Integrand size = 124, antiderivative size = 33 \[ \int \frac {-162 x^4+50 e x^4-246 x^5-4 x^6+e^9 \left (2 x-50 e x+306 x^2+4 x^3\right )+e^6 \left (54 x^2+150 e x^2-838 x^3-12 x^4\right )+e^3 \left (306 x^3-150 e x^3+778 x^4+12 x^5\right )}{25 e^9-75 e^6 x+75 e^3 x^2-25 x^3} \, dx=x^2 \left (-e+4 x+\left (\frac {2 x}{e^3-x}+\frac {1+x}{5}\right )^2\right ) \] Output:
x^2*(4*x-exp(1)+(2*x/(-x+exp(3))+1/5*x+1/5)^2)
Leaf count is larger than twice the leaf count of optimal. \(94\) vs. \(2(33)=66\).
Time = 0.05 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.85 \[ \int \frac {-162 x^4+50 e x^4-246 x^5-4 x^6+e^9 \left (2 x-50 e x+306 x^2+4 x^3\right )+e^6 \left (54 x^2+150 e x^2-838 x^3-12 x^4\right )+e^3 \left (306 x^3-150 e x^3+778 x^4+12 x^5\right )}{25 e^9-75 e^6 x+75 e^3 x^2-25 x^3} \, dx=\frac {1}{25} \left (25 e^7-e^{12} \left (1-\frac {100}{\left (e^3-x\right )^2}-\frac {20}{e^3-x}\right )-e^9 \left (42+\frac {380}{e^3-x}\right )-20 e^3 (-9+x) x-25 e x^2-e^6 (261+20 x)+x^2 \left (81+82 x+x^2\right )\right ) \] Input:
Integrate[(-162*x^4 + 50*E*x^4 - 246*x^5 - 4*x^6 + E^9*(2*x - 50*E*x + 306 *x^2 + 4*x^3) + E^6*(54*x^2 + 150*E*x^2 - 838*x^3 - 12*x^4) + E^3*(306*x^3 - 150*E*x^3 + 778*x^4 + 12*x^5))/(25*E^9 - 75*E^6*x + 75*E^3*x^2 - 25*x^3 ),x]
Output:
(25*E^7 - E^12*(1 - 100/(E^3 - x)^2 - 20/(E^3 - x)) - E^9*(42 + 380/(E^3 - x)) - 20*E^3*(-9 + x)*x - 25*E*x^2 - E^6*(261 + 20*x) + x^2*(81 + 82*x + x^2))/25
Leaf count is larger than twice the leaf count of optimal. \(84\) vs. \(2(33)=66\).
Time = 0.40 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.55, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {6, 2007, 2389, 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.
\(\displaystyle \int \frac {-4 x^6-246 x^5+50 e x^4-162 x^4+e^9 \left (4 x^3+306 x^2-50 e x+2 x\right )+e^3 \left (12 x^5+778 x^4-150 e x^3+306 x^3\right )+e^6 \left (-12 x^4-838 x^3+150 e x^2+54 x^2\right )}{-25 x^3+75 e^3 x^2-75 e^6 x+25 e^9} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {-4 x^6-246 x^5+(50 e-162) x^4+e^9 \left (4 x^3+306 x^2-50 e x+2 x\right )+e^3 \left (12 x^5+778 x^4-150 e x^3+306 x^3\right )+e^6 \left (-12 x^4-838 x^3+150 e x^2+54 x^2\right )}{-25 x^3+75 e^3 x^2-75 e^6 x+25 e^9}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-4 x^6-246 x^5+(50 e-162) x^4+e^9 \left (4 x^3+306 x^2-50 e x+2 x\right )+e^3 \left (12 x^5+778 x^4-150 e x^3+306 x^3\right )+e^6 \left (-12 x^4-838 x^3+150 e x^2+54 x^2\right )}{\left (5^{2/3} e^3-5^{2/3} x\right )^3}dx\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \int \left (\frac {4 x^3}{25}+\frac {246 x^2}{25}-\frac {2}{25} \left (-81+25 e+20 e^3\right ) x+\frac {4 e^9 \left (e^3-19\right )}{5 \left (e^3-x\right )^2}+\frac {8 e^{12}}{\left (e^3-x\right )^3}-\frac {4}{5} e^3 \left (e^3-9\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^4}{25}+\frac {82 x^3}{25}+\frac {1}{25} \left (81-25 e-20 e^3\right ) x^2+\frac {4}{5} e^3 \left (9-e^3\right ) x-\frac {4 e^9 \left (19-e^3\right )}{5 \left (e^3-x\right )}+\frac {4 e^{12}}{\left (e^3-x\right )^2}\) |
Input:
Int[(-162*x^4 + 50*E*x^4 - 246*x^5 - 4*x^6 + E^9*(2*x - 50*E*x + 306*x^2 + 4*x^3) + E^6*(54*x^2 + 150*E*x^2 - 838*x^3 - 12*x^4) + E^3*(306*x^3 - 150 *E*x^3 + 778*x^4 + 12*x^5))/(25*E^9 - 75*E^6*x + 75*E^3*x^2 - 25*x^3),x]
Output:
(4*E^12)/(E^3 - x)^2 - (4*E^9*(19 - E^3))/(5*(E^3 - x)) + (4*E^3*(9 - E^3) *x)/5 + ((81 - 25*E - 20*E^3)*x^2)/25 + (82*x^3)/25 + x^4/25
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, 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]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(30)=60\).
Time = 0.30 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.30
method | result | size |
risch | \(-\frac {4 x \,{\mathrm e}^{6}}{5}-\frac {4 x^{2} {\mathrm e}^{3}}{5}+\frac {36 x \,{\mathrm e}^{3}}{5}+\frac {x^{4}}{25}-x^{2} {\mathrm e}+\frac {82 x^{3}}{25}+\frac {81 x^{2}}{25}+\frac {\frac {\left (-20 \,{\mathrm e}^{12}+380 \,{\mathrm e}^{9}\right ) x}{25}+\frac {4 \,{\mathrm e}^{15}}{5}-\frac {56 \,{\mathrm e}^{12}}{5}}{{\mathrm e}^{6}-2 x \,{\mathrm e}^{3}+x^{2}}\) | \(76\) |
norman | \(\frac {\left (\frac {82}{25}-\frac {2 \,{\mathrm e}^{3}}{25}\right ) x^{5}+\left (2 \,{\mathrm e} \,{\mathrm e}^{3}+\frac {102 \,{\mathrm e}^{6}}{25}+\frac {18 \,{\mathrm e}^{3}}{25}\right ) x^{3}+\left (-2 \,{\mathrm e} \,{\mathrm e}^{9}+\frac {2 \,{\mathrm e}^{9}}{25}\right ) x +\left (\frac {{\mathrm e}^{6}}{25}-{\mathrm e}-\frac {184 \,{\mathrm e}^{3}}{25}+\frac {81}{25}\right ) x^{4}+\frac {x^{6}}{25}+\frac {{\mathrm e}^{12} \left (25 \,{\mathrm e}-1\right )}{25}}{\left (-x +{\mathrm e}^{3}\right )^{2}}\) | \(96\) |
gosper | \(\frac {{\mathrm e}^{6} x^{4}-2 x^{5} {\mathrm e}^{3}+x^{6}+25 \,{\mathrm e} \,{\mathrm e}^{12}-50 \,{\mathrm e} \,{\mathrm e}^{9} x +50 \,{\mathrm e} \,{\mathrm e}^{3} x^{3}-25 x^{4} {\mathrm e}+102 \,{\mathrm e}^{6} x^{3}-184 x^{4} {\mathrm e}^{3}+82 x^{5}-{\mathrm e}^{12}+2 x \,{\mathrm e}^{9}+18 x^{3} {\mathrm e}^{3}+81 x^{4}}{25 \,{\mathrm e}^{6}-50 x \,{\mathrm e}^{3}+25 x^{2}}\) | \(116\) |
parallelrisch | \(\frac {{\mathrm e}^{6} x^{4}-2 x^{5} {\mathrm e}^{3}+x^{6}+25 \,{\mathrm e} \,{\mathrm e}^{12}-50 \,{\mathrm e} \,{\mathrm e}^{9} x +50 \,{\mathrm e} \,{\mathrm e}^{3} x^{3}-25 x^{4} {\mathrm e}+102 \,{\mathrm e}^{6} x^{3}-184 x^{4} {\mathrm e}^{3}+82 x^{5}-{\mathrm e}^{12}+2 x \,{\mathrm e}^{9}+18 x^{3} {\mathrm e}^{3}+81 x^{4}}{25 \,{\mathrm e}^{6}-50 x \,{\mathrm e}^{3}+25 x^{2}}\) | \(116\) |
default | \(\frac {x^{4}}{25}-6 x \,{\mathrm e} \,{\mathrm e}^{3}-x^{2} {\mathrm e}-\frac {4 x \,{\mathrm e}^{6}}{5}-\frac {4 x^{2} {\mathrm e}^{3}}{5}+\frac {82 x^{3}}{25}+\frac {36 x \,{\mathrm e}^{3}}{5}+6 x \,{\mathrm e}^{4}+\frac {81 x^{2}}{25}-\frac {4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-{\mathrm e}^{9}+3 \textit {\_Z} \,{\mathrm e}^{6}-3 \textit {\_Z}^{2} {\mathrm e}^{3}+\textit {\_Z}^{3}\right )}{\sum }\frac {\left (-\textit {\_R} \,{\mathrm e}^{12}-9 \,{\mathrm e}^{9} {\mathrm e}^{3}+{\mathrm e}^{9} {\mathrm e}^{6}+19 \textit {\_R} \,{\mathrm e}^{9}\right ) \ln \left (x -\textit {\_R} \right )}{{\mathrm e}^{6}-2 \textit {\_R} \,{\mathrm e}^{3}+\textit {\_R}^{2}}\right )}{15}\) | \(128\) |
Input:
int(((-50*x*exp(1)+4*x^3+306*x^2+2*x)*exp(3)^3+(150*x^2*exp(1)-12*x^4-838* x^3+54*x^2)*exp(3)^2+(-150*x^3*exp(1)+12*x^5+778*x^4+306*x^3)*exp(3)+50*x^ 4*exp(1)-4*x^6-246*x^5-162*x^4)/(25*exp(3)^3-75*x*exp(3)^2+75*x^2*exp(3)-2 5*x^3),x,method=_RETURNVERBOSE)
Output:
-4/5*x*exp(6)-4/5*x^2*exp(3)+36/5*x*exp(3)+1/25*x^4-x^2*exp(1)+82/25*x^3+8 1/25*x^2+(1/25*(-20*exp(12)+380*exp(9))*x+4/5*exp(15)-56/5*exp(12))/(exp(6 )-2*x*exp(3)+x^2)
Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (29) = 58\).
Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.24 \[ \int \frac {-162 x^4+50 e x^4-246 x^5-4 x^6+e^9 \left (2 x-50 e x+306 x^2+4 x^3\right )+e^6 \left (54 x^2+150 e x^2-838 x^3-12 x^4\right )+e^3 \left (306 x^3-150 e x^3+778 x^4+12 x^5\right )}{25 e^9-75 e^6 x+75 e^3 x^2-25 x^3} \, dx=\frac {x^{6} + 82 \, x^{5} - 25 \, x^{4} e + 81 \, x^{4} + 50 \, x^{3} e^{4} - 25 \, x^{2} e^{7} - 40 \, {\left (x + 7\right )} e^{12} + 20 \, {\left (x^{2} + 28 \, x\right )} e^{9} + {\left (x^{4} + 102 \, x^{3} - 279 \, x^{2}\right )} e^{6} - 2 \, {\left (x^{5} + 92 \, x^{4} - 9 \, x^{3}\right )} e^{3} + 20 \, e^{15}}{25 \, {\left (x^{2} - 2 \, x e^{3} + e^{6}\right )}} \] Input:
integrate(((-50*exp(1)*x+4*x^3+306*x^2+2*x)*exp(3)^3+(150*x^2*exp(1)-12*x^ 4-838*x^3+54*x^2)*exp(3)^2+(-150*x^3*exp(1)+12*x^5+778*x^4+306*x^3)*exp(3) +50*x^4*exp(1)-4*x^6-246*x^5-162*x^4)/(25*exp(3)^3-75*x*exp(3)^2+75*x^2*ex p(3)-25*x^3),x, algorithm="fricas")
Output:
1/25*(x^6 + 82*x^5 - 25*x^4*e + 81*x^4 + 50*x^3*e^4 - 25*x^2*e^7 - 40*(x + 7)*e^12 + 20*(x^2 + 28*x)*e^9 + (x^4 + 102*x^3 - 279*x^2)*e^6 - 2*(x^5 + 92*x^4 - 9*x^3)*e^3 + 20*e^15)/(x^2 - 2*x*e^3 + e^6)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (26) = 52\).
Time = 0.33 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.48 \[ \int \frac {-162 x^4+50 e x^4-246 x^5-4 x^6+e^9 \left (2 x-50 e x+306 x^2+4 x^3\right )+e^6 \left (54 x^2+150 e x^2-838 x^3-12 x^4\right )+e^3 \left (306 x^3-150 e x^3+778 x^4+12 x^5\right )}{25 e^9-75 e^6 x+75 e^3 x^2-25 x^3} \, dx=\frac {x^{4}}{25} + \frac {82 x^{3}}{25} + x^{2} \left (- \frac {4 e^{3}}{5} - e + \frac {81}{25}\right ) + x \left (- \frac {4 e^{6}}{5} + \frac {36 e^{3}}{5}\right ) + \frac {x \left (- 4 e^{12} + 76 e^{9}\right ) - 56 e^{12} + 4 e^{15}}{5 x^{2} - 10 x e^{3} + 5 e^{6}} \] Input:
integrate(((-50*exp(1)*x+4*x**3+306*x**2+2*x)*exp(3)**3+(150*x**2*exp(1)-1 2*x**4-838*x**3+54*x**2)*exp(3)**2+(-150*x**3*exp(1)+12*x**5+778*x**4+306* x**3)*exp(3)+50*x**4*exp(1)-4*x**6-246*x**5-162*x**4)/(25*exp(3)**3-75*x*e xp(3)**2+75*x**2*exp(3)-25*x**3),x)
Output:
x**4/25 + 82*x**3/25 + x**2*(-4*exp(3)/5 - E + 81/25) + x*(-4*exp(6)/5 + 3 6*exp(3)/5) + (x*(-4*exp(12) + 76*exp(9)) - 56*exp(12) + 4*exp(15))/(5*x** 2 - 10*x*exp(3) + 5*exp(6))
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (29) = 58\).
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.09 \[ \int \frac {-162 x^4+50 e x^4-246 x^5-4 x^6+e^9 \left (2 x-50 e x+306 x^2+4 x^3\right )+e^6 \left (54 x^2+150 e x^2-838 x^3-12 x^4\right )+e^3 \left (306 x^3-150 e x^3+778 x^4+12 x^5\right )}{25 e^9-75 e^6 x+75 e^3 x^2-25 x^3} \, dx=\frac {1}{25} \, x^{4} + \frac {82}{25} \, x^{3} - \frac {1}{25} \, x^{2} {\left (20 \, e^{3} + 25 \, e - 81\right )} - \frac {4}{5} \, x {\left (e^{6} - 9 \, e^{3}\right )} - \frac {4 \, {\left (x {\left (e^{12} - 19 \, e^{9}\right )} - e^{15} + 14 \, e^{12}\right )}}{5 \, {\left (x^{2} - 2 \, x e^{3} + e^{6}\right )}} \] Input:
integrate(((-50*exp(1)*x+4*x^3+306*x^2+2*x)*exp(3)^3+(150*x^2*exp(1)-12*x^ 4-838*x^3+54*x^2)*exp(3)^2+(-150*x^3*exp(1)+12*x^5+778*x^4+306*x^3)*exp(3) +50*x^4*exp(1)-4*x^6-246*x^5-162*x^4)/(25*exp(3)^3-75*x*exp(3)^2+75*x^2*ex p(3)-25*x^3),x, algorithm="maxima")
Output:
1/25*x^4 + 82/25*x^3 - 1/25*x^2*(20*e^3 + 25*e - 81) - 4/5*x*(e^6 - 9*e^3) - 4/5*(x*(e^12 - 19*e^9) - e^15 + 14*e^12)/(x^2 - 2*x*e^3 + e^6)
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (29) = 58\).
Time = 0.11 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.06 \[ \int \frac {-162 x^4+50 e x^4-246 x^5-4 x^6+e^9 \left (2 x-50 e x+306 x^2+4 x^3\right )+e^6 \left (54 x^2+150 e x^2-838 x^3-12 x^4\right )+e^3 \left (306 x^3-150 e x^3+778 x^4+12 x^5\right )}{25 e^9-75 e^6 x+75 e^3 x^2-25 x^3} \, dx=\frac {1}{25} \, x^{4} + \frac {82}{25} \, x^{3} - \frac {4}{5} \, x^{2} e^{3} - x^{2} e + \frac {81}{25} \, x^{2} - \frac {4}{5} \, x e^{6} + \frac {36}{5} \, x e^{3} - \frac {4 \, {\left (x e^{12} - 19 \, x e^{9} - e^{15} + 14 \, e^{12}\right )}}{5 \, {\left (x - e^{3}\right )}^{2}} \] Input:
integrate(((-50*exp(1)*x+4*x^3+306*x^2+2*x)*exp(3)^3+(150*x^2*exp(1)-12*x^ 4-838*x^3+54*x^2)*exp(3)^2+(-150*x^3*exp(1)+12*x^5+778*x^4+306*x^3)*exp(3) +50*x^4*exp(1)-4*x^6-246*x^5-162*x^4)/(25*exp(3)^3-75*x*exp(3)^2+75*x^2*ex p(3)-25*x^3),x, algorithm="giac")
Output:
1/25*x^4 + 82/25*x^3 - 4/5*x^2*e^3 - x^2*e + 81/25*x^2 - 4/5*x*e^6 + 36/5* x*e^3 - 4/5*(x*e^12 - 19*x*e^9 - e^15 + 14*e^12)/(x - e^3)^2
Time = 2.54 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.21 \[ \int \frac {-162 x^4+50 e x^4-246 x^5-4 x^6+e^9 \left (2 x-50 e x+306 x^2+4 x^3\right )+e^6 \left (54 x^2+150 e x^2-838 x^3-12 x^4\right )+e^3 \left (306 x^3-150 e x^3+778 x^4+12 x^5\right )}{25 e^9-75 e^6 x+75 e^3 x^2-25 x^3} \, dx=\frac {4\,{\mathrm {e}}^{15}-56\,{\mathrm {e}}^{12}+x\,\left (76\,{\mathrm {e}}^9-4\,{\mathrm {e}}^{12}\right )}{5\,x^2-10\,{\mathrm {e}}^3\,x+5\,{\mathrm {e}}^6}-x^2\,\left (\mathrm {e}+\frac {4\,{\mathrm {e}}^3}{5}-\frac {81}{25}\right )+\frac {82\,x^3}{25}+\frac {x^4}{25}-x\,\left (\frac {738\,{\mathrm {e}}^6}{25}-\frac {4\,{\mathrm {e}}^9}{25}-\frac {2\,{\mathrm {e}}^3\,\left (75\,\mathrm {e}+419\,{\mathrm {e}}^3-2\,{\mathrm {e}}^6-153\right )}{25}+3\,{\mathrm {e}}^3\,\left (2\,\mathrm {e}+\frac {8\,{\mathrm {e}}^3}{5}-\frac {162}{25}\right )\right ) \] Input:
int((exp(6)*(150*x^2*exp(1) + 54*x^2 - 838*x^3 - 12*x^4) + exp(3)*(306*x^3 - 150*x^3*exp(1) + 778*x^4 + 12*x^5) + 50*x^4*exp(1) + exp(9)*(2*x - 50*x *exp(1) + 306*x^2 + 4*x^3) - 162*x^4 - 246*x^5 - 4*x^6)/(25*exp(9) - 75*x* exp(6) + 75*x^2*exp(3) - 25*x^3),x)
Output:
(4*exp(15) - 56*exp(12) + x*(76*exp(9) - 4*exp(12)))/(5*exp(6) - 10*x*exp( 3) + 5*x^2) - x^2*(exp(1) + (4*exp(3))/5 - 81/25) + (82*x^3)/25 + x^4/25 - x*((738*exp(6))/25 - (4*exp(9))/25 - (2*exp(3)*(75*exp(1) + 419*exp(3) - 2*exp(6) - 153))/25 + 3*exp(3)*(2*exp(1) + (8*exp(3))/5 - 162/25))
Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.79 \[ \int \frac {-162 x^4+50 e x^4-246 x^5-4 x^6+e^9 \left (2 x-50 e x+306 x^2+4 x^3\right )+e^6 \left (54 x^2+150 e x^2-838 x^3-12 x^4\right )+e^3 \left (306 x^3-150 e x^3+778 x^4+12 x^5\right )}{25 e^9-75 e^6 x+75 e^3 x^2-25 x^3} \, dx=\frac {x^{2} \left (e^{6} x^{2}-25 e^{7}+102 e^{6} x +e^{6}-2 e^{3} x^{3}+50 e^{4} x -184 e^{3} x^{2}+18 e^{3} x +x^{4}-25 e \,x^{2}+82 x^{3}+81 x^{2}\right )}{25 e^{6}-50 e^{3} x +25 x^{2}} \] Input:
int(((-50*exp(1)*x+4*x^3+306*x^2+2*x)*exp(3)^3+(150*x^2*exp(1)-12*x^4-838* x^3+54*x^2)*exp(3)^2+(-150*x^3*exp(1)+12*x^5+778*x^4+306*x^3)*exp(3)+50*x^ 4*exp(1)-4*x^6-246*x^5-162*x^4)/(25*exp(3)^3-75*x*exp(3)^2+75*x^2*exp(3)-2 5*x^3),x)
Output:
(x**2*( - 25*e**7 + e**6*x**2 + 102*e**6*x + e**6 + 50*e**4*x - 2*e**3*x** 3 - 184*e**3*x**2 + 18*e**3*x - 25*e*x**2 + x**4 + 82*x**3 + 81*x**2))/(25 *(e**6 - 2*e**3*x + x**2))