Integrand size = 178, antiderivative size = 28 \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=\frac {x^2}{\left (-2 x-\frac {6 x}{5 e^9 \left (e+e^3\right )}+x^2\right )^2} \] Output:
x^2/(x^2-2*x-6/5*x/(exp(1)+exp(3))/exp(9))^2
Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=\frac {25 e^{20} \left (1+e^2\right )^2}{\left (-6+5 e^{10} (-2+x)+5 e^{12} (-2+x)\right )^2} \] Input:
Integrate[(E^27*(-250*E^3 - 750*E^5 - 750*E^7 - 250*E^9))/(-216 + E^9*(E*( -1080 + 540*x) + E^3*(-1080 + 540*x)) + E^18*(E^4*(-3600 + 3600*x - 900*x^ 2) + E^2*(-1800 + 1800*x - 450*x^2) + E^6*(-1800 + 1800*x - 450*x^2)) + E^ 27*(E^3*(-1000 + 1500*x - 750*x^2 + 125*x^3) + E^9*(-1000 + 1500*x - 750*x ^2 + 125*x^3) + E^5*(-3000 + 4500*x - 2250*x^2 + 375*x^3) + E^7*(-3000 + 4 500*x - 2250*x^2 + 375*x^3))),x]
Output:
(25*E^20*(1 + E^2)^2)/(-6 + 5*E^10*(-2 + x) + 5*E^12*(-2 + x))^2
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {27, 2007, 17}
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.
| \(\displaystyle \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{e^{18} \left (e^4 \left (-900 x^2+3600 x-3600\right )+e^6 \left (-450 x^2+1800 x-1800\right )+e^2 \left (-450 x^2+1800 x-1800\right )\right )+e^{27} \left (e^9 \left (125 x^3-750 x^2+1500 x-1000\right )+e^3 \left (125 x^3-750 x^2+1500 x-1000\right )+e^7 \left (375 x^3-2250 x^2+4500 x-3000\right )+e^5 \left (375 x^3-2250 x^2+4500 x-3000\right )\right )+e^9 \left (e^3 (540 x-1080)+e (540 x-1080)\right )-216} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -250 e^{30} \left (1+e^2\right )^3 \int \frac {1}{-540 \left (e^{12} (2-x)+e^{10} (2-x)\right )-450 \left (e^{24} \left (x^2-4 x+4\right )+2 e^{22} \left (x^2-4 x+4\right )+e^{20} \left (x^2-4 x+4\right )\right )-125 \left (e^{36} \left (-x^3+6 x^2-12 x+8\right )+3 e^{34} \left (-x^3+6 x^2-12 x+8\right )+3 e^{32} \left (-x^3+6 x^2-12 x+8\right )+e^{30} \left (-x^3+6 x^2-12 x+8\right )\right )-216}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle -250 e^{30} \left (1+e^2\right )^3 \int \frac {1}{\left (5 e^{10} \left (1+e^2\right ) x-2 \left (3+5 e^{10}+5 e^{12}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {25 e^{20} \left (1+e^2\right )^2}{\left (2 \left (3+5 e^{10}+5 e^{12}\right )-5 e^{10} \left (1+e^2\right ) x\right )^2}\) |
Input:
Int[(E^27*(-250*E^3 - 750*E^5 - 750*E^7 - 250*E^9))/(-216 + E^9*(E*(-1080 + 540*x) + E^3*(-1080 + 540*x)) + E^18*(E^4*(-3600 + 3600*x - 900*x^2) + E ^2*(-1800 + 1800*x - 450*x^2) + E^6*(-1800 + 1800*x - 450*x^2)) + E^27*(E^ 3*(-1000 + 1500*x - 750*x^2 + 125*x^3) + E^9*(-1000 + 1500*x - 750*x^2 + 1 25*x^3) + E^5*(-3000 + 4500*x - 2250*x^2 + 375*x^3) + E^7*(-3000 + 4500*x - 2250*x^2 + 375*x^3))),x]
Output:
(25*E^20*(1 + E^2)^2)/(2*(3 + 5*E^10 + 5*E^12) - 5*E^10*(1 + E^2)*x)^2
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
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[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Time = 0.57 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86
| method | result | size |
| norman | \(\frac {25 \,{\mathrm e}^{18} \left ({\mathrm e}^{6}+2 \,{\mathrm e} \,{\mathrm e}^{3}+{\mathrm e}^{2}\right )}{\left (5 \,{\mathrm e}^{3} {\mathrm e}^{9} x +5 \,{\mathrm e} \,{\mathrm e}^{9} x -10 \,{\mathrm e}^{3} {\mathrm e}^{9}-10 \,{\mathrm e} \,{\mathrm e}^{9}-6\right )^{2}}\) | \(52\) |
| gosper | \(\frac {25 \,{\mathrm e}^{18} \left ({\mathrm e}^{6}+2 \,{\mathrm e} \,{\mathrm e}^{3}+{\mathrm e}^{2}\right )}{25 \,{\mathrm e}^{6} {\mathrm e}^{18} x^{2}+50 \,{\mathrm e}^{3} {\mathrm e} \,{\mathrm e}^{18} x^{2}+25 \,{\mathrm e}^{2} {\mathrm e}^{18} x^{2}-100 \,{\mathrm e}^{6} {\mathrm e}^{18} x -200 \,{\mathrm e}^{3} {\mathrm e} \,{\mathrm e}^{18} x -100 \,{\mathrm e}^{2} {\mathrm e}^{18} x +100 \,{\mathrm e}^{6} {\mathrm e}^{18}+200 \,{\mathrm e}^{3} {\mathrm e} \,{\mathrm e}^{18}+100 \,{\mathrm e}^{2} {\mathrm e}^{18}-60 \,{\mathrm e}^{3} {\mathrm e}^{9} x -60 \,{\mathrm e} \,{\mathrm e}^{9} x +120 \,{\mathrm e}^{3} {\mathrm e}^{9}+120 \,{\mathrm e} \,{\mathrm e}^{9}+36}\) | \(154\) |
| parallelrisch | \(\frac {\left (-250 \,{\mathrm e}^{9}-750 \,{\mathrm e} \,{\mathrm e}^{6}-750 \,{\mathrm e}^{2} {\mathrm e}^{3}-250 \,{\mathrm e}^{3}\right ) {\mathrm e}^{9} \left (-5 \,{\mathrm e}^{3} {\mathrm e}^{9}-5 \,{\mathrm e} \,{\mathrm e}^{9}\right )}{50 \left ({\mathrm e}+{\mathrm e}^{3}\right )^{2} \left (25 \,{\mathrm e}^{6} {\mathrm e}^{18} x^{2}+50 \,{\mathrm e}^{3} {\mathrm e} \,{\mathrm e}^{18} x^{2}+25 \,{\mathrm e}^{2} {\mathrm e}^{18} x^{2}-100 \,{\mathrm e}^{6} {\mathrm e}^{18} x -200 \,{\mathrm e}^{3} {\mathrm e} \,{\mathrm e}^{18} x -100 \,{\mathrm e}^{2} {\mathrm e}^{18} x +100 \,{\mathrm e}^{6} {\mathrm e}^{18}+200 \,{\mathrm e}^{3} {\mathrm e} \,{\mathrm e}^{18}+100 \,{\mathrm e}^{2} {\mathrm e}^{18}-60 \,{\mathrm e}^{3} {\mathrm e}^{9} x -60 \,{\mathrm e} \,{\mathrm e}^{9} x +120 \,{\mathrm e}^{3} {\mathrm e}^{9}+120 \,{\mathrm e} \,{\mathrm e}^{9}+36\right )}\) | \(186\) |
| default | \(\frac {50 \left (-{\mathrm e}^{9}-3 \,{\mathrm e} \,{\mathrm e}^{6}-3 \,{\mathrm e}^{2} {\mathrm e}^{3}-{\mathrm e}^{3}\right ) {\mathrm e}^{27} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (125 \,{\mathrm e}^{30}+125 \,{\mathrm e}^{36}+375 \,{\mathrm e}^{34}+375 \,{\mathrm e}^{32}\right ) \textit {\_Z}^{3}+\left (-450 \,{\mathrm e}^{20}-750 \,{\mathrm e}^{30}-750 \,{\mathrm e}^{36}-2250 \,{\mathrm e}^{34}-2250 \,{\mathrm e}^{32}-450 \,{\mathrm e}^{24}-900 \,{\mathrm e}^{22}\right ) \textit {\_Z}^{2}+\left (1800 \,{\mathrm e}^{20}+4500 \,{\mathrm e}^{32}+540 \,{\mathrm e}^{10}+1500 \,{\mathrm e}^{36}+1500 \,{\mathrm e}^{30}+4500 \,{\mathrm e}^{34}+1800 \,{\mathrm e}^{24}+540 \,{\mathrm e}^{12}+3600 \,{\mathrm e}^{22}\right ) \textit {\_Z} -216-1800 \,{\mathrm e}^{24}-1080 \,{\mathrm e}^{10}-1000 \,{\mathrm e}^{36}-3000 \,{\mathrm e}^{32}-1800 \,{\mathrm e}^{20}-1080 \,{\mathrm e}^{12}-3000 \,{\mathrm e}^{34}-1000 \,{\mathrm e}^{30}-3600 \,{\mathrm e}^{22}\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{120 \,{\mathrm e}^{20}+300 \,{\mathrm e}^{32}-60 \textit {\_R} \,{\mathrm e}^{20}+36 \,{\mathrm e}^{10}-100 \textit {\_R} \,{\mathrm e}^{30}+100 \,{\mathrm e}^{36}-100 \textit {\_R} \,{\mathrm e}^{36}+100 \,{\mathrm e}^{30}+25 \textit {\_R}^{2} {\mathrm e}^{30}+25 \textit {\_R}^{2} {\mathrm e}^{36}-300 \textit {\_R} \,{\mathrm e}^{34}-300 \,{\mathrm e}^{32} \textit {\_R} +300 \,{\mathrm e}^{34}+75 \textit {\_R}^{2} {\mathrm e}^{34}+75 \textit {\_R}^{2} {\mathrm e}^{32}+120 \,{\mathrm e}^{24}-60 \,{\mathrm e}^{24} \textit {\_R} +36 \,{\mathrm e}^{12}-120 \textit {\_R} \,{\mathrm e}^{22}+240 \,{\mathrm e}^{22}}\right )}{3}\) | \(280\) |
| risch | \(\frac {{\mathrm e}^{17} {\mathrm e}^{9}}{\left ({\mathrm e}^{2}+1\right ) \left ({\mathrm e}^{24} x^{2}-4 \,{\mathrm e}^{24} x +4 \,{\mathrm e}^{24}+2 x^{2} {\mathrm e}^{22}-8 x \,{\mathrm e}^{22}+8 \,{\mathrm e}^{22}+{\mathrm e}^{20} x^{2}-4 x \,{\mathrm e}^{20}+4 \,{\mathrm e}^{20}-\frac {12 x \,{\mathrm e}^{12}}{5}+\frac {24 \,{\mathrm e}^{12}}{5}-\frac {12 x \,{\mathrm e}^{10}}{5}+\frac {24 \,{\mathrm e}^{10}}{5}+\frac {36}{25}\right )}+\frac {3 \,{\mathrm e}^{17} {\mathrm e}^{7}}{\left ({\mathrm e}^{2}+1\right ) \left ({\mathrm e}^{24} x^{2}-4 \,{\mathrm e}^{24} x +4 \,{\mathrm e}^{24}+2 x^{2} {\mathrm e}^{22}-8 x \,{\mathrm e}^{22}+8 \,{\mathrm e}^{22}+{\mathrm e}^{20} x^{2}-4 x \,{\mathrm e}^{20}+4 \,{\mathrm e}^{20}-\frac {12 x \,{\mathrm e}^{12}}{5}+\frac {24 \,{\mathrm e}^{12}}{5}-\frac {12 x \,{\mathrm e}^{10}}{5}+\frac {24 \,{\mathrm e}^{10}}{5}+\frac {36}{25}\right )}+\frac {3 \,{\mathrm e}^{17} {\mathrm e}^{5}}{\left ({\mathrm e}^{2}+1\right ) \left ({\mathrm e}^{24} x^{2}-4 \,{\mathrm e}^{24} x +4 \,{\mathrm e}^{24}+2 x^{2} {\mathrm e}^{22}-8 x \,{\mathrm e}^{22}+8 \,{\mathrm e}^{22}+{\mathrm e}^{20} x^{2}-4 x \,{\mathrm e}^{20}+4 \,{\mathrm e}^{20}-\frac {12 x \,{\mathrm e}^{12}}{5}+\frac {24 \,{\mathrm e}^{12}}{5}-\frac {12 x \,{\mathrm e}^{10}}{5}+\frac {24 \,{\mathrm e}^{10}}{5}+\frac {36}{25}\right )}+\frac {{\mathrm e}^{17} {\mathrm e}^{3}}{\left ({\mathrm e}^{2}+1\right ) \left ({\mathrm e}^{24} x^{2}-4 \,{\mathrm e}^{24} x +4 \,{\mathrm e}^{24}+2 x^{2} {\mathrm e}^{22}-8 x \,{\mathrm e}^{22}+8 \,{\mathrm e}^{22}+{\mathrm e}^{20} x^{2}-4 x \,{\mathrm e}^{20}+4 \,{\mathrm e}^{20}-\frac {12 x \,{\mathrm e}^{12}}{5}+\frac {24 \,{\mathrm e}^{12}}{5}-\frac {12 x \,{\mathrm e}^{10}}{5}+\frac {24 \,{\mathrm e}^{10}}{5}+\frac {36}{25}\right )}\) | \(320\) |
Input:
int((-250*exp(3)^3-750*exp(1)*exp(3)^2-750*exp(1)^2*exp(3)-250*exp(1)^3)*e xp(9)^3/(((125*x^3-750*x^2+1500*x-1000)*exp(3)^3+(375*x^3-2250*x^2+4500*x- 3000)*exp(1)*exp(3)^2+(375*x^3-2250*x^2+4500*x-3000)*exp(1)^2*exp(3)+(125* x^3-750*x^2+1500*x-1000)*exp(1)^3)*exp(9)^3+((-450*x^2+1800*x-1800)*exp(3) ^2+(-900*x^2+3600*x-3600)*exp(1)*exp(3)+(-450*x^2+1800*x-1800)*exp(1)^2)*e xp(9)^2+((540*x-1080)*exp(3)+(540*x-1080)*exp(1))*exp(9)-216),x,method=_RE TURNVERBOSE)
Output:
25*exp(9)^2*(exp(3)^2+2*exp(1)*exp(3)+exp(1)^2)/(5*exp(3)*exp(9)*x+5*exp(1 )*exp(9)*x-10*exp(3)*exp(9)-10*exp(1)*exp(9)-6)^2
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (28) = 56\).
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32 \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=\frac {25 \, {\left (e^{24} + 2 \, e^{22} + e^{20}\right )}}{25 \, {\left (x^{2} - 4 \, x + 4\right )} e^{24} + 50 \, {\left (x^{2} - 4 \, x + 4\right )} e^{22} + 25 \, {\left (x^{2} - 4 \, x + 4\right )} e^{20} - 60 \, {\left (x - 2\right )} e^{12} - 60 \, {\left (x - 2\right )} e^{10} + 36} \] Input:
integrate((-250*exp(3)^3-750*exp(1)*exp(3)^2-750*exp(1)^2*exp(3)-250*exp(1 )^3)*exp(9)^3/(((125*x^3-750*x^2+1500*x-1000)*exp(3)^3+(375*x^3-2250*x^2+4 500*x-3000)*exp(1)*exp(3)^2+(375*x^3-2250*x^2+4500*x-3000)*exp(1)^2*exp(3) +(125*x^3-750*x^2+1500*x-1000)*exp(1)^3)*exp(9)^3+((-450*x^2+1800*x-1800)* exp(3)^2+(-900*x^2+3600*x-3600)*exp(1)*exp(3)+(-450*x^2+1800*x-1800)*exp(1 )^2)*exp(9)^2+((540*x-1080)*exp(3)+(540*x-1080)*exp(1))*exp(9)-216),x, alg orithm="fricas")
Output:
25*(e^24 + 2*e^22 + e^20)/(25*(x^2 - 4*x + 4)*e^24 + 50*(x^2 - 4*x + 4)*e^ 22 + 25*(x^2 - 4*x + 4)*e^20 - 60*(x - 2)*e^12 - 60*(x - 2)*e^10 + 36)
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (26) = 52\).
Time = 0.66 (sec) , antiderivative size = 131, normalized size of antiderivative = 4.68 \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=- \frac {- 250 e^{36} - 750 e^{34} - 750 e^{32} - 250 e^{30}}{x^{2} \cdot \left (250 e^{30} + 750 e^{32} + 750 e^{34} + 250 e^{36}\right ) + x \left (- 1000 e^{36} - 3000 e^{34} - 3000 e^{32} - 1000 e^{30} - 600 e^{24} - 1200 e^{22} - 600 e^{20}\right ) + 360 e^{10} + 360 e^{12} + 1200 e^{20} + 2400 e^{22} + 1200 e^{24} + 1000 e^{30} + 3000 e^{32} + 3000 e^{34} + 1000 e^{36}} \] Input:
integrate((-250*exp(3)**3-750*exp(1)*exp(3)**2-750*exp(1)**2*exp(3)-250*ex p(1)**3)*exp(9)**3/(((125*x**3-750*x**2+1500*x-1000)*exp(3)**3+(375*x**3-2 250*x**2+4500*x-3000)*exp(1)*exp(3)**2+(375*x**3-2250*x**2+4500*x-3000)*ex p(1)**2*exp(3)+(125*x**3-750*x**2+1500*x-1000)*exp(1)**3)*exp(9)**3+((-450 *x**2+1800*x-1800)*exp(3)**2+(-900*x**2+3600*x-3600)*exp(1)*exp(3)+(-450*x **2+1800*x-1800)*exp(1)**2)*exp(9)**2+((540*x-1080)*exp(3)+(540*x-1080)*ex p(1))*exp(9)-216),x)
Output:
-(-250*exp(36) - 750*exp(34) - 750*exp(32) - 250*exp(30))/(x**2*(250*exp(3 0) + 750*exp(32) + 750*exp(34) + 250*exp(36)) + x*(-1000*exp(36) - 3000*ex p(34) - 3000*exp(32) - 1000*exp(30) - 600*exp(24) - 1200*exp(22) - 600*exp (20)) + 360*exp(10) + 360*exp(12) + 1200*exp(20) + 2400*exp(22) + 1200*exp (24) + 1000*exp(30) + 3000*exp(32) + 3000*exp(34) + 1000*exp(36))
Leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (28) = 56\).
Time = 0.03 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.79 \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=\frac {25 \, {\left (e^{9} + 3 \, e^{7} + 3 \, e^{5} + e^{3}\right )} e^{27}}{25 \, x^{2} {\left (e^{36} + 3 \, e^{34} + 3 \, e^{32} + e^{30}\right )} - 20 \, x {\left (5 \, e^{36} + 15 \, e^{34} + 15 \, e^{32} + 5 \, e^{30} + 3 \, e^{24} + 6 \, e^{22} + 3 \, e^{20}\right )} + 100 \, e^{36} + 300 \, e^{34} + 300 \, e^{32} + 100 \, e^{30} + 120 \, e^{24} + 240 \, e^{22} + 120 \, e^{20} + 36 \, e^{12} + 36 \, e^{10}} \] Input:
integrate((-250*exp(3)^3-750*exp(1)*exp(3)^2-750*exp(1)^2*exp(3)-250*exp(1 )^3)*exp(9)^3/(((125*x^3-750*x^2+1500*x-1000)*exp(3)^3+(375*x^3-2250*x^2+4 500*x-3000)*exp(1)*exp(3)^2+(375*x^3-2250*x^2+4500*x-3000)*exp(1)^2*exp(3) +(125*x^3-750*x^2+1500*x-1000)*exp(1)^3)*exp(9)^3+((-450*x^2+1800*x-1800)* exp(3)^2+(-900*x^2+3600*x-3600)*exp(1)*exp(3)+(-450*x^2+1800*x-1800)*exp(1 )^2)*exp(9)^2+((540*x-1080)*exp(3)+(540*x-1080)*exp(1))*exp(9)-216),x, alg orithm="maxima")
Output:
25*(e^9 + 3*e^7 + 3*e^5 + e^3)*e^27/(25*x^2*(e^36 + 3*e^34 + 3*e^32 + e^30 ) - 20*x*(5*e^36 + 15*e^34 + 15*e^32 + 5*e^30 + 3*e^24 + 6*e^22 + 3*e^20) + 100*e^36 + 300*e^34 + 300*e^32 + 100*e^30 + 120*e^24 + 240*e^22 + 120*e^ 20 + 36*e^12 + 36*e^10)
Time = 0.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=\frac {25 \, {\left (e^{9} + 3 \, e^{7} + 3 \, e^{5} + e^{3}\right )} e^{27}}{{\left (5 \, x e^{12} + 5 \, x e^{10} - 10 \, e^{12} - 10 \, e^{10} - 6\right )}^{2} {\left (e^{12} + e^{10}\right )}} \] Input:
integrate((-250*exp(3)^3-750*exp(1)*exp(3)^2-750*exp(1)^2*exp(3)-250*exp(1 )^3)*exp(9)^3/(((125*x^3-750*x^2+1500*x-1000)*exp(3)^3+(375*x^3-2250*x^2+4 500*x-3000)*exp(1)*exp(3)^2+(375*x^3-2250*x^2+4500*x-3000)*exp(1)^2*exp(3) +(125*x^3-750*x^2+1500*x-1000)*exp(1)^3)*exp(9)^3+((-450*x^2+1800*x-1800)* exp(3)^2+(-900*x^2+3600*x-3600)*exp(1)*exp(3)+(-450*x^2+1800*x-1800)*exp(1 )^2)*exp(9)^2+((540*x-1080)*exp(3)+(540*x-1080)*exp(1))*exp(9)-216),x, alg orithm="giac")
Output:
25*(e^9 + 3*e^7 + 3*e^5 + e^3)*e^27/((5*x*e^12 + 5*x*e^10 - 10*e^12 - 10*e ^10 - 6)^2*(e^12 + e^10))
Time = 3.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.68 \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=\frac {25\,{\mathrm {e}}^{20}\,{\left ({\mathrm {e}}^2+1\right )}^2}{\left (25\,{\mathrm {e}}^{20}+50\,{\mathrm {e}}^{22}+25\,{\mathrm {e}}^{24}\right )\,x^2+\left (-60\,{\mathrm {e}}^{10}-60\,{\mathrm {e}}^{12}-100\,{\mathrm {e}}^{20}-200\,{\mathrm {e}}^{22}-100\,{\mathrm {e}}^{24}\right )\,x+120\,{\mathrm {e}}^{10}+120\,{\mathrm {e}}^{12}+100\,{\mathrm {e}}^{20}+200\,{\mathrm {e}}^{22}+100\,{\mathrm {e}}^{24}+36} \] Input:
int(-(exp(27)*(250*exp(3) + 750*exp(5) + 750*exp(7) + 250*exp(9)))/(exp(9) *(exp(1)*(540*x - 1080) + exp(3)*(540*x - 1080)) - exp(18)*(exp(2)*(450*x^ 2 - 1800*x + 1800) + exp(6)*(450*x^2 - 1800*x + 1800) + exp(4)*(900*x^2 - 3600*x + 3600)) + exp(27)*(exp(3)*(1500*x - 750*x^2 + 125*x^3 - 1000) + ex p(9)*(1500*x - 750*x^2 + 125*x^3 - 1000) + exp(5)*(4500*x - 2250*x^2 + 375 *x^3 - 3000) + exp(7)*(4500*x - 2250*x^2 + 375*x^3 - 3000)) - 216),x)
Output:
(25*exp(20)*(exp(2) + 1)^2)/(120*exp(10) + 120*exp(12) + 100*exp(20) + 200 *exp(22) + 100*exp(24) - x*(60*exp(10) + 60*exp(12) + 100*exp(20) + 200*ex p(22) + 100*exp(24)) + x^2*(25*exp(20) + 50*exp(22) + 25*exp(24)) + 36)
Time = 0.17 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.50 \[ \int \frac {e^{27} \left (-250 e^3-750 e^5-750 e^7-250 e^9\right )}{-216+e^9 \left (e (-1080+540 x)+e^3 (-1080+540 x)\right )+e^{18} \left (e^4 \left (-3600+3600 x-900 x^2\right )+e^2 \left (-1800+1800 x-450 x^2\right )+e^6 \left (-1800+1800 x-450 x^2\right )\right )+e^{27} \left (e^3 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^9 \left (-1000+1500 x-750 x^2+125 x^3\right )+e^5 \left (-3000+4500 x-2250 x^2+375 x^3\right )+e^7 \left (-3000+4500 x-2250 x^2+375 x^3\right )\right )} \, dx=\frac {25 e^{20} \left (e^{4}+2 e^{2}+1\right )}{25 e^{24} x^{2}-100 e^{24} x +100 e^{24}+50 e^{22} x^{2}-200 e^{22} x +200 e^{22}+25 e^{20} x^{2}-100 e^{20} x +100 e^{20}-60 e^{12} x +120 e^{12}-60 e^{10} x +120 e^{10}+36} \] Input:
int((-250*exp(3)^3-750*exp(1)*exp(3)^2-750*exp(1)^2*exp(3)-250*exp(1)^3)*e xp(9)^3/(((125*x^3-750*x^2+1500*x-1000)*exp(3)^3+(375*x^3-2250*x^2+4500*x- 3000)*exp(1)*exp(3)^2+(375*x^3-2250*x^2+4500*x-3000)*exp(1)^2*exp(3)+(125* x^3-750*x^2+1500*x-1000)*exp(1)^3)*exp(9)^3+((-450*x^2+1800*x-1800)*exp(3) ^2+(-900*x^2+3600*x-3600)*exp(1)*exp(3)+(-450*x^2+1800*x-1800)*exp(1)^2)*e xp(9)^2+((540*x-1080)*exp(3)+(540*x-1080)*exp(1))*exp(9)-216),x)
Output:
(25*e**20*(e**4 + 2*e**2 + 1))/(25*e**24*x**2 - 100*e**24*x + 100*e**24 + 50*e**22*x**2 - 200*e**22*x + 200*e**22 + 25*e**20*x**2 - 100*e**20*x + 10 0*e**20 - 60*e**12*x + 120*e**12 - 60*e**10*x + 120*e**10 + 36)