\(\int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x))+e^{-\frac {25 x}{-21+25 e^4-25 x}} (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x))}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 (-9450 x^3-11250 x^4)} \, dx\) [3866]

Optimal result

Integrand size = 150, antiderivative size = 27 \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=\frac {\left (3+e^{\frac {x}{\frac {21}{25}-e^4+x}}\right )^2}{9 x^2} \]

[Out]

1/9*(exp(x/(21/25+x-exp(4)))+3)^2/x^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(421\) vs. \(2(27)=54\).

Time = 4.70 (sec) , antiderivative size = 421, normalized size of antiderivative = 15.59, number of steps used = 107, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {6, 6873, 12, 6874, 46, 78, 2262, 2240, 2241, 2264, 2263, 2265, 2209} \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=\frac {2 e^{\frac {25 x}{25 x-25 e^4+21}}}{3 x^2}+\frac {e^{\frac {50 x}{25 x-25 e^4+21}}}{9 x^2}+\frac {441+625 e^8}{\left (21-25 e^4\right )^2 x^2}-\frac {1050 e^4}{\left (21-25 e^4\right )^2 x^2}-\frac {100 \left (441+625 e^8\right )}{\left (21-25 e^4\right )^3 x}+\frac {2500 e^4 \left (21+25 e^4\right )}{\left (21-25 e^4\right )^3 x}+\frac {2100}{\left (21-25 e^4\right )^2 x}-\frac {1250 \left (441+625 e^8\right )}{\left (21-25 e^4\right )^3 \left (25 x-25 e^4+21\right )}-\frac {1250}{\left (21-25 e^4\right ) \left (25 x-25 e^4+21\right )}+\frac {52500}{\left (21-25 e^4\right )^2 \left (25 x-25 e^4+21\right )}+\frac {1562500 e^8}{\left (21-25 e^4\right )^3 \left (25 x-25 e^4+21\right )}-\frac {3750 \left (441+625 e^8\right ) \log (x)}{\left (21-25 e^4\right )^4}+\frac {62500 e^4 \left (21+50 e^4\right ) \log (x)}{\left (21-25 e^4\right )^4}-\frac {1250 \log (x)}{\left (21-25 e^4\right )^2}+\frac {105000 \log (x)}{\left (21-25 e^4\right )^3}+\frac {3750 \left (441+625 e^8\right ) \log \left (25 x-25 e^4+21\right )}{\left (21-25 e^4\right )^4}-\frac {62500 e^4 \left (21+50 e^4\right ) \log \left (25 x-25 e^4+21\right )}{\left (21-25 e^4\right )^4}+\frac {1250 \log \left (25 x-25 e^4+21\right )}{\left (21-25 e^4\right )^2}-\frac {105000 \log \left (25 x-25 e^4+21\right )}{\left (21-25 e^4\right )^3} \]

[In]

Int[(-7938 - 11250*E^8 - 18900*x - 11250*x^2 + E^4*(18900 + 22500*x) + (-882 - 1250*E^8 - 1050*x - 1250*x^2 +
E^4*(2100 + 1250*x))/E^((50*x)/(-21 + 25*E^4 - 25*x)) + (-5292 - 7500*E^8 - 9450*x - 7500*x^2 + E^4*(12600 + 1
1250*x))/E^((25*x)/(-21 + 25*E^4 - 25*x)))/(3969*x^3 + 5625*E^8*x^3 + 9450*x^4 + 5625*x^5 + E^4*(-9450*x^3 - 1
1250*x^4)),x]

[Out]

(2*E^((25*x)/(21 - 25*E^4 + 25*x)))/(3*x^2) + E^((50*x)/(21 - 25*E^4 + 25*x))/(9*x^2) - (1050*E^4)/((21 - 25*E
^4)^2*x^2) + (441 + 625*E^8)/((21 - 25*E^4)^2*x^2) + 2100/((21 - 25*E^4)^2*x) + (2500*E^4*(21 + 25*E^4))/((21
- 25*E^4)^3*x) - (100*(441 + 625*E^8))/((21 - 25*E^4)^3*x) + (1562500*E^8)/((21 - 25*E^4)^3*(21 - 25*E^4 + 25*
x)) + 52500/((21 - 25*E^4)^2*(21 - 25*E^4 + 25*x)) - 1250/((21 - 25*E^4)*(21 - 25*E^4 + 25*x)) - (1250*(441 +
625*E^8))/((21 - 25*E^4)^3*(21 - 25*E^4 + 25*x)) + (105000*Log[x])/(21 - 25*E^4)^3 - (1250*Log[x])/(21 - 25*E^
4)^2 + (62500*E^4*(21 + 50*E^4)*Log[x])/(21 - 25*E^4)^4 - (3750*(441 + 625*E^8)*Log[x])/(21 - 25*E^4)^4 - (105
000*Log[21 - 25*E^4 + 25*x])/(21 - 25*E^4)^3 + (1250*Log[21 - 25*E^4 + 25*x])/(21 - 25*E^4)^2 - (62500*E^4*(21
+ 50*E^4)*Log[21 - 25*E^4 + 25*x])/(21 - 25*E^4)^4 + (3750*(441 + 625*E^8)*Log[21 - 25*E^4 + 25*x])/(21 - 25*
E^4)^4

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2241

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[F^a*(ExpIntegralEi[
b*(c + d*x)^n*Log[F]]/(f*n)), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2262

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :>
Int[(g + h*x)^m*F^((d*e + b*f)/d - f*((b*c - a*d)/(d*(c + d*x)))), x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m},
x] && NeQ[b*c - a*d, 0] && EqQ[d*g - c*h, 0]

Rule 2263

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))/((g_.) + (h_.)*(x_)), x_Symbol] :> Dist[d
/h, Int[F^(e + f*((a + b*x)/(c + d*x)))/(c + d*x), x], x] - Dist[(d*g - c*h)/h, Int[F^(e + f*((a + b*x)/(c + d
*x)))/((c + d*x)*(g + h*x)), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, h}, x] && NeQ[b*c - a*d, 0] && NeQ[d*g -
c*h, 0]

Rule 2264

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))*((g_.) + (h_.)*(x_))^(m_), x_Symbol] :> S
imp[(g + h*x)^(m + 1)*(F^(e + f*((a + b*x)/(c + d*x)))/(h*(m + 1))), x] - Dist[f*(b*c - a*d)*(Log[F]/(h*(m + 1
))), Int[(g + h*x)^(m + 1)*(F^(e + f*((a + b*x)/(c + d*x)))/(c + d*x)^2), x], x] /; FreeQ[{F, a, b, c, d, e, f
, g, h}, x] && NeQ[b*c - a*d, 0] && NeQ[d*g - c*h, 0] && ILtQ[m, -1]

Rule 2265

Int[(F_)^((e_.) + ((f_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_)))/(((g_.) + (h_.)*(x_))*((i_.) + (j_.)*(x_)
)), x_Symbol] :> Dist[-d/(h*(d*i - c*j)), Subst[Int[F^(e + f*((b*i - a*j)/(d*i - c*j)) - (b*c - a*d)*f*(x/(d*i
- c*j)))/x, x], x, (i + j*x)/(c + d*x)], x] /; FreeQ[{F, a, b, c, d, e, f, g, h}, x] && EqQ[d*g - c*h, 0]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{\left (3969+5625 e^8\right ) x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx \\ & = \int \frac {-7938 \left (1+\frac {625 e^8}{441}\right )-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{9 x^3 \left (21-25 e^4+25 x\right )^2} \, dx \\ & = \frac {1}{9} \int \frac {-7938 \left (1+\frac {625 e^8}{441}\right )-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{x^3 \left (21-25 e^4+25 x\right )^2} \, dx \\ & = \frac {1}{9} \int \left (-\frac {18 \left (441+625 e^8\right )}{\left (-21+25 e^4-25 x\right )^2 x^3}+\frac {900 e^4 (21+25 x)}{\left (-21+25 e^4-25 x\right )^2 x^3}-\frac {18900}{x^2 \left (21-25 e^4+25 x\right )^2}-\frac {11250}{x \left (21-25 e^4+25 x\right )^2}+\frac {6 e^{\frac {25 x}{21-25 e^4+25 x}} \left (-2 \left (21-25 e^4\right )^2-75 \left (21-25 e^4\right ) x-1250 x^2\right )}{x^3 \left (21-25 e^4+25 x\right )^2}+\frac {2 e^{\frac {50 x}{21-25 e^4+25 x}} \left (-\left (21-25 e^4\right )^2-25 \left (21-25 e^4\right ) x-625 x^2\right )}{x^3 \left (21-25 e^4+25 x\right )^2}\right ) \, dx \\ & = \frac {2}{9} \int \frac {e^{\frac {50 x}{21-25 e^4+25 x}} \left (-\left (21-25 e^4\right )^2-25 \left (21-25 e^4\right ) x-625 x^2\right )}{x^3 \left (21-25 e^4+25 x\right )^2} \, dx+\frac {2}{3} \int \frac {e^{\frac {25 x}{21-25 e^4+25 x}} \left (-2 \left (21-25 e^4\right )^2-75 \left (21-25 e^4\right ) x-1250 x^2\right )}{x^3 \left (21-25 e^4+25 x\right )^2} \, dx-1250 \int \frac {1}{x \left (21-25 e^4+25 x\right )^2} \, dx-2100 \int \frac {1}{x^2 \left (21-25 e^4+25 x\right )^2} \, dx+\left (100 e^4\right ) \int \frac {21+25 x}{\left (-21+25 e^4-25 x\right )^2 x^3} \, dx-\left (2 \left (441+625 e^8\right )\right ) \int \frac {1}{\left (-21+25 e^4-25 x\right )^2 x^3} \, dx \\ & = \frac {2}{9} \int \left (-\frac {15625 e^{\frac {50 x}{21-25 e^4+25 x}}}{\left (-21+25 e^4\right ) \left (-21+25 e^4-25 x\right )^2}-\frac {31250 e^{\frac {50 x}{21-25 e^4+25 x}}}{\left (-21+25 e^4\right )^2 \left (-21+25 e^4-25 x\right )}-\frac {e^{\frac {50 x}{21-25 e^4+25 x}}}{x^3}-\frac {25 e^{\frac {50 x}{21-25 e^4+25 x}}}{\left (-21+25 e^4\right ) x^2}-\frac {1250 e^{\frac {50 x}{21-25 e^4+25 x}}}{\left (-21+25 e^4\right )^2 x}\right ) \, dx+\frac {2}{3} \int \left (-\frac {15625 e^{\frac {25 x}{21-25 e^4+25 x}}}{\left (-21+25 e^4\right ) \left (-21+25 e^4-25 x\right )^2}-\frac {31250 e^{\frac {25 x}{21-25 e^4+25 x}}}{\left (-21+25 e^4\right )^2 \left (-21+25 e^4-25 x\right )}-\frac {2 e^{\frac {25 x}{21-25 e^4+25 x}}}{x^3}-\frac {25 e^{\frac {25 x}{21-25 e^4+25 x}}}{\left (-21+25 e^4\right ) x^2}-\frac {1250 e^{\frac {25 x}{21-25 e^4+25 x}}}{\left (-21+25 e^4\right )^2 x}\right ) \, dx-1250 \int \left (\frac {25}{\left (-21+25 e^4\right ) \left (-21+25 e^4-25 x\right )^2}+\frac {25}{\left (-21+25 e^4\right )^2 \left (-21+25 e^4-25 x\right )}+\frac {1}{\left (-21+25 e^4\right )^2 x}\right ) \, dx-2100 \int \left (\frac {625}{\left (-21+25 e^4\right )^2 \left (-21+25 e^4-25 x\right )^2}+\frac {1250}{\left (-21+25 e^4\right )^3 \left (-21+25 e^4-25 x\right )}+\frac {1}{\left (-21+25 e^4\right )^2 x^2}+\frac {50}{\left (-21+25 e^4\right )^3 x}\right ) \, dx+\left (100 e^4\right ) \int \left (\frac {390625 e^4}{\left (-21+25 e^4\right )^3 \left (-21+25 e^4-25 x\right )^2}+\frac {15625 \left (21+50 e^4\right )}{\left (-21+25 e^4\right )^4 \left (-21+25 e^4-25 x\right )}+\frac {21}{\left (-21+25 e^4\right )^2 x^3}+\frac {25 \left (21+25 e^4\right )}{\left (-21+25 e^4\right )^3 x^2}+\frac {625 \left (21+50 e^4\right )}{\left (-21+25 e^4\right )^4 x}\right ) \, dx-\left (2 \left (441+625 e^8\right )\right ) \int \left (\frac {15625}{\left (-21+25 e^4\right )^3 \left (-21+25 e^4-25 x\right )^2}+\frac {46875}{\left (-21+25 e^4\right )^4 \left (-21+25 e^4-25 x\right )}+\frac {1}{\left (-21+25 e^4\right )^2 x^3}+\frac {50}{\left (-21+25 e^4\right )^3 x^2}+\frac {1875}{\left (-21+25 e^4\right )^4 x}\right ) \, dx \\ & = -\frac {1050 e^4}{\left (21-25 e^4\right )^2 x^2}+\frac {441+625 e^8}{\left (21-25 e^4\right )^2 x^2}+\frac {2100}{\left (21-25 e^4\right )^2 x}+\frac {2500 e^4 \left (21+25 e^4\right )}{\left (21-25 e^4\right )^3 x}-\frac {100 \left (441+625 e^8\right )}{\left (21-25 e^4\right )^3 x}+\frac {1562500 e^8}{\left (21-25 e^4\right )^3 \left (21-25 e^4+25 x\right )}+\frac {52500}{\left (21-25 e^4\right )^2 \left (21-25 e^4+25 x\right )}-\frac {1250}{\left (21-25 e^4\right ) \left (21-25 e^4+25 x\right )}-\frac {1250 \left (441+625 e^8\right )}{\left (21-25 e^4\right )^3 \left (21-25 e^4+25 x\right )}+\frac {105000 \log (x)}{\left (21-25 e^4\right )^3}-\frac {1250 \log (x)}{\left (21-25 e^4\right )^2}+\frac {62500 e^4 \left (21+50 e^4\right ) \log (x)}{\left (21-25 e^4\right )^4}-\frac {3750 \left (441+625 e^8\right ) \log (x)}{\left (21-25 e^4\right )^4}-\frac {105000 \log \left (21-25 e^4+25 x\right )}{\left (21-25 e^4\right )^3}+\frac {1250 \log \left (21-25 e^4+25 x\right )}{\left (21-25 e^4\right )^2}-\frac {62500 e^4 \left (21+50 e^4\right ) \log \left (21-25 e^4+25 x\right )}{\left (21-25 e^4\right )^4}+\frac {3750 \left (441+625 e^8\right ) \log \left (21-25 e^4+25 x\right )}{\left (21-25 e^4\right )^4}-\frac {2}{9} \int \frac {e^{\frac {50 x}{21-25 e^4+25 x}}}{x^3} \, dx-\frac {4}{3} \int \frac {e^{\frac {25 x}{21-25 e^4+25 x}}}{x^3} \, dx-\frac {2500 \int \frac {e^{\frac {50 x}{21-25 e^4+25 x}}}{x} \, dx}{9 \left (21-25 e^4\right )^2}-\frac {2500 \int \frac {e^{\frac {25 x}{21-25 e^4+25 x}}}{x} \, dx}{3 \left (21-25 e^4\right )^2}-\frac {62500 \int \frac {e^{\frac {50 x}{21-25 e^4+25 x}}}{-21+25 e^4-25 x} \, dx}{9 \left (21-25 e^4\right )^2}-\frac {62500 \int \frac {e^{\frac {25 x}{21-25 e^4+25 x}}}{-21+25 e^4-25 x} \, dx}{3 \left (21-25 e^4\right )^2}+\frac {50 \int \frac {e^{\frac {50 x}{21-25 e^4+25 x}}}{x^2} \, dx}{9 \left (21-25 e^4\right )}+\frac {50 \int \frac {e^{\frac {25 x}{21-25 e^4+25 x}}}{x^2} \, dx}{3 \left (21-25 e^4\right )}+\frac {31250 \int \frac {e^{\frac {50 x}{21-25 e^4+25 x}}}{\left (-21+25 e^4-25 x\right )^2} \, dx}{9 \left (21-25 e^4\right )}+\frac {31250 \int \frac {e^{\frac {25 x}{21-25 e^4+25 x}}}{\left (-21+25 e^4-25 x\right )^2} \, dx}{3 \left (21-25 e^4\right )} \\ & = \frac {2 e^{\frac {25 x}{21-25 e^4+25 x}}}{3 x^2}+\frac {e^{\frac {50 x}{21-25 e^4+25 x}}}{9 x^2}-\frac {1050 e^4}{\left (21-25 e^4\right )^2 x^2}+\frac {441+625 e^8}{\left (21-25 e^4\right )^2 x^2}+\frac {2100}{\left (21-25 e^4\right )^2 x}-\frac {50 e^{\frac {25 x}{21-25 e^4+25 x}}}{3 \left (21-25 e^4\right ) x}-\frac {50 e^{\frac {50 x}{21-25 e^4+25 x}}}{9 \left (21-25 e^4\right ) x}+\frac {2500 e^4 \left (21+25 e^4\right )}{\left (21-25 e^4\right )^3 x}-\frac {100 \left (441+625 e^8\right )}{\left (21-25 e^4\right )^3 x}+\frac {1562500 e^8}{\left (21-25 e^4\right )^3 \left (21-25 e^4+25 x\right )}+\frac {52500}{\left (21-25 e^4\right )^2 \left (21-25 e^4+25 x\right )}-\frac {1250}{\left (21-25 e^4\right ) \left (21-25 e^4+25 x\right )}-\frac {1250 \left (441+625 e^8\right )}{\left (21-25 e^4\right )^3 \left (21-25 e^4+25 x\right )}+\frac {105000 \log (x)}{\left (21-25 e^4\right )^3}-\frac {1250 \log (x)}{\left (21-25 e^4\right )^2}+\frac {62500 e^4 \left (21+50 e^4\right ) \log (x)}{\left (21-25 e^4\right )^4}-\frac {3750 \left (441+625 e^8\right ) \log (x)}{\left (21-25 e^4\right )^4}-\frac {105000 \log \left (21-25 e^4+25 x\right )}{\left (21-25 e^4\right )^3}+\frac {1250 \log \left (21-25 e^4+25 x\right )}{\left (21-25 e^4\right )^2}-\frac {62500 e^4 \left (21+50 e^4\right ) \log \left (21-25 e^4+25 x\right )}{\left (21-25 e^4\right )^4}+\frac {3750 \left (441+625 e^8\right ) \log \left (21-25 e^4+25 x\right )}{\left (21-25 e^4\right )^4}+\frac {2500}{9} \int \frac {e^{\frac {50 x}{21-25 e^4+25 x}}}{x \left (21-25 e^4+25 x\right )^2} \, dx+\frac {1250}{3} \int \frac {e^{\frac {25 x}{21-25 e^4+25 x}}}{x \left (21-25 e^4+25 x\right )^2} \, dx-\frac {62500 \int \frac {e^{2-\frac {2 \left (21-25 e^4\right )}{21-25 e^4+25 x}}}{-21+25 e^4-25 x} \, dx}{9 \left (21-25 e^4\right )^2}-\frac {62500 \int \frac {e^{\frac {50 x}{21-25 e^4+25 x}}}{21-25 e^4+25 x} \, dx}{9 \left (21-25 e^4\right )^2}-\frac {62500 \int \frac {e^{1-\frac {21-25 e^4}{21-25 e^4+25 x}}}{-21+25 e^4-25 x} \, dx}{3 \left (21-25 e^4\right )^2}-\frac {62500 \int \frac {e^{\frac {25 x}{21-25 e^4+25 x}}}{21-25 e^4+25 x} \, dx}{3 \left (21-25 e^4\right )^2}-\frac {2500 \int \frac {e^{\frac {50 x}{21-25 e^4+25 x}}}{x \left (21-25 e^4+25 x\right )} \, dx}{9 \left (21-25 e^4\right )}-\frac {2500 \int \frac {e^{\frac {25 x}{21-25 e^4+25 x}}}{x \left (21-25 e^4+25 x\right )} \, dx}{3 \left (21-25 e^4\right )}+\frac {31250 \int \frac {e^{2-\frac {2 \left (21-25 e^4\right )}{21-25 e^4+25 x}}}{\left (-21+25 e^4-25 x\right )^2} \, dx}{9 \left (21-25 e^4\right )}+\frac {31250 \int \frac {e^{1-\frac {21-25 e^4}{21-25 e^4+25 x}}}{\left (-21+25 e^4-25 x\right )^2} \, dx}{3 \left (21-25 e^4\right )}-\frac {1}{9} \left (50 \left (21-25 e^4\right )\right ) \int \frac {e^{\frac {50 x}{21-25 e^4+25 x}}}{x^2 \left (21-25 e^4+25 x\right )^2} \, dx-\frac {1}{3} \left (50 \left (21-25 e^4\right )\right ) \int \frac {e^{\frac {25 x}{21-25 e^4+25 x}}}{x^2 \left (21-25 e^4+25 x\right )^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=\frac {\left (3+e^{\frac {25 x}{21-25 e^4+25 x}}\right )^2}{9 x^2} \]

[In]

Integrate[(-7938 - 11250*E^8 - 18900*x - 11250*x^2 + E^4*(18900 + 22500*x) + (-882 - 1250*E^8 - 1050*x - 1250*
x^2 + E^4*(2100 + 1250*x))/E^((50*x)/(-21 + 25*E^4 - 25*x)) + (-5292 - 7500*E^8 - 9450*x - 7500*x^2 + E^4*(126
00 + 11250*x))/E^((25*x)/(-21 + 25*E^4 - 25*x)))/(3969*x^3 + 5625*E^8*x^3 + 9450*x^4 + 5625*x^5 + E^4*(-9450*x
^3 - 11250*x^4)),x]

[Out]

(3 + E^((25*x)/(21 - 25*E^4 + 25*x)))^2/(9*x^2)

Maple [A] (verified)

Time = 4.56 (sec) , antiderivative size = 4, normalized size of antiderivative = 0.15

method	result	size
parts	\(\frac {1}{x^{2}}\)	\(4\)
parallelrisch	\(\frac {3515625+390625 \,{\mathrm e}^{-\frac {50 x}{25 \,{\mathrm e}^{4}-25 x -21}}+2343750 \,{\mathrm e}^{-\frac {25 x}{25 \,{\mathrm e}^{4}-25 x -21}}}{3515625 x^{2}}\)	\(44\)
risch	\(\frac {1}{x^{2}}+\frac {{\mathrm e}^{-\frac {50 x}{25 \,{\mathrm e}^{4}-25 x -21}}}{9 x^{2}}+\frac {2 \,{\mathrm e}^{-\frac {25 x}{25 \,{\mathrm e}^{4}-25 x -21}}}{3 x^{2}}\)	\(45\)
norman	\(\frac {\left (\frac {25 \,{\mathrm e}^{4}}{9}-\frac {7}{3}\right ) {\mathrm e}^{-\frac {50 x}{25 \,{\mathrm e}^{4}-25 x -21}}+\left (\frac {50 \,{\mathrm e}^{4}}{3}-14\right ) {\mathrm e}^{-\frac {25 x}{25 \,{\mathrm e}^{4}-25 x -21}}-25 x -\frac {50 \,{\mathrm e}^{-\frac {25 x}{25 \,{\mathrm e}^{4}-25 x -21}} x}{3}-\frac {25 \,{\mathrm e}^{-\frac {50 x}{25 \,{\mathrm e}^{4}-25 x -21}} x}{9}-21+25 \,{\mathrm e}^{4}}{x^{2} \left (25 \,{\mathrm e}^{4}-25 x -21\right )}\)	\(109\)
derivativedivides	\(-\frac {\left (25 \,{\mathrm e}^{4}-21\right ) \left (-\frac {13781250 \left (-\frac {1}{1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}}+\frac {1}{2 \left (1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}\right )^{2}}\right )}{9765625 \,{\mathrm e}^{20}-41015625 \,{\mathrm e}^{16}[phone] \,{\mathrm e}^{12}-57881250 \,{\mathrm e}^{8}[phone] \,{\mathrm e}^{4}-4084101}+\frac {32812500 \,{\mathrm e}^{4} \left (-\frac {1}{1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}}+\frac {1}{2 \left (1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}\right )^{2}}\right )}{9765625 \,{\mathrm e}^{20}-41015625 \,{\mathrm e}^{16}[phone] \,{\mathrm e}^{12}-57881250 \,{\mathrm e}^{8}[phone] \,{\mathrm e}^{4}-4084101}-\frac {19531250 \,{\mathrm e}^{8} \left (-\frac {1}{1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}}+\frac {1}{2 \left (1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}\right )^{2}}\right )}{9765625 \,{\mathrm e}^{20}-41015625 \,{\mathrm e}^{16}[phone] \,{\mathrm e}^{12}-57881250 \,{\mathrm e}^{8}[phone] \,{\mathrm e}^{4}-4084101}\right )}{25}\)	\(265\)
default	\(-\frac {\left (25 \,{\mathrm e}^{4}-21\right ) \left (-\frac {13781250 \left (-\frac {1}{1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}}+\frac {1}{2 \left (1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}\right )^{2}}\right )}{9765625 \,{\mathrm e}^{20}-41015625 \,{\mathrm e}^{16}[phone] \,{\mathrm e}^{12}-57881250 \,{\mathrm e}^{8}[phone] \,{\mathrm e}^{4}-4084101}+\frac {32812500 \,{\mathrm e}^{4} \left (-\frac {1}{1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}}+\frac {1}{2 \left (1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}\right )^{2}}\right )}{9765625 \,{\mathrm e}^{20}-41015625 \,{\mathrm e}^{16}[phone] \,{\mathrm e}^{12}-57881250 \,{\mathrm e}^{8}[phone] \,{\mathrm e}^{4}-4084101}-\frac {19531250 \,{\mathrm e}^{8} \left (-\frac {1}{1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}}+\frac {1}{2 \left (1+\frac {25 \,{\mathrm e}^{4}-21}{-25 \,{\mathrm e}^{4}+25 x +21}\right )^{2}}\right )}{9765625 \,{\mathrm e}^{20}-41015625 \,{\mathrm e}^{16}[phone] \,{\mathrm e}^{12}-57881250 \,{\mathrm e}^{8}[phone] \,{\mathrm e}^{4}-4084101}\right )}{25}\)	\(265\)

[In]

int(((-1250*exp(4)^2+(1250*x+2100)*exp(4)-1250*x^2-1050*x-882)*exp(-25*x/(25*exp(4)-25*x-21))^2+(-7500*exp(4)^
2+(11250*x+12600)*exp(4)-7500*x^2-9450*x-5292)*exp(-25*x/(25*exp(4)-25*x-21))-11250*exp(4)^2+(22500*x+18900)*e
xp(4)-11250*x^2-18900*x-7938)/(5625*x^3*exp(4)^2+(-11250*x^4-9450*x^3)*exp(4)+5625*x^5+9450*x^4+3969*x^3),x,me
thod=_RETURNVERBOSE)

[Out]

1/x^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=\frac {e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21}\right )} + 6 \, e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21}\right )} + 9}{9 \, x^{2}} \]

[In]

integrate(((-1250*exp(4)^2+(1250*x+2100)*exp(4)-1250*x^2-1050*x-882)*exp(-25*x/(25*exp(4)-25*x-21))^2+(-7500*e
xp(4)^2+(11250*x+12600)*exp(4)-7500*x^2-9450*x-5292)*exp(-25*x/(25*exp(4)-25*x-21))-11250*exp(4)^2+(22500*x+18
900)*exp(4)-11250*x^2-18900*x-7938)/(5625*x^3*exp(4)^2+(-11250*x^4-9450*x^3)*exp(4)+5625*x^5+9450*x^4+3969*x^3
),x, algorithm="fricas")

[Out]

1/9*(e^(50*x/(25*x - 25*e^4 + 21)) + 6*e^(25*x/(25*x - 25*e^4 + 21)) + 9)/x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (19) = 38\).

Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=\frac {1}{x^{2}} + \frac {18 x^{2} e^{- \frac {25 x}{- 25 x - 21 + 25 e^{4}}} + 3 x^{2} e^{- \frac {50 x}{- 25 x - 21 + 25 e^{4}}}}{27 x^{4}} \]

[In]

integrate(((-1250*exp(4)**2+(1250*x+2100)*exp(4)-1250*x**2-1050*x-882)*exp(-25*x/(25*exp(4)-25*x-21))**2+(-750
0*exp(4)**2+(11250*x+12600)*exp(4)-7500*x**2-9450*x-5292)*exp(-25*x/(25*exp(4)-25*x-21))-11250*exp(4)**2+(2250
0*x+18900)*exp(4)-11250*x**2-18900*x-7938)/(5625*x**3*exp(4)**2+(-11250*x**4-9450*x**3)*exp(4)+5625*x**5+9450*
x**4+3969*x**3),x)

[Out]

x**(-2) + (18*x**2*exp(-25*x/(-25*x - 21 + 25*exp(4))) + 3*x**2*exp(-50*x/(-25*x - 21 + 25*exp(4))))/(27*x**4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 716 vs. \(2 (24) = 48\).

Time = 0.37 (sec) , antiderivative size = 716, normalized size of antiderivative = 26.52 \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=\text {Too large to display} \]

[In]

integrate(((-1250*exp(4)^2+(1250*x+2100)*exp(4)-1250*x^2-1050*x-882)*exp(-25*x/(25*exp(4)-25*x-21))^2+(-7500*e
xp(4)^2+(11250*x+12600)*exp(4)-7500*x^2-9450*x-5292)*exp(-25*x/(25*exp(4)-25*x-21))-11250*exp(4)^2+(22500*x+18
900)*exp(4)-11250*x^2-18900*x-7938)/(5625*x^3*exp(4)^2+(-11250*x^4-9450*x^3)*exp(4)+5625*x^5+9450*x^4+3969*x^3
),x, algorithm="maxima")

[Out]

625*((3750*x^2 - 75*x*(25*e^4 - 21) - 625*e^8 + 1050*e^4 - 441)/(25*x^3*(15625*e^12 - 39375*e^8 + 33075*e^4 -
9261) - x^2*(390625*e^16 - 1312500*e^12 + 1653750*e^8 - 926100*e^4 + 194481)) + 3750*log(25*x - 25*e^4 + 21)/(
390625*e^16 - 1312500*e^12 + 1653750*e^8 - 926100*e^4 + 194481) - 3750*log(x)/(390625*e^16 - 1312500*e^12 + 16
53750*e^8 - 926100*e^4 + 194481))*e^8 - 1050*((3750*x^2 - 75*x*(25*e^4 - 21) - 625*e^8 + 1050*e^4 - 441)/(25*x
^3*(15625*e^12 - 39375*e^8 + 33075*e^4 - 9261) - x^2*(390625*e^16 - 1312500*e^12 + 1653750*e^8 - 926100*e^4 +
194481)) + 3750*log(25*x - 25*e^4 + 21)/(390625*e^16 - 1312500*e^12 + 1653750*e^8 - 926100*e^4 + 194481) - 375
0*log(x)/(390625*e^16 - 1312500*e^12 + 1653750*e^8 - 926100*e^4 + 194481))*e^4 - 2500*((50*x - 25*e^4 + 21)/(2
5*x^2*(625*e^8 - 1050*e^4 + 441) - x*(15625*e^12 - 39375*e^8 + 33075*e^4 - 9261)) + 50*log(25*x - 25*e^4 + 21)
/(15625*e^12 - 39375*e^8 + 33075*e^4 - 9261) - 50*log(x)/(15625*e^12 - 39375*e^8 + 33075*e^4 - 9261))*e^4 + 44
1*(3750*x^2 - 75*x*(25*e^4 - 21) - 625*e^8 + 1050*e^4 - 441)/(25*x^3*(15625*e^12 - 39375*e^8 + 33075*e^4 - 926
1) - x^2*(390625*e^16 - 1312500*e^12 + 1653750*e^8 - 926100*e^4 + 194481)) + 2100*(50*x - 25*e^4 + 21)/(25*x^2
*(625*e^8 - 1050*e^4 + 441) - x*(15625*e^12 - 39375*e^8 + 33075*e^4 - 9261)) + 1/9*(e^(50*e^4/(25*x - 25*e^4 +
21) + 2) + 6*e^(25*e^4/(25*x - 25*e^4 + 21) + 21/(25*x - 25*e^4 + 21) + 1))*e^(-42/(25*x - 25*e^4 + 21))/x^2
+ 1653750*log(25*x - 25*e^4 + 21)/(390625*e^16 - 1312500*e^12 + 1653750*e^8 - 926100*e^4 + 194481) + 105000*lo
g(25*x - 25*e^4 + 21)/(15625*e^12 - 39375*e^8 + 33075*e^4 - 9261) + 1250*log(25*x - 25*e^4 + 21)/(625*e^8 - 10
50*e^4 + 441) - 1653750*log(x)/(390625*e^16 - 1312500*e^12 + 1653750*e^8 - 926100*e^4 + 194481) - 105000*log(x
)/(15625*e^12 - 39375*e^8 + 33075*e^4 - 9261) - 1250*log(x)/(625*e^8 - 1050*e^4 + 441) + 1250/(25*x*(25*e^4 -
21) - 625*e^8 + 1050*e^4 - 441)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (24) = 48\).

Time = 3.06 (sec) , antiderivative size = 625, normalized size of antiderivative = 23.15 \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=-\frac {\frac {281250 \, x e^{8}}{25 \, x - 25 \, e^{4} + 21} - \frac {472500 \, x e^{4}}{25 \, x - 25 \, e^{4} + 21} + \frac {22050 \, x e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21}\right )}}{25 \, x - 25 \, e^{4} + 21} - \frac {275625 \, x^{2} e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21}\right )}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} + \frac {132300 \, x e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21}\right )}}{25 \, x - 25 \, e^{4} + 21} - \frac {1653750 \, x^{2} e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21}\right )}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} + \frac {31250 \, x e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21} + 8\right )}}{25 \, x - 25 \, e^{4} + 21} - \frac {390625 \, x^{2} e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21} + 8\right )}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} - \frac {52500 \, x e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21} + 4\right )}}{25 \, x - 25 \, e^{4} + 21} + \frac {656250 \, x^{2} e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21} + 4\right )}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} + \frac {187500 \, x e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21} + 8\right )}}{25 \, x - 25 \, e^{4} + 21} - \frac {2343750 \, x^{2} e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21} + 8\right )}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} - \frac {315000 \, x e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21} + 4\right )}}{25 \, x - 25 \, e^{4} + 21} + \frac {3937500 \, x^{2} e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21} + 4\right )}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} + \frac {198450 \, x}{25 \, x - 25 \, e^{4} + 21} - 5625 \, e^{8} + 9450 \, e^{4} - 441 \, e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21}\right )} - 2646 \, e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21}\right )} - 625 \, e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21} + 8\right )} + 1050 \, e^{\left (\frac {50 \, x}{25 \, x - 25 \, e^{4} + 21} + 4\right )} - 3750 \, e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21} + 8\right )} + 6300 \, e^{\left (\frac {25 \, x}{25 \, x - 25 \, e^{4} + 21} + 4\right )} - 3969}{9 \, {\left (\frac {15625 \, x^{2} e^{12}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} - \frac {39375 \, x^{2} e^{8}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} + \frac {33075 \, x^{2} e^{4}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}} - \frac {9261 \, x^{2}}{{\left (25 \, x - 25 \, e^{4} + 21\right )}^{2}}\right )} {\left (25 \, e^{4} - 21\right )}} \]

[In]

integrate(((-1250*exp(4)^2+(1250*x+2100)*exp(4)-1250*x^2-1050*x-882)*exp(-25*x/(25*exp(4)-25*x-21))^2+(-7500*e
xp(4)^2+(11250*x+12600)*exp(4)-7500*x^2-9450*x-5292)*exp(-25*x/(25*exp(4)-25*x-21))-11250*exp(4)^2+(22500*x+18
900)*exp(4)-11250*x^2-18900*x-7938)/(5625*x^3*exp(4)^2+(-11250*x^4-9450*x^3)*exp(4)+5625*x^5+9450*x^4+3969*x^3
),x, algorithm="giac")

[Out]

-1/9*(281250*x*e^8/(25*x - 25*e^4 + 21) - 472500*x*e^4/(25*x - 25*e^4 + 21) + 22050*x*e^(50*x/(25*x - 25*e^4 +
21))/(25*x - 25*e^4 + 21) - 275625*x^2*e^(50*x/(25*x - 25*e^4 + 21))/(25*x - 25*e^4 + 21)^2 + 132300*x*e^(25*
x/(25*x - 25*e^4 + 21))/(25*x - 25*e^4 + 21) - 1653750*x^2*e^(25*x/(25*x - 25*e^4 + 21))/(25*x - 25*e^4 + 21)^
2 + 31250*x*e^(50*x/(25*x - 25*e^4 + 21) + 8)/(25*x - 25*e^4 + 21) - 390625*x^2*e^(50*x/(25*x - 25*e^4 + 21) +
8)/(25*x - 25*e^4 + 21)^2 - 52500*x*e^(50*x/(25*x - 25*e^4 + 21) + 4)/(25*x - 25*e^4 + 21) + 656250*x^2*e^(50
*x/(25*x - 25*e^4 + 21) + 4)/(25*x - 25*e^4 + 21)^2 + 187500*x*e^(25*x/(25*x - 25*e^4 + 21) + 8)/(25*x - 25*e^
4 + 21) - 2343750*x^2*e^(25*x/(25*x - 25*e^4 + 21) + 8)/(25*x - 25*e^4 + 21)^2 - 315000*x*e^(25*x/(25*x - 25*e
^4 + 21) + 4)/(25*x - 25*e^4 + 21) + 3937500*x^2*e^(25*x/(25*x - 25*e^4 + 21) + 4)/(25*x - 25*e^4 + 21)^2 + 19
8450*x/(25*x - 25*e^4 + 21) - 5625*e^8 + 9450*e^4 - 441*e^(50*x/(25*x - 25*e^4 + 21)) - 2646*e^(25*x/(25*x - 2
5*e^4 + 21)) - 625*e^(50*x/(25*x - 25*e^4 + 21) + 8) + 1050*e^(50*x/(25*x - 25*e^4 + 21) + 4) - 3750*e^(25*x/(
25*x - 25*e^4 + 21) + 8) + 6300*e^(25*x/(25*x - 25*e^4 + 21) + 4) - 3969)/((15625*x^2*e^12/(25*x - 25*e^4 + 21
)^2 - 39375*x^2*e^8/(25*x - 25*e^4 + 21)^2 + 33075*x^2*e^4/(25*x - 25*e^4 + 21)^2 - 9261*x^2/(25*x - 25*e^4 +
21)^2)*(25*e^4 - 21))

Mupad [B] (verification not implemented)

Time = 10.16 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {-7938-11250 e^8-18900 x-11250 x^2+e^4 (18900+22500 x)+e^{-\frac {50 x}{-21+25 e^4-25 x}} \left (-882-1250 e^8-1050 x-1250 x^2+e^4 (2100+1250 x)\right )+e^{-\frac {25 x}{-21+25 e^4-25 x}} \left (-5292-7500 e^8-9450 x-7500 x^2+e^4 (12600+11250 x)\right )}{3969 x^3+5625 e^8 x^3+9450 x^4+5625 x^5+e^4 \left (-9450 x^3-11250 x^4\right )} \, dx=\frac {{\left ({\mathrm {e}}^{\frac {25\,x}{25\,x-25\,{\mathrm {e}}^4+21}}+3\right )}^2}{9\,x^2} \]

[In]

int(-(18900*x + 11250*exp(8) + exp((50*x)/(25*x - 25*exp(4) + 21))*(1050*x + 1250*exp(8) + 1250*x^2 - exp(4)*(
1250*x + 2100) + 882) + exp((25*x)/(25*x - 25*exp(4) + 21))*(9450*x + 7500*exp(8) + 7500*x^2 - exp(4)*(11250*x
+ 12600) + 5292) + 11250*x^2 - exp(4)*(22500*x + 18900) + 7938)/(5625*x^3*exp(8) - exp(4)*(9450*x^3 + 11250*x
^4) + 3969*x^3 + 9450*x^4 + 5625*x^5),x)

[Out]

(exp((25*x)/(25*x - 25*exp(4) + 21)) + 3)^2/(9*x^2)