\(\int \frac {-1296 x^3+\frac {5}{4} e^x (324 x^3-324 x^4)}{-25600+40000 e^x-25000 e^{2 x}+\frac {15625 e^{3 x}}{2}-\frac {78125 e^{4 x}}{64}+\frac {78125 e^{5 x}}{1024}} \, dx\) [282]

Optimal result

Integrand size = 68, antiderivative size = 18 \[ \int \frac {-1296 x^3+\frac {5}{4} e^x \left (324 x^3-324 x^4\right )}{-25600+40000 e^x-25000 e^{2 x}+\frac {15625 e^{3 x}}{2}-\frac {78125 e^{4 x}}{64}+\frac {78125 e^{5 x}}{1024}} \, dx=\frac {81 x^4}{25 \left (-4+\frac {5 e^x}{4}\right )^4} \]

[Out]

81/25*x^4/(exp(ln(5/4)+x)-4)^4

Rubi [A] (verified)

Time = 5.79 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89, number of steps used = 226, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6820, 12, 6874, 2216, 2215, 2221, 2611, 6744, 2320, 6724, 2222, 2317, 2438, 36, 29, 31} \[ \int \frac {-1296 x^3+\frac {5}{4} e^x \left (324 x^3-324 x^4\right )}{-25600+40000 e^x-25000 e^{2 x}+\frac {15625 e^{3 x}}{2}-\frac {78125 e^{4 x}}{64}+\frac {78125 e^{5 x}}{1024}} \, dx=\frac {20736 x^4}{25 \left (16-5 e^x\right )^4} \]

[In]

Int[(-1296*x^3 + (5*E^x*(324*x^3 - 324*x^4))/4)/(-25600 + 40000*E^x - 25000*E^(2*x) + (15625*E^(3*x))/2 - (781
25*E^(4*x))/64 + (78125*E^(5*x))/1024),x]

[Out]

(20736*x^4)/(25*(16 - 5*E^x)^4)

Rule 12

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 2215

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

Rule 2216

Int[((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.))^(p_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dis
t[1/a, Int[(c + d*x)^m*(a + b*(F^(g*(e + f*x)))^n)^(p + 1), x], x] - Dist[b/a, Int[(c + d*x)^m*(F^(g*(e + f*x)
))^n*(a + b*(F^(g*(e + f*x)))^n)^p, x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && ILtQ[p, 0] && IGtQ[m, 0
]

Rule 2221

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

Rule 2222

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

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

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

Rule 6820

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {82944 \left (16+5 e^x (-1+x)\right ) x^3}{25 \left (16-5 e^x\right )^5} \, dx \\ & = \frac {82944}{25} \int \frac {\left (16+5 e^x (-1+x)\right ) x^3}{\left (16-5 e^x\right )^5} \, dx \\ & = \frac {82944}{25} \int \left (-\frac {(-1+x) x^3}{\left (-16+5 e^x\right )^4}-\frac {16 x^4}{\left (-16+5 e^x\right )^5}\right ) \, dx \\ & = -\left (\frac {82944}{25} \int \frac {(-1+x) x^3}{\left (-16+5 e^x\right )^4} \, dx\right )-\frac {1327104}{25} \int \frac {x^4}{\left (-16+5 e^x\right )^5} \, dx \\ & = \frac {82944}{25} \int \frac {x^4}{\left (-16+5 e^x\right )^4} \, dx-\frac {82944}{25} \int \left (-\frac {x^3}{\left (-16+5 e^x\right )^4}+\frac {x^4}{\left (-16+5 e^x\right )^4}\right ) \, dx-\frac {82944}{5} \int \frac {e^x x^4}{\left (-16+5 e^x\right )^5} \, dx \\ & = \frac {20736 x^4}{25 \left (16-5 e^x\right )^4}-\frac {5184}{25} \int \frac {x^4}{\left (-16+5 e^x\right )^3} \, dx+\frac {5184}{5} \int \frac {e^x x^4}{\left (-16+5 e^x\right )^4} \, dx-\frac {82944}{25} \int \frac {x^4}{\left (-16+5 e^x\right )^4} \, dx \\ & = \frac {20736 x^4}{25 \left (16-5 e^x\right )^4}+\frac {1728 x^4}{25 \left (16-5 e^x\right )^3}+\frac {324}{25} \int \frac {x^4}{\left (-16+5 e^x\right )^2} \, dx-\frac {324}{5} \int \frac {e^x x^4}{\left (-16+5 e^x\right )^3} \, dx+\frac {5184}{25} \int \frac {x^4}{\left (-16+5 e^x\right )^3} \, dx+\frac {6912}{25} \int \frac {x^3}{\left (-16+5 e^x\right )^3} \, dx-\frac {5184}{5} \int \frac {e^x x^4}{\left (-16+5 e^x\right )^4} \, dx \\ & = \frac {20736 x^4}{25 \left (16-5 e^x\right )^4}+\frac {162 x^4}{25 \left (16-5 e^x\right )^2}-\frac {81}{100} \int \frac {x^4}{-16+5 e^x} \, dx+\frac {81}{20} \int \frac {e^x x^4}{\left (-16+5 e^x\right )^2} \, dx-\frac {324}{25} \int \frac {x^4}{\left (-16+5 e^x\right )^2} \, dx-\frac {432}{25} \int \frac {x^3}{\left (-16+5 e^x\right )^2} \, dx-\frac {648}{25} \int \frac {x^3}{\left (-16+5 e^x\right )^2} \, dx+\frac {324}{5} \int \frac {e^x x^4}{\left (-16+5 e^x\right )^3} \, dx+\frac {432}{5} \int \frac {e^x x^3}{\left (-16+5 e^x\right )^3} \, dx-\frac {6912}{25} \int \frac {x^3}{\left (-16+5 e^x\right )^3} \, dx \\ & = -\frac {216 x^3}{25 \left (16-5 e^x\right )^2}+\frac {20736 x^4}{25 \left (16-5 e^x\right )^4}+\frac {81 x^4}{100 \left (16-5 e^x\right )}+\frac {81 x^5}{8000}-\frac {81}{320} \int \frac {e^x x^4}{-16+5 e^x} \, dx+\frac {81}{100} \int \frac {x^4}{-16+5 e^x} \, dx+\frac {27}{25} \int \frac {x^3}{-16+5 e^x} \, dx+\frac {81}{50} \int \frac {x^3}{-16+5 e^x} \, dx+\frac {81}{25} \int \frac {x^3}{-16+5 e^x} \, dx-\frac {81}{20} \int \frac {e^x x^4}{\left (-16+5 e^x\right )^2} \, dx-\frac {27}{5} \int \frac {e^x x^3}{\left (-16+5 e^x\right )^2} \, dx-\frac {81}{10} \int \frac {e^x x^3}{\left (-16+5 e^x\right )^2} \, dx+\frac {432}{25} \int \frac {x^3}{\left (-16+5 e^x\right )^2} \, dx+\frac {648}{25} \int \frac {x^2}{\left (-16+5 e^x\right )^2} \, dx+\frac {648}{25} \int \frac {x^3}{\left (-16+5 e^x\right )^2} \, dx-\frac {432}{5} \int \frac {e^x x^3}{\left (-16+5 e^x\right )^3} \, dx \\ & = -\frac {27 x^3}{10 \left (16-5 e^x\right )}-\frac {297 x^4}{3200}+\frac {20736 x^4}{25 \left (16-5 e^x\right )^4}-\frac {81 x^4 \log \left (1-\frac {5 e^x}{16}\right )}{1600}+\frac {81}{400} \int x^3 \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {81}{320} \int \frac {e^x x^4}{-16+5 e^x} \, dx+\frac {27}{80} \int \frac {e^x x^3}{-16+5 e^x} \, dx+\frac {81}{160} \int \frac {e^x x^3}{-16+5 e^x} \, dx+\frac {81}{80} \int \frac {e^x x^3}{-16+5 e^x} \, dx-\frac {27}{25} \int \frac {x^3}{-16+5 e^x} \, dx-\frac {81}{50} \int \frac {x^2}{-16+5 e^x} \, dx-\frac {81}{50} \int \frac {x^3}{-16+5 e^x} \, dx-\frac {81}{25} \int \frac {x^2}{-16+5 e^x} \, dx-\frac {81}{25} \int \frac {x^3}{-16+5 e^x} \, dx-\frac {243}{50} \int \frac {x^2}{-16+5 e^x} \, dx+\frac {27}{5} \int \frac {e^x x^3}{\left (-16+5 e^x\right )^2} \, dx+\frac {81}{10} \int \frac {e^x x^2}{\left (-16+5 e^x\right )^2} \, dx+\frac {81}{10} \int \frac {e^x x^3}{\left (-16+5 e^x\right )^2} \, dx-\frac {648}{25} \int \frac {x^2}{\left (-16+5 e^x\right )^2} \, dx \\ & = \frac {81 x^2}{50 \left (16-5 e^x\right )}+\frac {81 x^3}{400}+\frac {20736 x^4}{25 \left (16-5 e^x\right )^4}+\frac {297}{800} x^3 \log \left (1-\frac {5 e^x}{16}\right )-\frac {81}{400} x^3 \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right )-\frac {81}{400} \int x^2 \log \left (1-\frac {5 e^x}{16}\right ) \, dx-\frac {81}{400} \int x^3 \log \left (1-\frac {5 e^x}{16}\right ) \, dx-\frac {243}{800} \int x^2 \log \left (1-\frac {5 e^x}{16}\right ) \, dx-\frac {27}{80} \int \frac {e^x x^3}{-16+5 e^x} \, dx-\frac {81}{160} \int \frac {e^x x^2}{-16+5 e^x} \, dx-\frac {81}{160} \int \frac {e^x x^3}{-16+5 e^x} \, dx-\frac {243}{400} \int x^2 \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {243}{400} \int x^2 \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right ) \, dx-\frac {81}{80} \int \frac {e^x x^2}{-16+5 e^x} \, dx-\frac {81}{80} \int \frac {e^x x^3}{-16+5 e^x} \, dx-\frac {243}{160} \int \frac {e^x x^2}{-16+5 e^x} \, dx+\frac {81}{50} \int \frac {x^2}{-16+5 e^x} \, dx+\frac {81}{25} \int \frac {x}{-16+5 e^x} \, dx+\frac {81}{25} \int \frac {x^2}{-16+5 e^x} \, dx+\frac {243}{50} \int \frac {x^2}{-16+5 e^x} \, dx-\frac {81}{10} \int \frac {e^x x^2}{\left (-16+5 e^x\right )^2} \, dx \\ & = -\frac {81 x^2}{800}+\frac {20736 x^4}{25 \left (16-5 e^x\right )^4}-\frac {243}{400} x^2 \log \left (1-\frac {5 e^x}{16}\right )+\frac {891}{800} x^2 \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right )+\frac {243}{400} x^2 \operatorname {PolyLog}\left (3,\frac {5 e^x}{16}\right )+\frac {81}{400} \int x \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {81}{400} \int x^2 \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {243}{800} \int x^2 \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {81}{200} \int x \log \left (1-\frac {5 e^x}{16}\right ) \, dx-\frac {81}{200} \int x \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right ) \, dx+\frac {81}{160} \int \frac {e^x x^2}{-16+5 e^x} \, dx+\frac {243}{400} \int x \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {243}{400} \int x^2 \log \left (1-\frac {5 e^x}{16}\right ) \, dx-\frac {243}{400} \int x \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right ) \, dx-\frac {243}{400} \int x^2 \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right ) \, dx+\frac {81}{80} \int \frac {e^x x}{-16+5 e^x} \, dx+\frac {81}{80} \int \frac {e^x x^2}{-16+5 e^x} \, dx-\frac {243}{200} \int x \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right ) \, dx-\frac {243}{200} \int x \operatorname {PolyLog}\left (3,\frac {5 e^x}{16}\right ) \, dx+\frac {243}{160} \int \frac {e^x x^2}{-16+5 e^x} \, dx-\frac {81}{25} \int \frac {x}{-16+5 e^x} \, dx \\ & = \frac {20736 x^4}{25 \left (16-5 e^x\right )^4}+\frac {81}{400} x \log \left (1-\frac {5 e^x}{16}\right )-\frac {243}{200} x \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right )-\frac {891}{400} x \operatorname {PolyLog}\left (3,\frac {5 e^x}{16}\right )-\frac {243}{200} x \operatorname {PolyLog}\left (4,\frac {5 e^x}{16}\right )-\frac {81}{400} \int \log \left (1-\frac {5 e^x}{16}\right ) \, dx-\frac {81}{400} \int x \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {81}{400} \int \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right ) \, dx-\frac {81}{200} \int x \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {81}{200} \int \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right ) \, dx+\frac {81}{200} \int x \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right ) \, dx+\frac {81}{200} \int \operatorname {PolyLog}\left (3,\frac {5 e^x}{16}\right ) \, dx-\frac {243}{400} \int x \log \left (1-\frac {5 e^x}{16}\right ) \, dx+\frac {243}{400} \int \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right ) \, dx+\frac {243}{400} \int x \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right ) \, dx+\frac {243}{400} \int \operatorname {PolyLog}\left (3,\frac {5 e^x}{16}\right ) \, dx-\frac {81}{80} \int \frac {e^x x}{-16+5 e^x} \, dx+\frac {243}{200} \int x \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right ) \, dx+\frac {243}{200} \int \operatorname {PolyLog}\left (3,\frac {5 e^x}{16}\right ) \, dx+\frac {243}{200} \int x \operatorname {PolyLog}\left (3,\frac {5 e^x}{16}\right ) \, dx+\frac {243}{200} \int \operatorname {PolyLog}\left (4,\frac {5 e^x}{16}\right ) \, dx \\ & = \frac {20736 x^4}{25 \left (16-5 e^x\right )^4}+\frac {81}{400} \int \log \left (1-\frac {5 e^x}{16}\right ) \, dx-\frac {81}{400} \int \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right ) \, dx-\frac {81}{400} \text {Subst}\left (\int \frac {\log \left (1-\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )+\frac {81}{400} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {81}{200} \int \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right ) \, dx-\frac {81}{200} \int \operatorname {PolyLog}\left (3,\frac {5 e^x}{16}\right ) \, dx+\frac {81}{200} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )+\frac {81}{200} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {243}{400} \int \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right ) \, dx-\frac {243}{400} \int \operatorname {PolyLog}\left (3,\frac {5 e^x}{16}\right ) \, dx+\frac {243}{400} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )+\frac {243}{400} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {243}{200} \int \operatorname {PolyLog}\left (3,\frac {5 e^x}{16}\right ) \, dx-\frac {243}{200} \int \operatorname {PolyLog}\left (4,\frac {5 e^x}{16}\right ) \, dx+\frac {243}{200} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )+\frac {243}{200} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (4,\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right ) \\ & = \frac {20736 x^4}{25 \left (16-5 e^x\right )^4}+\frac {81}{400} \operatorname {PolyLog}\left (2,\frac {5 e^x}{16}\right )+\frac {243}{200} \operatorname {PolyLog}\left (3,\frac {5 e^x}{16}\right )+\frac {891}{400} \operatorname {PolyLog}\left (4,\frac {5 e^x}{16}\right )+\frac {243}{200} \operatorname {PolyLog}\left (5,\frac {5 e^x}{16}\right )+\frac {81}{400} \text {Subst}\left (\int \frac {\log \left (1-\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {81}{400} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {81}{200} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {81}{200} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {243}{400} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {243}{400} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {243}{200} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right )-\frac {243}{200} \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (4,\frac {5 x}{16}\right )}{x} \, dx,x,e^x\right ) \\ & = \frac {20736 x^4}{25 \left (16-5 e^x\right )^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \frac {-1296 x^3+\frac {5}{4} e^x \left (324 x^3-324 x^4\right )}{-25600+40000 e^x-25000 e^{2 x}+\frac {15625 e^{3 x}}{2}-\frac {78125 e^{4 x}}{64}+\frac {78125 e^{5 x}}{1024}} \, dx=\frac {20736 x^4}{25 \left (-16+5 e^x\right )^4} \]

[In]

Integrate[(-1296*x^3 + (5*E^x*(324*x^3 - 324*x^4))/4)/(-25600 + 40000*E^x - 25000*E^(2*x) + (15625*E^(3*x))/2
- (78125*E^(4*x))/64 + (78125*E^(5*x))/1024),x]

[Out]

(20736*x^4)/(25*(-16 + 5*E^x)^4)

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78

method	result	size
risch	\(\frac {81 x^{4}}{25 \left (\frac {5 \,{\mathrm e}^{x}}{4}-4\right )^{4}}\)	\(14\)
norman	\(\frac {81 x^{4}}{25 \left ({\mathrm e}^{\ln \left (\frac {5}{4}\right )+x}-4\right )^{4}}\)	\(15\)
parallelrisch	\(\frac {81 x^{4}}{25 \left (\frac {625 \,{\mathrm e}^{4 x}}{256}-\frac {125 \,{\mathrm e}^{3 x}}{4}+150 \,{\mathrm e}^{2 x}-256 \,{\mathrm e}^{\ln \left (\frac {5}{4}\right )+x}+256\right )}\)	\(42\)

[In]

int(((-324*x^4+324*x^3)*exp(ln(5/4)+x)-1296*x^3)/(25*exp(ln(5/4)+x)^5-500*exp(ln(5/4)+x)^4+4000*exp(ln(5/4)+x)
^3-16000*exp(ln(5/4)+x)^2+32000*exp(ln(5/4)+x)-25600),x,method=_RETURNVERBOSE)

[Out]

81/25*x^4/(5/4*exp(x)-4)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (14) = 28\).

Time = 0.24 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.61 \[ \int \frac {-1296 x^3+\frac {5}{4} e^x \left (324 x^3-324 x^4\right )}{-25600+40000 e^x-25000 e^{2 x}+\frac {15625 e^{3 x}}{2}-\frac {78125 e^{4 x}}{64}+\frac {78125 e^{5 x}}{1024}} \, dx=\frac {81 \, x^{4}}{25 \, {\left (e^{\left (4 \, x + 4 \, \log \left (\frac {5}{4}\right )\right )} - 16 \, e^{\left (3 \, x + 3 \, \log \left (\frac {5}{4}\right )\right )} + 96 \, e^{\left (2 \, x + 2 \, \log \left (\frac {5}{4}\right )\right )} - 256 \, e^{\left (x + \log \left (\frac {5}{4}\right )\right )} + 256\right )}} \]

[In]

integrate(((-324*x^4+324*x^3)*exp(log(5/4)+x)-1296*x^3)/(25*exp(log(5/4)+x)^5-500*exp(log(5/4)+x)^4+4000*exp(l
og(5/4)+x)^3-16000*exp(log(5/4)+x)^2+32000*exp(log(5/4)+x)-25600),x, algorithm="fricas")

[Out]

81/25*x^4/(e^(4*x + 4*log(5/4)) - 16*e^(3*x + 3*log(5/4)) + 96*e^(2*x + 2*log(5/4)) - 256*e^(x + log(5/4)) + 2
56)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).

Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {-1296 x^3+\frac {5}{4} e^x \left (324 x^3-324 x^4\right )}{-25600+40000 e^x-25000 e^{2 x}+\frac {15625 e^{3 x}}{2}-\frac {78125 e^{4 x}}{64}+\frac {78125 e^{5 x}}{1024}} \, dx=\frac {20736 x^{4}}{15625 e^{4 x} - 200000 e^{3 x} + 960000 e^{2 x} - 2048000 e^{x} + 1638400} \]

[In]

integrate(((-324*x**4+324*x**3)*exp(ln(5/4)+x)-1296*x**3)/(25*exp(ln(5/4)+x)**5-500*exp(ln(5/4)+x)**4+4000*exp
(ln(5/4)+x)**3-16000*exp(ln(5/4)+x)**2+32000*exp(ln(5/4)+x)-25600),x)

[Out]

20736*x**4/(15625*exp(4*x) - 200000*exp(3*x) + 960000*exp(2*x) - 2048000*exp(x) + 1638400)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (14) = 28\).

Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {-1296 x^3+\frac {5}{4} e^x \left (324 x^3-324 x^4\right )}{-25600+40000 e^x-25000 e^{2 x}+\frac {15625 e^{3 x}}{2}-\frac {78125 e^{4 x}}{64}+\frac {78125 e^{5 x}}{1024}} \, dx=\frac {20736 \, x^{4}}{25 \, {\left (625 \, e^{\left (4 \, x\right )} - 8000 \, e^{\left (3 \, x\right )} + 38400 \, e^{\left (2 \, x\right )} - 81920 \, e^{x} + 65536\right )}} \]

[In]

integrate(((-324*x^4+324*x^3)*exp(log(5/4)+x)-1296*x^3)/(25*exp(log(5/4)+x)^5-500*exp(log(5/4)+x)^4+4000*exp(l
og(5/4)+x)^3-16000*exp(log(5/4)+x)^2+32000*exp(log(5/4)+x)-25600),x, algorithm="maxima")

[Out]

20736/25*x^4/(625*e^(4*x) - 8000*e^(3*x) + 38400*e^(2*x) - 81920*e^x + 65536)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (14) = 28\).

Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {-1296 x^3+\frac {5}{4} e^x \left (324 x^3-324 x^4\right )}{-25600+40000 e^x-25000 e^{2 x}+\frac {15625 e^{3 x}}{2}-\frac {78125 e^{4 x}}{64}+\frac {78125 e^{5 x}}{1024}} \, dx=\frac {20736 \, x^{4}}{25 \, {\left (625 \, e^{\left (4 \, x\right )} - 8000 \, e^{\left (3 \, x\right )} + 38400 \, e^{\left (2 \, x\right )} - 81920 \, e^{x} + 65536\right )}} \]

[In]

integrate(((-324*x^4+324*x^3)*exp(log(5/4)+x)-1296*x^3)/(25*exp(log(5/4)+x)^5-500*exp(log(5/4)+x)^4+4000*exp(l
og(5/4)+x)^3-16000*exp(log(5/4)+x)^2+32000*exp(log(5/4)+x)-25600),x, algorithm="giac")

[Out]

20736/25*x^4/(625*e^(4*x) - 8000*e^(3*x) + 38400*e^(2*x) - 81920*e^x + 65536)

Mupad [B] (verification not implemented)

Time = 9.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.72 \[ \int \frac {-1296 x^3+\frac {5}{4} e^x \left (324 x^3-324 x^4\right )}{-25600+40000 e^x-25000 e^{2 x}+\frac {15625 e^{3 x}}{2}-\frac {78125 e^{4 x}}{64}+\frac {78125 e^{5 x}}{1024}} \, dx=\frac {81\,x^4}{25\,\left (150\,{\mathrm {e}}^{2\,x}-\frac {125\,{\mathrm {e}}^{3\,x}}{4}+\frac {625\,{\mathrm {e}}^{4\,x}}{256}-320\,{\mathrm {e}}^x+256\right )} \]

[In]

int(-(exp(x + log(5/4))*(324*x^3 - 324*x^4) - 1296*x^3)/(16000*exp(2*x + 2*log(5/4)) - 4000*exp(3*x + 3*log(5/
4)) + 500*exp(4*x + 4*log(5/4)) - 25*exp(5*x + 5*log(5/4)) - 32000*exp(x + log(5/4)) + 25600),x)

[Out]

(81*x^4)/(25*(150*exp(2*x) - (125*exp(3*x))/4 + (625*exp(4*x))/256 - 320*exp(x) + 256))