} x^{4}-150 x^{2} {\mathrm e}^{13}-150 x^{4} {\mathrm e}^{8}+500 x^{2} {\mathrm e}^{9}+150 \,{\mathrm e}^{8} x^{3}-500 x \,{\mathrm e}^{9}-500 x^{3} {\mathrm e}^{4}+625 x \,{\mathrm e}^{5}+500 x^{2} {\mathrm e}^{4}-625 \,{\mathrm e}^{5}-625 x^{2}+625 x}\)	\(134\)
gosper	\(-\frac {x \,{\mathrm e}^{5}}{{\mathrm e}^{5} {\mathrm e}^{16} x^{5}-{\mathrm e}^{16} x^{6}-{\mathrm e}^{5} {\mathrm e}^{16} x^{4}+{\mathrm e}^{16} x^{5}+20 \,{\mathrm e}^{5} {\mathrm e}^{12} x^{4}-20 \,{\mathrm e}^{12} x^{5}-20 \,{\mathrm e}^{5} {\mathrm e}^{12} x^{3}+20 \,{\mathrm e}^{12} x^{4}+150 \,{\mathrm e}^{5} {\mathrm e}^{8} x^{3}-150 x^{4} {\mathrm e}^{8}-150 \,{\mathrm e}^{5} {\mathrm e}^{8} x^{2}+150 \,{\mathrm e}^{8} x^{3}+500 x^{2} {\mathrm e}^{4} {\mathrm e}^{5}-500 x^{3} {\mathrm e}^{4}-500 x \,{\mathrm e}^{4} {\mathrm e}^{5}+500 x^{2} {\mathrm e}^{4}+625 x \,{\mathrm e}^{5}-625 x^{2}-625 \,{\mathrm e}^{5}+625 x}\)	\(174\)
default	\(\text {Expression too large to display}\)	\(1549\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x^2-3*x)*exp(4)+5)*exp(5)^2+((-5*x^3+4*x^2)*exp(4)-5*x^2)*exp(5))/(((x^7-2*x^6+x^5)*exp(4)^5+(25*x^6-
50*x^5+25*x^4)*exp(4)^4+(250*x^5-500*x^4+250*x^3)*exp(4)^3+(1250*x^4-2500*x^3+1250*x^2)*exp(4)^2+(3125*x^3-625
0*x^2+3125*x)*exp(4)+3125*x^2-6250*x+3125)*exp(5)^2+((-2*x^8+4*x^7-2*x^6)*exp(4)^5+(-50*x^7+100*x^6-50*x^5)*ex
p(4)^4+(-500*x^6+1000*x^5-500*x^4)*exp(4)^3+(-2500*x^5+5000*x^4-2500*x^3)*exp(4)^2+(-6250*x^4+12500*x^3-6250*x
^2)*exp(4)-6250*x^3+12500*x^2-6250*x)*exp(5)+(x^9-2*x^8+x^7)*exp(4)^5+(25*x^8-50*x^7+25*x^6)*exp(4)^4+(250*x^7
-500*x^6+250*x^5)*exp(4)^3+(1250*x^6-2500*x^5+1250*x^4)*exp(4)^2+(3125*x^5-6250*x^4+3125*x^3)*exp(4)+3125*x^4-
6250*x^3+3125*x^2),x,method=_RETURNVERBOSE)

[Out]

exp(5)*(-1/(-exp(20)+2*exp(25)-25*exp(16)-exp(30)+50*exp(21)-250*exp(12)-1250*exp(8)+500*exp(17)-25*exp(26)-31
25*exp(4)-250*exp(22)+2500*exp(13)-1250*exp(18)+6250*exp(9)-3125+6250*exp(5)-3125*exp(14)-3125*exp(10))^2*sum(
(-332031250*exp(14)-300000*exp(48)[phone]*exp(29)[phone]*exp(23)-62109375*exp(36)[phone]*exp(35)+164062
50*exp(21)-164062500*exp(32)-322265625*exp(18)[phone]*exp(13)+653125*exp(49)[phone]*exp(27)-64453125*exp(
26)-179687500*exp(22)+3843755*exp(45)-214843750*exp(24)[phone]*exp(15)+3928125*exp(39)-48828125*exp(20)+393
7500*exp(25)[phone]*exp(9)-146484375*exp(10)[phone]*exp(17)[phone]*exp(5)-15656250*exp(30)-2621875*exp(3
4)-20*exp(60)+5*exp(65)-2615625*exp(44)+75000*exp(53)[phone]*exp(37)-15468750*exp(40)[phone]*exp(33)+5625*
exp(57)-263671875*exp(28)-20*exp(50)[phone]*exp(19)+(-56250*exp(29)+375*exp(36)-15625*exp(35)+2500*exp(32)-
3*exp(49)-90*exp(45)+18750*exp(24)-18750*exp(39)+15625*exp(20)-46875*exp(25)+46875*exp(30)+56250*exp(34)+exp(4
4)-7500*exp(37)+30*exp(40)-28125*exp(33)+9375*exp(28)+90*exp(50)+28125*exp(38)-exp(59)+7500*exp(42)-9375*exp(4
3)-375*exp(51)-2500*exp(47)-1125*exp(41)-30*exp(55)+1125*exp(46)+3*exp(54))*_R^5+(18750*exp(48)+2*exp(64)-8531
25*exp(29)+1500*exp(36)-593750*exp(35)-1171875*exp(21)+14375*exp(32)-8*exp(49)+1171875*exp(26)-315*exp(45)+271
875*exp(24)-384375*exp(39)+500000*exp(20)+390625*exp(16)-1531250*exp(25)+1593750*exp(30)+928125*exp(34)+60*exp
(60)+37502*exp(44)-48125*exp(37)+31335*exp(40)-262500*exp(33)+81250*exp(28)+435*exp(50)+300000*exp(38)-8*exp(5
9)+750*exp(56)+58125*exp(42)-137500*exp(43)-3750*exp(51)-29375*exp(47)+5000*exp(52)-5250*exp(41)-265*exp(55)+6
750*exp(46)-390625*exp(31)+12*exp(54))*_R^4+(190625*exp(48)+7*exp(64)-3993750*exp(29)+784125*exp(36)-6375000*e
xp(35)-17187500*exp(21)+33750*exp(32)-18763*exp(49)-3906250*exp(27)[phone]*exp(26)[phone]*exp(22)+3906250*
exp(12)-16215*exp(45)+1150000*exp(24)-2837500*exp(39)+3328125*exp(20)+5468750*exp(16)-10984375*exp(25)-1171875
0*exp(17)[phone]*exp(30)+5100000*exp(34)+260*exp(60)-30*exp(65)+600003*exp(44)-9375*exp(53)-130000*exp(37)+1
047015*exp(40)-903125*exp(33)-2500*exp(57)+246875*exp(28)+960*exp(50)-exp(69)+1237500*exp(38)-18*exp(59)-375*e
xp(61)+4125*exp(56)+190000*exp(42)-762500*exp(43)-13000*exp(51)-127500*exp(47)+36250*exp(52)-11625*exp(41)-740
*exp(55)+18000*exp(46)-7812500*exp(31)+22*exp(54))*_R^3+(737500*exp(48)+12*exp(64)-9790625*exp(29)-19531250*ex
p(23)[phone]*exp(36)-30750000*exp(35)-77734375*exp(21)+7866875*exp(32)[phone]*exp(18)-58593750*exp(13)-271
893*exp(49)-54687500*exp(27)[phone]*exp(26)[phone]*exp(22)[phone]*exp(12)-500865*exp(45)+2496875*exp(2
4)-10212500*exp(39)+9281250*exp(20)[phone]*exp(8)[phone]*exp(16)-34500000*exp(25)-101562500*exp(17)+483125
00*exp(30)[phone]*exp(34)+535*exp(60)-85*exp(65)+3115629*exp(44)-81250*exp(53)-230625*exp(37)+8156445*exp(40
)-1843750*exp(33)-14375*exp(57)+450000*exp(28)+1510*exp(50)-2*exp(69)+2912500*exp(38)-28*exp(59)-1500*exp(61)+
10250*exp(56)+380000*exp(42)-2175000*exp(43)-26000*exp(51)-300000*exp(47)+110625*exp(52)-409125*exp(41)-1290*e
xp(55)+31500*exp(46)-56250000*exp(31)+32*exp(54))*_R^2+(-146484375*exp(14)+984375*exp(48)+12*exp(64)-1056250*e
xp(29)[phone]*exp(23)[phone]*exp(36)-16453125*exp(35)+2343750*exp(21)-13671875*exp(32)-175781250*exp(18)+
58593750*exp(13)-1150003*exp(49)[phone]*exp(27)-1562500*exp(26)-72265625*exp(22)-3328265*exp(45)-48828125*ex
p(24)-6618750*exp(39)-2156250*exp(25)[phone]*exp(9)[phone]*exp(17)+9796875*exp(30)+4318750*exp(34)+560*exp
(60)-140*exp(65)+4506250*exp(44)-246875*exp(53)-3940000*exp(37)[phone]*exp(40)-243750*exp(33)-33750*exp(57)-
58593750*exp(28)+560*exp(50)-3*exp(69)[phone]*exp(19)+978125*exp(38)-18*exp(59)-2875*exp(61)+11500*exp(56)+
135000*exp(42)-1471875*exp(43)-17250*exp(51)-202500*exp(47)+135000*exp(52)-5471625*exp(41)-840*exp(55)+11500*e
xp(46)-9375000*exp(31)+12*exp(54))*_R-300000*exp(38)+250*exp(61)-1000*exp(56)-22500*exp(42)+450000*exp(43)+150
0*exp(51)+33750*exp(47)-22500*exp(52)[phone]*exp(41)+30*exp(55)-1000*exp(46)[phone]*exp(31))/(-7*exp(20)*_
R^6+12*_R^5*exp(25)-150*exp(16)*_R^5-5*_R^4*exp(30)+250*_R^4*exp(21)-1250*exp(12)*_R^4-5000*_R^3*exp(8)+2000*_
R^3*exp(17)-100*_R^3*exp(26)-9375*_R^2*exp(4)-750*_R^2*exp(22)+7500*_R^2*exp(13)-2500*exp(18)*_R+12500*_R*exp(
9)+6250*exp(5)-3125*exp(14)-6250*_R)*ln(x-_R),_R=RootOf(exp(20)*_Z^7-(2*exp(25)-25*exp(16))*_Z^6-(-exp(30)+50*
exp(21)-250*exp(12))*_Z^5-(-1250*exp(8)+500*exp(17)-25*exp(26))*_Z^4-(-3125*exp(4)-250*exp(22)+2500*exp(13))*_
Z^3-(-1250*exp(18)+6250*exp(9)-3125)*_Z^2-(6250*exp(5)-3125*exp(14))*_Z+3125*exp(10)))-1/(-exp(20)+2*exp(25)-2
5*exp(16)-exp(30)+50*exp(21)-250*exp(12)-1250*exp(8)+500*exp(17)-25*exp(26)-3125*exp(4)-250*exp(22)+2500*exp(1
3)-1250*exp(18)+6250*exp(9)-3125+6250*exp(5)-3125*exp(14)-3125*exp(10))^2*(-15625-56250*exp(14)+3*exp(29)+9375
*exp(23)+30*exp(35)+1125*exp(21)-28125*exp(18)+28125*exp(13)+2500*exp(27)-1125*exp(26)-7500*exp(22)-2500*exp(1
2)-exp(24)+15625*exp(15)+exp(39)-30*exp(20)-937...

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (27) = 54\).
time = 0.63, size = 104, normalized size = 3.47 \begin {gather*} \frac {x e^{5}}{x^{6} e^{16} - x^{5} {\left (e^{21} + e^{16} - 20 \, e^{12}\right )} + x^{4} {\left (e^{21} - 20 \, e^{17} - 20 \, e^{12} + 150 \, e^{8}\right )} + 10 \, x^{3} {\left (2 \, e^{17} - 15 \, e^{13} - 15 \, e^{8} + 50 \, e^{4}\right )} + 25 \, x^{2} {\left (6 \, e^{13} - 20 \, e^{9} - 20 \, e^{4} + 25\right )} + 125 \, x {\left (4 \, e^{9} - 5 \, e^{5} - 5\right )} + 625 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2-3*x)*exp(4)+5)*exp(5)^2+((-5*x^3+4*x^2)*exp(4)-5*x^2)*exp(5))/(((x^7-2*x^6+x^5)*exp(4)^5+(2
5*x^6-50*x^5+25*x^4)*exp(4)^4+(250*x^5-500*x^4+250*x^3)*exp(4)^3+(1250*x^4-2500*x^3+1250*x^2)*exp(4)^2+(3125*x
^3-6250*x^2+3125*x)*exp(4)+3125*x^2-6250*x+3125)*exp(5)^2+((-2*x^8+4*x^7-2*x^6)*exp(4)^5+(-50*x^7+100*x^6-50*x
^5)*exp(4)^4+(-500*x^6+1000*x^5-500*x^4)*exp(4)^3+(-2500*x^5+5000*x^4-2500*x^3)*exp(4)^2+(-6250*x^4+12500*x^3-
6250*x^2)*exp(4)-6250*x^3+12500*x^2-6250*x)*exp(5)+(x^9-2*x^8+x^7)*exp(4)^5+(25*x^8-50*x^7+25*x^6)*exp(4)^4+(2
50*x^7-500*x^6+250*x^5)*exp(4)^3+(1250*x^6-2500*x^5+1250*x^4)*exp(4)^2+(3125*x^5-6250*x^4+3125*x^3)*exp(4)+312
5*x^4-6250*x^3+3125*x^2),x, algorithm="maxima")

[Out]

x*e^5/(x^6*e^16 - x^5*(e^21 + e^16 - 20*e^12) + x^4*(e^21 - 20*e^17 - 20*e^12 + 150*e^8) + 10*x^3*(2*e^17 - 15
*e^13 - 15*e^8 + 50*e^4) + 25*x^2*(6*e^13 - 20*e^9 - 20*e^4 + 25) + 125*x*(4*e^9 - 5*e^5 - 5) + 625*e^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (27) = 54\).
time = 0.39, size = 123, normalized size = 4.10 \begin {gather*} \frac {x e^{5}}{625 \, x^{2} - {\left (x^{5} - x^{4}\right )} e^{21} - 20 \, {\left (x^{4} - x^{3}\right )} e^{17} + {\left (x^{6} - x^{5}\right )} e^{16} - 150 \, {\left (x^{3} - x^{2}\right )} e^{13} + 20 \, {\left (x^{5} - x^{4}\right )} e^{12} - 500 \, {\left (x^{2} - x\right )} e^{9} + 150 \, {\left (x^{4} - x^{3}\right )} e^{8} - 625 \, {\left (x - 1\right )} e^{5} + 500 \, {\left (x^{3} - x^{2}\right )} e^{4} - 625 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2-3*x)*exp(4)+5)*exp(5)^2+((-5*x^3+4*x^2)*exp(4)-5*x^2)*exp(5))/(((x^7-2*x^6+x^5)*exp(4)^5+(2
5*x^6-50*x^5+25*x^4)*exp(4)^4+(250*x^5-500*x^4+250*x^3)*exp(4)^3+(1250*x^4-2500*x^3+1250*x^2)*exp(4)^2+(3125*x
^3-6250*x^2+3125*x)*exp(4)+3125*x^2-6250*x+3125)*exp(5)^2+((-2*x^8+4*x^7-2*x^6)*exp(4)^5+(-50*x^7+100*x^6-50*x
^5)*exp(4)^4+(-500*x^6+1000*x^5-500*x^4)*exp(4)^3+(-2500*x^5+5000*x^4-2500*x^3)*exp(4)^2+(-6250*x^4+12500*x^3-
6250*x^2)*exp(4)-6250*x^3+12500*x^2-6250*x)*exp(5)+(x^9-2*x^8+x^7)*exp(4)^5+(25*x^8-50*x^7+25*x^6)*exp(4)^4+(2
50*x^7-500*x^6+250*x^5)*exp(4)^3+(1250*x^6-2500*x^5+1250*x^4)*exp(4)^2+(3125*x^5-6250*x^4+3125*x^3)*exp(4)+312
5*x^4-6250*x^3+3125*x^2),x, algorithm="fricas")

[Out]

x*e^5/(625*x^2 - (x^5 - x^4)*e^21 - 20*(x^4 - x^3)*e^17 + (x^6 - x^5)*e^16 - 150*(x^3 - x^2)*e^13 + 20*(x^5 -
x^4)*e^12 - 500*(x^2 - x)*e^9 + 150*(x^4 - x^3)*e^8 - 625*(x - 1)*e^5 + 500*(x^3 - x^2)*e^4 - 625*x)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (20) = 40\).
time = 14.84, size = 110, normalized size = 3.67 \begin {gather*} \frac {x e^{5}}{x^{6} e^{16} + x^{5} \left (- e^{21} - e^{16} + 20 e^{12}\right ) + x^{4} \left (- 20 e^{17} - 20 e^{12} + 150 e^{8} + e^{21}\right ) + x^{3} \left (- 150 e^{13} - 150 e^{8} + 500 e^{4} + 20 e^{17}\right ) + x^{2} \left (- 500 e^{9} - 500 e^{4} + 625 + 150 e^{13}\right ) + x \left (- 625 e^{5} - 625 + 500 e^{9}\right ) + 625 e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x**2-3*x)*exp(4)+5)*exp(5)**2+((-5*x**3+4*x**2)*exp(4)-5*x**2)*exp(5))/(((x**7-2*x**6+x**5)*exp
(4)**5+(25*x**6-50*x**5+25*x**4)*exp(4)**4+(250*x**5-500*x**4+250*x**3)*exp(4)**3+(1250*x**4-2500*x**3+1250*x*
*2)*exp(4)**2+(3125*x**3-6250*x**2+3125*x)*exp(4)+3125*x**2-6250*x+3125)*exp(5)**2+((-2*x**8+4*x**7-2*x**6)*ex
p(4)**5+(-50*x**7+100*x**6-50*x**5)*exp(4)**4+(-500*x**6+1000*x**5-500*x**4)*exp(4)**3+(-2500*x**5+5000*x**4-2
500*x**3)*exp(4)**2+(-6250*x**4+12500*x**3-6250*x**2)*exp(4)-6250*x**3+12500*x**2-6250*x)*exp(5)+(x**9-2*x**8+
x**7)*exp(4)**5+(25*x**8-50*x**7+25*x**6)*exp(4)**4+(250*x**7-500*x**6+250*x**5)*exp(4)**3+(1250*x**6-2500*x**
5+1250*x**4)*exp(4)**2+(3125*x**5-6250*x**4+3125*x**3)*exp(4)+3125*x**4-6250*x**3+3125*x**2),x)

[Out]

x*exp(5)/(x**6*exp(16) + x**5*(-exp(21) - exp(16) + 20*exp(12)) + x**4*(-20*exp(17) - 20*exp(12) + 150*exp(8)
+ exp(21)) + x**3*(-150*exp(13) - 150*exp(8) + 500*exp(4) + 20*exp(17)) + x**2*(-500*exp(9) - 500*exp(4) + 625
+ 150*exp(13)) + x*(-625*exp(5) - 625 + 500*exp(9)) + 625*exp(5))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (27) = 54\).
time = 0.56, size = 528, normalized size = 17.60 \begin {gather*} -\frac {x e^{36} + x e^{31} + 20 \, x e^{27} + x e^{26} + 20 \, x e^{22} + 150 \, x e^{18} - 625 \, x e^{5} - e^{41} - e^{36} - 20 \, e^{32} - e^{31} - 20 \, e^{27} - e^{26} - 150 \, e^{23} - 20 \, e^{22} - 150 \, e^{18} - 500 \, e^{14}}{{\left (x^{2} - x e^{5} - x + e^{5}\right )} {\left (e^{52} + 20 \, e^{48} + 150 \, e^{44} + 20 \, e^{43} + 500 \, e^{40} + 400 \, e^{39} + 625 \, e^{36} + 3000 \, e^{35} + 150 \, e^{34} + 10000 \, e^{31} + 3000 \, e^{30} + 12500 \, e^{27} + 22500 \, e^{26} + 500 \, e^{25} + 75000 \, e^{22} + 10000 \, e^{21} + 93750 \, e^{18} + 75000 \, e^{17} + 625 \, e^{16} + 250000 \, e^{13} + 12500 \, e^{12} + 312500 \, e^{9} + 93750 \, e^{8} + 312500 \, e^{4} + 390625\right )}} + \frac {x^{3} e^{52} + x^{3} e^{47} + 20 \, x^{3} e^{43} + x^{3} e^{42} + 20 \, x^{3} e^{38} + 150 \, x^{3} e^{34} - 625 \, x^{3} e^{21} + x^{2} e^{52} + 20 \, x^{2} e^{48} + x^{2} e^{47} + 40 \, x^{2} e^{43} + 400 \, x^{2} e^{39} + 20 \, x^{2} e^{38} + 400 \, x^{2} e^{34} + 2500 \, x^{2} e^{30} - 625 \, x^{2} e^{26} - 625 \, x^{2} e^{21} - 12500 \, x^{2} e^{17} + x e^{52} + 20 \, x e^{48} + 150 \, x e^{44} + 20 \, x e^{43} + 400 \, x e^{39} + 2500 \, x e^{35} + 150 \, x e^{34} - 625 \, x e^{31} + 2500 \, x e^{30} + 11875 \, x e^{26} - 12500 \, x e^{22} - 625 \, x e^{21} - 12500 \, x e^{17} - 93750 \, x e^{13} - 625 \, e^{36} - 625 \, e^{31} - 12500 \, e^{27} - 625 \, e^{26} - 12500 \, e^{22} - 625 \, e^{21} - 93750 \, e^{18} - 12500 \, e^{17} - 93750 \, e^{13} - 312500 \, e^{9}}{{\left (x e^{4} + 5\right )}^{4} {\left (e^{52} + 20 \, e^{48} + 150 \, e^{44} + 20 \, e^{43} + 500 \, e^{40} + 400 \, e^{39} + 625 \, e^{36} + 3000 \, e^{35} + 150 \, e^{34} + 10000 \, e^{31} + 3000 \, e^{30} + 12500 \, e^{27} + 22500 \, e^{26} + 500 \, e^{25} + 75000 \, e^{22} + 10000 \, e^{21} + 93750 \, e^{18} + 75000 \, e^{17} + 625 \, e^{16} + 250000 \, e^{13} + 12500 \, e^{12} + 312500 \, e^{9} + 93750 \, e^{8} + 312500 \, e^{4} + 390625\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2-3*x)*exp(4)+5)*exp(5)^2+((-5*x^3+4*x^2)*exp(4)-5*x^2)*exp(5))/(((x^7-2*x^6+x^5)*exp(4)^5+(2
5*x^6-50*x^5+25*x^4)*exp(4)^4+(250*x^5-500*x^4+250*x^3)*exp(4)^3+(1250*x^4-2500*x^3+1250*x^2)*exp(4)^2+(3125*x
^3-6250*x^2+3125*x)*exp(4)+3125*x^2-6250*x+3125)*exp(5)^2+((-2*x^8+4*x^7-2*x^6)*exp(4)^5+(-50*x^7+100*x^6-50*x
^5)*exp(4)^4+(-500*x^6+1000*x^5-500*x^4)*exp(4)^3+(-2500*x^5+5000*x^4-2500*x^3)*exp(4)^2+(-6250*x^4+12500*x^3-
6250*x^2)*exp(4)-6250*x^3+12500*x^2-6250*x)*exp(5)+(x^9-2*x^8+x^7)*exp(4)^5+(25*x^8-50*x^7+25*x^6)*exp(4)^4+(2
50*x^7-500*x^6+250*x^5)*exp(4)^3+(1250*x^6-2500*x^5+1250*x^4)*exp(4)^2+(3125*x^5-6250*x^4+3125*x^3)*exp(4)+312
5*x^4-6250*x^3+3125*x^2),x, algorithm="giac")

[Out]

-(x*e^36 + x*e^31 + 20*x*e^27 + x*e^26 + 20*x*e^22 + 150*x*e^18 - 625*x*e^5 - e^41 - e^36 - 20*e^32 - e^31 - 2
0*e^27 - e^26 - 150*e^23 - 20*e^22 - 150*e^18 - 500*e^14)/((x^2 - x*e^5 - x + e^5)*(e^52 + 20*e^48 + 150*e^44
+ 20*e^43 + 500*e^40 + 400*e^39 + 625*e^36 + 3000*e^35 + 150*e^34 + 10000*e^31 + 3000*e^30 + 12500*e^27 + 2250
0*e^26 + 500*e^25 + 75000*e^22 + 10000*e^21 + 93750*e^18 + 75000*e^17 + 625*e^16 + 250000*e^13 + 12500*e^12 +
312500*e^9 + 93750*e^8 + 312500*e^4 + 390625)) + (x^3*e^52 + x^3*e^47 + 20*x^3*e^43 + x^3*e^42 + 20*x^3*e^38 +
150*x^3*e^34 - 625*x^3*e^21 + x^2*e^52 + 20*x^2*e^48 + x^2*e^47 + 40*x^2*e^43 + 400*x^2*e^39 + 20*x^2*e^38 +
400*x^2*e^34 + 2500*x^2*e^30 - 625*x^2*e^26 - 625*x^2*e^21 - 12500*x^2*e^17 + x*e^52 + 20*x*e^48 + 150*x*e^44
+ 20*x*e^43 + 400*x*e^39 + 2500*x*e^35 + 150*x*e^34 - 625*x*e^31 + 2500*x*e^30 + 11875*x*e^26 - 12500*x*e^22 -
625*x*e^21 - 12500*x*e^17 - 93750*x*e^13 - 625*e^36 - 625*e^31 - 12500*e^27 - 625*e^26 - 12500*e^22 - 625*e^2
1 - 93750*e^18 - 12500*e^17 - 93750*e^13 - 312500*e^9)/((x*e^4 + 5)^4*(e^52 + 20*e^48 + 150*e^44 + 20*e^43 + 5
00*e^40 + 400*e^39 + 625*e^36 + 3000*e^35 + 150*e^34 + 10000*e^31 + 3000*e^30 + 12500*e^27 + 22500*e^26 + 500*
e^25 + 75000*e^22 + 10000*e^21 + 93750*e^18 + 75000*e^17 + 625*e^16 + 250000*e^13 + 12500*e^12 + 312500*e^9 +
93750*e^8 + 312500*e^4 + 390625))

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Mupad [B]
time = 0.99, size = 25, normalized size = 0.83 \begin {gather*} \frac {x\,{\mathrm {e}}^5}{\left (x-{\mathrm {e}}^5\right )\,{\left (x\,{\mathrm {e}}^4+5\right )}^4\,\left (x-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(10)*(exp(4)*(3*x - 4*x^2) - 5) - exp(5)*(exp(4)*(4*x^2 - 5*x^3) - 5*x^2))/(exp(10)*(exp(20)*(x^5 - 2
*x^6 + x^7) - 6250*x + exp(4)*(3125*x - 6250*x^2 + 3125*x^3) + exp(16)*(25*x^4 - 50*x^5 + 25*x^6) + exp(12)*(2
50*x^3 - 500*x^4 + 250*x^5) + exp(8)*(1250*x^2 - 2500*x^3 + 1250*x^4) + 3125*x^2 + 3125) + exp(20)*(x^7 - 2*x^
8 + x^9) - exp(5)*(6250*x + exp(20)*(2*x^6 - 4*x^7 + 2*x^8) + exp(16)*(50*x^5 - 100*x^6 + 50*x^7) + exp(12)*(5
00*x^4 - 1000*x^5 + 500*x^6) + exp(8)*(2500*x^3 - 5000*x^4 + 2500*x^5) + exp(4)*(6250*x^2 - 12500*x^3 + 6250*x
^4) - 12500*x^2 + 6250*x^3) + exp(16)*(25*x^6 - 50*x^7 + 25*x^8) + exp(12)*(250*x^5 - 500*x^6 + 250*x^7) + exp
(8)*(1250*x^4 - 2500*x^5 + 1250*x^6) + exp(4)*(3125*x^3 - 6250*x^4 + 3125*x^5) + 3125*x^2 - 6250*x^3 + 3125*x^
4),x)

[Out]

(x*exp(5))/((x - exp(5))*(x*exp(4) + 5)^4*(x - 1))

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