3.8.30 \(\int \frac {e^{5+x} (-80 x+(-960 x+384 x^2+48 x^3) \log (2))+e^{5+x} (1280-1200 x-80 x^2+(-1024+1984 x-1152 x^2+176 x^3+16 x^4) \log (2)) \log (-80-5 x+(64-60 x+12 x^2+x^3) \log (2))}{-80 x^2-5 x^3+(64 x^2-60 x^3+12 x^4+x^5) \log (2)} \, dx\)

Optimal. Leaf size=27 \[ \frac {16 e^{5+x} \log \left ((16+x) \left (-5+(2-x)^2 \log (2)\right )\right )}{x} \]

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Rubi [C]  time = 13.08, antiderivative size = 633, normalized size of antiderivative = 23.44, number of steps used = 21, number of rules used = 7, integrand size = 125, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6741, 6688, 6728, 2178, 2270, 2197, 2554} \begin {gather*} -\frac {8 e^{7+\frac {\sqrt {\log (32)}}{\log (2)}} \left (2400 \log ^3(2)+20 \log ^2(2) \left (155-27 \log (16)-60 \sqrt {\log (32)}\right )-\log (8) (5-\log (16))^2+4 \sqrt {\log (32)} (\log (32)-\log (16) \log (64))+\log (2) (25+8 \log (32)-\log (16) (5+8 \log (64)))\right ) \text {Ei}\left (\frac {\log (2) x-\sqrt {\log (32)}-\log (4)}{\log (2)}\right )}{(5-\log (16)) (5-320 \log (2)-\log (16)) \sqrt {\log (32)}}+\frac {8 e^{7-\frac {\sqrt {\log (32)}}{\log (2)}} \left (2400 \log ^3(2)+20 \log ^2(2) \left (155-27 \log (16)+60 \sqrt {\log (32)}\right )-\log (8) (5-\log (16))^2-4 \sqrt {\log (32)} (\log (32)-\log (16) \log (64))+\log (2) (25+8 \log (32)-\log (16) (5+8 \log (64)))\right ) \text {Ei}\left (\frac {2 \log (2) x+2 \sqrt {\log (32)}-\log (16)}{2 \log (2)}\right )}{(5-\log (16)) (5-320 \log (2)-\log (16)) \sqrt {\log (32)}}-\frac {8 e^{7-\frac {\sqrt {\log (32)}}{\log (2)}} \left (2560 \log ^3(2)+\log (2) \left (25+\log ^2(16)\right )+8 \log ^2(2) \left (390-79 \log (16)+160 \sqrt {\log (32)}\right )+\left (\log ^2(16)-4 \log (32)\right ) \sqrt {\log (32)}-\log (8) (5-\log (16))^2\right ) \text {Ei}\left (\frac {2 \log (2) x+2 \sqrt {\log (32)}-\log (16)}{2 \log (2)}\right )}{(5-\log (16)) (5-320 \log (2)-\log (16)) \sqrt {\log (32)}}+\frac {8 e^{7+\frac {\sqrt {\log (32)}}{\log (2)}} \left (4 \log ^{\frac {3}{2}}(32)+2560 \log ^3(2)+8 \log ^2(2) \left (390-79 \log (16)-160 \sqrt {\log (32)}\right )+\log (2) \left (25+\log ^2(16)-4 \log (16) \sqrt {\log (32)}\right )-\log (8) (5-\log (16))^2\right ) \text {Ei}\left (\frac {\log (2) x-\sqrt {\log (32)}-\log (4)}{\log (2)}\right )}{(5-\log (16)) (5-320 \log (2)-\log (16)) \sqrt {\log (32)}}+\frac {5 e^5 (1+\log (4096)) \text {Ei}(x)}{5-\log (16)}-\frac {e^5 (5+64 \log (2)-\log (16)) \text {Ei}(x)}{5-\log (16)}+\frac {16 e^{x+5} \log \left (-\left ((x+16) \left (x^2 (-\log (2))+4 x \log (2)+5-\log (16)\right )\right )\right )}{x} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(E^(5 + x)*(-80*x + (-960*x + 384*x^2 + 48*x^3)*Log[2]) + E^(5 + x)*(1280 - 1200*x - 80*x^2 + (-1024 + 198
4*x - 1152*x^2 + 176*x^3 + 16*x^4)*Log[2])*Log[-80 - 5*x + (64 - 60*x + 12*x^2 + x^3)*Log[2]])/(-80*x^2 - 5*x^
3 + (64*x^2 - 60*x^3 + 12*x^4 + x^5)*Log[2]),x]

[Out]

-((E^5*ExpIntegralEi[x]*(5 + 64*Log[2] - Log[16]))/(5 - Log[16])) - (8*E^(7 - Sqrt[Log[32]]/Log[2])*ExpIntegra
lEi[(2*x*Log[2] - Log[16] + 2*Sqrt[Log[32]])/(2*Log[2])]*(2560*Log[2]^3 - Log[8]*(5 - Log[16])^2 + Log[2]*(25
+ Log[16]^2) + 8*Log[2]^2*(390 - 79*Log[16] + 160*Sqrt[Log[32]]) + (Log[16]^2 - 4*Log[32])*Sqrt[Log[32]]))/((5
- Log[16])*(5 - 320*Log[2] - Log[16])*Sqrt[Log[32]]) + (8*E^(7 + Sqrt[Log[32]]/Log[2])*ExpIntegralEi[(x*Log[2
] - Log[4] - Sqrt[Log[32]])/Log[2]]*(2560*Log[2]^3 - Log[8]*(5 - Log[16])^2 + 8*Log[2]^2*(390 - 79*Log[16] - 1
60*Sqrt[Log[32]]) + Log[2]*(25 + Log[16]^2 - 4*Log[16]*Sqrt[Log[32]]) + 4*Log[32]^(3/2)))/((5 - Log[16])*(5 -
320*Log[2] - Log[16])*Sqrt[Log[32]]) + (8*E^(7 - Sqrt[Log[32]]/Log[2])*ExpIntegralEi[(2*x*Log[2] - Log[16] + 2
*Sqrt[Log[32]])/(2*Log[2])]*(2400*Log[2]^3 - Log[8]*(5 - Log[16])^2 + 20*Log[2]^2*(155 - 27*Log[16] + 60*Sqrt[
Log[32]]) - 4*Sqrt[Log[32]]*(Log[32] - Log[16]*Log[64]) + Log[2]*(25 + 8*Log[32] - Log[16]*(5 + 8*Log[64]))))/
((5 - Log[16])*(5 - 320*Log[2] - Log[16])*Sqrt[Log[32]]) - (8*E^(7 + Sqrt[Log[32]]/Log[2])*ExpIntegralEi[(x*Lo
g[2] - Log[4] - Sqrt[Log[32]])/Log[2]]*(2400*Log[2]^3 - Log[8]*(5 - Log[16])^2 + 20*Log[2]^2*(155 - 27*Log[16]
- 60*Sqrt[Log[32]]) + 4*Sqrt[Log[32]]*(Log[32] - Log[16]*Log[64]) + Log[2]*(25 + 8*Log[32] - Log[16]*(5 + 8*L
og[64]))))/((5 - Log[16])*(5 - 320*Log[2] - Log[16])*Sqrt[Log[32]]) + (5*E^5*ExpIntegralEi[x]*(1 + Log[4096]))
/(5 - Log[16]) + (16*E^(5 + x)*Log[-((16 + x)*(5 + 4*x*Log[2] - x^2*Log[2] - Log[16]))])/x

Rule 2178

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

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rule 2270

Int[((F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))*(u_)^(m_.))/((a_.) + (b_.)*(x_) + (c_)*(x_)^2), x_Symbol] :> Int[
ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x] && Poly
nomialQ[u, x] && IntegerQ[m]

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6688

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

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^{5+x} \left (-80 x+\left (-960 x+384 x^2+48 x^3\right ) \log (2)\right )-e^{5+x} \left (1280-1200 x-80 x^2+\left (-1024+1984 x-1152 x^2+176 x^3+16 x^4\right ) \log (2)\right ) \log \left (-80-5 x+\left (64-60 x+12 x^2+x^3\right ) \log (2)\right )}{x^2 (16+x) \left (5+4 x \log (2)-x^2 \log (2)-\log (16)\right )} \, dx\\ &=\int \frac {e^{5+x} \left (80 x-48 x \left (-20+8 x+x^2\right ) \log (2)-16 \left (-16+15 x+x^2\right ) \left (-5-4 x \log (2)+x^2 \log (2)+\log (16)\right ) \log \left ((16+x) \left (-5-4 x \log (2)+x^2 \log (2)+\log (16)\right )\right )\right )}{x^2 (16+x) \left (5+4 x \log (2)-x^2 \log (2)-\log (16)\right )} \, dx\\ &=\int \left (\frac {16 e^{5+x} \left (-24 x \log (2)+5 (1+12 \log (2))-x^2 \log (8)\right )}{x (16+x) \left (5+4 x \log (2)-x^2 \log (2)-\log (16)\right )}+\frac {16 e^{5+x} (-1+x) \log \left ((16+x) \left (-5-4 x \log (2)+x^2 \log (2)+\log (16)\right )\right )}{x^2}\right ) \, dx\\ &=16 \int \frac {e^{5+x} \left (-24 x \log (2)+5 (1+12 \log (2))-x^2 \log (8)\right )}{x (16+x) \left (5+4 x \log (2)-x^2 \log (2)-\log (16)\right )} \, dx+16 \int \frac {e^{5+x} (-1+x) \log \left ((16+x) \left (-5-4 x \log (2)+x^2 \log (2)+\log (16)\right )\right )}{x^2} \, dx\\ &=\frac {16 e^{5+x} \log \left (-\left ((16+x) \left (5+4 x \log (2)-x^2 \log (2)-\log (16)\right )\right )\right )}{x}-16 \int \frac {e^{5+x} \left (5+64 \log (2)-24 x \log (2)-x^2 \log (8)-\log (16)\right )}{x (16+x) \left (5+4 x \log (2)-x^2 \log (2)-\log (16)\right )} \, dx+16 \int \left (\frac {e^{5+x} (5+444 \log (2)-256 \log (8))}{16 (16+x) (-5+320 \log (2)+\log (16))}+\frac {e^{5+x} \left (4800 \log ^3(2)+20 \log ^2(2) (155-27 \log (16))+5 \log (2) (5-\log (16))-\log (8) (5-\log (16))^2-4 x \log (2) \left (300 \log ^2(2)-\log (32)+\log (16) \log (64)\right )\right )}{(5-\log (16)) (5-320 \log (2)-\log (16)) \left (5+4 x \log (2)-x^2 \log (2)-\log (16)\right )}-\frac {5 e^{5+x} (1+\log (4096))}{16 x (-5+\log (16))}\right ) \, dx\\ &=\frac {16 e^{5+x} \log \left (-\left ((16+x) \left (5+4 x \log (2)-x^2 \log (2)-\log (16)\right )\right )\right )}{x}-16 \int \left (\frac {e^{5+x} (-5-64 \log (2)+\log (16))}{16 x (-5+\log (16))}+\frac {e^{5+x} (5+448 \log (2)-256 \log (8)-\log (16))}{16 (16+x) (-5+320 \log (2)+\log (16))}+\frac {e^{5+x} \left (5120 \log ^3(2)+624 \log ^2(2) (5-\log (16))+\log (2) (5-\log (16))^2-\log (8) (5-\log (16))^2-4 x \log (2) \left (320 \log ^2(2)+\log (2) \log (16)-\log (32)\right )\right )}{(5-\log (16)) (5-320 \log (2)-\log (16)) \left (5+4 x \log (2)-x^2 \log (2)-\log (16)\right )}\right ) \, dx+\frac {16 \int \frac {e^{5+x} \left (4800 \log ^3(2)+20 \log ^2(2) (155-27 \log (16))+5 \log (2) (5-\log (16))-\log (8) (5-\log (16))^2-4 x \log (2) \left (300 \log ^2(2)-\log (32)+\log (16) \log (64)\right )\right )}{5+4 x \log (2)-x^2 \log (2)-\log (16)} \, dx}{(5-\log (16)) (5-320 \log (2)-\log (16))}+\frac {(5 (1+\log (4096))) \int \frac {e^{5+x}}{x} \, dx}{5-\log (16)}-\int \frac {e^{5+x}}{16+x} \, dx\\ &=-\frac {\text {Ei}(16+x)}{e^{11}}+\frac {5 e^5 \text {Ei}(x) (1+\log (4096))}{5-\log (16)}+\frac {16 e^{5+x} \log \left (-\left ((16+x) \left (5+4 x \log (2)-x^2 \log (2)-\log (16)\right )\right )\right )}{x}-\frac {16 \int \frac {e^{5+x} \left (5120 \log ^3(2)+624 \log ^2(2) (5-\log (16))+\log (2) (5-\log (16))^2-\log (8) (5-\log (16))^2-4 x \log (2) \left (320 \log ^2(2)+\log (2) \log (16)-\log (32)\right )\right )}{5+4 x \log (2)-x^2 \log (2)-\log (16)} \, dx}{(5-\log (16)) (5-320 \log (2)-\log (16))}+\frac {16 \int \left (\frac {e^{5+x} \left (-4 \log (2) \left (300 \log ^2(2)-\log (32)+\log (16) \log (64)\right )-\frac {\log (2) \left (25 \log (2)+3100 \log ^2(2)+2400 \log ^3(2)-25 \log (8)-5 \log (2) \log (16)-540 \log ^2(2) \log (16)+10 \log (8) \log (16)-\log (8) \log ^2(16)+8 \log (2) \log (32)-8 \log (2) \log (16) \log (64)\right )}{\sqrt {\log (32)}}\right )}{4 \log (2)-2 x \log (2)-2 \sqrt {\log (32)}}+\frac {e^{5+x} \left (-4 \log (2) \left (300 \log ^2(2)-\log (32)+\log (16) \log (64)\right )+\frac {\log (2) \left (25 \log (2)+3100 \log ^2(2)+2400 \log ^3(2)-25 \log (8)-5 \log (2) \log (16)-540 \log ^2(2) \log (16)+10 \log (8) \log (16)-\log (8) \log ^2(16)+8 \log (2) \log (32)-8 \log (2) \log (16) \log (64)\right )}{\sqrt {\log (32)}}\right )}{4 \log (2)-2 x \log (2)+2 \sqrt {\log (32)}}\right ) \, dx}{(5-\log (16)) (5-320 \log (2)-\log (16))}-\frac {(5+64 \log (2)-\log (16)) \int \frac {e^{5+x}}{x} \, dx}{5-\log (16)}+\int \frac {e^{5+x}}{16+x} \, dx\\ &=-\frac {e^5 \text {Ei}(x) (5+64 \log (2)-\log (16))}{5-\log (16)}+\frac {5 e^5 \text {Ei}(x) (1+\log (4096))}{5-\log (16)}+\frac {16 e^{5+x} \log \left (-\left ((16+x) \left (5+4 x \log (2)-x^2 \log (2)-\log (16)\right )\right )\right )}{x}-\frac {16 \int \left (\frac {e^{5+x} \left (-4 \log (2) \left (320 \log ^2(2)+\log (2) \log (16)-\log (32)\right )-\frac {\log (2) \left (25 \log (2)+3120 \log ^2(2)+2560 \log ^3(2)-25 \log (8)-632 \log ^2(2) \log (16)+10 \log (8) \log (16)+\log (2) \log ^2(16)-\log (8) \log ^2(16)\right )}{\sqrt {\log (32)}}\right )}{4 \log (2)-2 x \log (2)-2 \sqrt {\log (32)}}+\frac {e^{5+x} \left (-4 \log (2) \left (320 \log ^2(2)+\log (2) \log (16)-\log (32)\right )+\frac {\log (2) \left (25 \log (2)+3120 \log ^2(2)+2560 \log ^3(2)-25 \log (8)-632 \log ^2(2) \log (16)+10 \log (8) \log (16)+\log (2) \log ^2(16)-\log (8) \log ^2(16)\right )}{\sqrt {\log (32)}}\right )}{4 \log (2)-2 x \log (2)+2 \sqrt {\log (32)}}\right ) \, dx}{(5-\log (16)) (5-320 \log (2)-\log (16))}-\frac {\left (16 \log (2) \left (2400 \log ^3(2)-\log (8) (5-\log (16))^2+20 \log ^2(2) \left (155-27 \log (16)+60 \sqrt {\log (32)}\right )-4 \sqrt {\log (32)} (\log (32)-\log (16) \log (64))+\log (2) (25+8 \log (32)-\log (16) (5+8 \log (64)))\right )\right ) \int \frac {e^{5+x}}{4 \log (2)-2 x \log (2)-2 \sqrt {\log (32)}} \, dx}{(5-\log (16)) (5-320 \log (2)-\log (16)) \sqrt {\log (32)}}+\frac {\left (16 \log (2) \left (2400 \log ^3(2)-\log (8) (5-\log (16))^2+20 \log ^2(2) \left (155-27 \log (16)-60 \sqrt {\log (32)}\right )+4 \sqrt {\log (32)} (\log (32)-\log (16) \log (64))+\log (2) (25+8 \log (32)-\log (16) (5+8 \log (64)))\right )\right ) \int \frac {e^{5+x}}{4 \log (2)-2 x \log (2)+2 \sqrt {\log (32)}} \, dx}{(5-\log (16)) (5-320 \log (2)-\log (16)) \sqrt {\log (32)}}\\ &=-\frac {e^5 \text {Ei}(x) (5+64 \log (2)-\log (16))}{5-\log (16)}+\frac {8 e^{7-\frac {\sqrt {\log (32)}}{\log (2)}} \text {Ei}\left (\frac {2 x \log (2)-\log (16)+2 \sqrt {\log (32)}}{2 \log (2)}\right ) \left (2400 \log ^3(2)-\log (8) (5-\log (16))^2+20 \log ^2(2) \left (155-27 \log (16)+60 \sqrt {\log (32)}\right )-4 \sqrt {\log (32)} (\log (32)-\log (16) \log (64))+\log (2) (25+8 \log (32)-\log (16) (5+8 \log (64)))\right )}{(5-\log (16)) (5-320 \log (2)-\log (16)) \sqrt {\log (32)}}-\frac {8 e^{7+\frac {\sqrt {\log (32)}}{\log (2)}} \text {Ei}\left (\frac {x \log (2)-\log (4)-\sqrt {\log (32)}}{\log (2)}\right ) \left (2400 \log ^3(2)-\log (8) (5-\log (16))^2+20 \log ^2(2) \left (155-27 \log (16)-60 \sqrt {\log (32)}\right )+4 \sqrt {\log (32)} (\log (32)-\log (16) \log (64))+\log (2) (25+8 \log (32)-\log (16) (5+8 \log (64)))\right )}{(5-\log (16)) (5-320 \log (2)-\log (16)) \sqrt {\log (32)}}+\frac {5 e^5 \text {Ei}(x) (1+\log (4096))}{5-\log (16)}+\frac {16 e^{5+x} \log \left (-\left ((16+x) \left (5+4 x \log (2)-x^2 \log (2)-\log (16)\right )\right )\right )}{x}+\frac {\left (16 \log (2) \left (2560 \log ^3(2)-\log (8) (5-\log (16))^2+\log (2) \left (25+\log ^2(16)\right )+8 \log ^2(2) \left (390-79 \log (16)+160 \sqrt {\log (32)}\right )+\left (\log ^2(16)-4 \log (32)\right ) \sqrt {\log (32)}\right )\right ) \int \frac {e^{5+x}}{4 \log (2)-2 x \log (2)-2 \sqrt {\log (32)}} \, dx}{(5-\log (16)) (5-320 \log (2)-\log (16)) \sqrt {\log (32)}}-\frac {\left (16 \log (2) \left (2560 \log ^3(2)-\log (8) (5-\log (16))^2+8 \log ^2(2) \left (390-79 \log (16)-160 \sqrt {\log (32)}\right )+\log (2) \left (25+\log ^2(16)-4 \log (16) \sqrt {\log (32)}\right )+4 \log ^{\frac {3}{2}}(32)\right )\right ) \int \frac {e^{5+x}}{4 \log (2)-2 x \log (2)+2 \sqrt {\log (32)}} \, dx}{(5-\log (16)) (5-320 \log (2)-\log (16)) \sqrt {\log (32)}}\\ &=-\frac {e^5 \text {Ei}(x) (5+64 \log (2)-\log (16))}{5-\log (16)}-\frac {8 e^{7-\frac {\sqrt {\log (32)}}{\log (2)}} \text {Ei}\left (\frac {2 x \log (2)-\log (16)+2 \sqrt {\log (32)}}{2 \log (2)}\right ) \left (2560 \log ^3(2)-\log (8) (5-\log (16))^2+\log (2) \left (25+\log ^2(16)\right )+8 \log ^2(2) \left (390-79 \log (16)+160 \sqrt {\log (32)}\right )+\left (\log ^2(16)-4 \log (32)\right ) \sqrt {\log (32)}\right )}{(5-\log (16)) (5-320 \log (2)-\log (16)) \sqrt {\log (32)}}+\frac {8 e^{7+\frac {\sqrt {\log (32)}}{\log (2)}} \text {Ei}\left (\frac {x \log (2)-\log (4)-\sqrt {\log (32)}}{\log (2)}\right ) \left (2560 \log ^3(2)-\log (8) (5-\log (16))^2+8 \log ^2(2) \left (390-79 \log (16)-160 \sqrt {\log (32)}\right )+\log (2) \left (25+\log ^2(16)-4 \log (16) \sqrt {\log (32)}\right )+4 \log ^{\frac {3}{2}}(32)\right )}{(5-\log (16)) (5-320 \log (2)-\log (16)) \sqrt {\log (32)}}+\frac {8 e^{7-\frac {\sqrt {\log (32)}}{\log (2)}} \text {Ei}\left (\frac {2 x \log (2)-\log (16)+2 \sqrt {\log (32)}}{2 \log (2)}\right ) \left (2400 \log ^3(2)-\log (8) (5-\log (16))^2+20 \log ^2(2) \left (155-27 \log (16)+60 \sqrt {\log (32)}\right )-4 \sqrt {\log (32)} (\log (32)-\log (16) \log (64))+\log (2) (25+8 \log (32)-\log (16) (5+8 \log (64)))\right )}{(5-\log (16)) (5-320 \log (2)-\log (16)) \sqrt {\log (32)}}-\frac {8 e^{7+\frac {\sqrt {\log (32)}}{\log (2)}} \text {Ei}\left (\frac {x \log (2)-\log (4)-\sqrt {\log (32)}}{\log (2)}\right ) \left (2400 \log ^3(2)-\log (8) (5-\log (16))^2+20 \log ^2(2) \left (155-27 \log (16)-60 \sqrt {\log (32)}\right )+4 \sqrt {\log (32)} (\log (32)-\log (16) \log (64))+\log (2) (25+8 \log (32)-\log (16) (5+8 \log (64)))\right )}{(5-\log (16)) (5-320 \log (2)-\log (16)) \sqrt {\log (32)}}+\frac {5 e^5 \text {Ei}(x) (1+\log (4096))}{5-\log (16)}+\frac {16 e^{5+x} \log \left (-\left ((16+x) \left (5+4 x \log (2)-x^2 \log (2)-\log (16)\right )\right )\right )}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.01, size = 30, normalized size = 1.11 \begin {gather*} \frac {16 e^{5+x} \log \left ((16+x) \left (-5-4 x \log (2)+x^2 \log (2)+\log (16)\right )\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(5 + x)*(-80*x + (-960*x + 384*x^2 + 48*x^3)*Log[2]) + E^(5 + x)*(1280 - 1200*x - 80*x^2 + (-1024
+ 1984*x - 1152*x^2 + 176*x^3 + 16*x^4)*Log[2])*Log[-80 - 5*x + (64 - 60*x + 12*x^2 + x^3)*Log[2]])/(-80*x^2
- 5*x^3 + (64*x^2 - 60*x^3 + 12*x^4 + x^5)*Log[2]),x]

[Out]

(16*E^(5 + x)*Log[(16 + x)*(-5 - 4*x*Log[2] + x^2*Log[2] + Log[16])])/x

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fricas [A]  time = 0.72, size = 31, normalized size = 1.15 \begin {gather*} \frac {16 \, e^{\left (x + 5\right )} \log \left ({\left (x^{3} + 12 \, x^{2} - 60 \, x + 64\right )} \log \relax (2) - 5 \, x - 80\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x^4+176*x^3-1152*x^2+1984*x-1024)*log(2)-80*x^2-1200*x+1280)*exp(5+x)*log((x^3+12*x^2-60*x+64)
*log(2)-5*x-80)+((48*x^3+384*x^2-960*x)*log(2)-80*x)*exp(5+x))/((x^5+12*x^4-60*x^3+64*x^2)*log(2)-5*x^3-80*x^2
),x, algorithm="fricas")

[Out]

16*e^(x + 5)*log((x^3 + 12*x^2 - 60*x + 64)*log(2) - 5*x - 80)/x

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giac [A]  time = 0.58, size = 37, normalized size = 1.37 \begin {gather*} \frac {16 \, e^{\left (x + 5\right )} \log \left (x^{3} \log \relax (2) + 12 \, x^{2} \log \relax (2) - 60 \, x \log \relax (2) - 5 \, x + 64 \, \log \relax (2) - 80\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x^4+176*x^3-1152*x^2+1984*x-1024)*log(2)-80*x^2-1200*x+1280)*exp(5+x)*log((x^3+12*x^2-60*x+64)
*log(2)-5*x-80)+((48*x^3+384*x^2-960*x)*log(2)-80*x)*exp(5+x))/((x^5+12*x^4-60*x^3+64*x^2)*log(2)-5*x^3-80*x^2
),x, algorithm="giac")

[Out]

16*e^(x + 5)*log(x^3*log(2) + 12*x^2*log(2) - 60*x*log(2) - 5*x + 64*log(2) - 80)/x

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maple [C]  time = 0.15, size = 191, normalized size = 7.07

method	result	size
risch	\(\frac {16 \,{\mathrm e}^{5+x} \ln \left (-5+\left (x^{2}-4 x +4\right ) \ln \relax (2)\right )}{x}+\frac {8 \,{\mathrm e}^{5+x} \left (-i \pi \,\mathrm {csgn}\left (i \left (-5+\left (x^{2}-4 x +4\right ) \ln \relax (2)\right )\right ) \mathrm {csgn}\left (i \left (x +16\right )\right ) \mathrm {csgn}\left (i \left (-5+\left (x^{2}-4 x +4\right ) \ln \relax (2)\right ) \left (x +16\right )\right )+i \pi \,\mathrm {csgn}\left (i \left (-5+\left (x^{2}-4 x +4\right ) \ln \relax (2)\right )\right ) \mathrm {csgn}\left (i \left (-5+\left (x^{2}-4 x +4\right ) \ln \relax (2)\right ) \left (x +16\right )\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (x +16\right )\right ) \mathrm {csgn}\left (i \left (-5+\left (x^{2}-4 x +4\right ) \ln \relax (2)\right ) \left (x +16\right )\right )^{2}-i \pi \mathrm {csgn}\left (i \left (-5+\left (x^{2}-4 x +4\right ) \ln \relax (2)\right ) \left (x +16\right )\right )^{3}+2 \ln \left (x +16\right )\right )}{x}\)	\(191\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((16*x^4+176*x^3-1152*x^2+1984*x-1024)*ln(2)-80*x^2-1200*x+1280)*exp(5+x)*ln((x^3+12*x^2-60*x+64)*ln(2)-5
*x-80)+((48*x^3+384*x^2-960*x)*ln(2)-80*x)*exp(5+x))/((x^5+12*x^4-60*x^3+64*x^2)*ln(2)-5*x^3-80*x^2),x,method=
_RETURNVERBOSE)

[Out]

16/x*exp(5+x)*ln(-5+(x^2-4*x+4)*ln(2))+8*exp(5+x)*(-I*Pi*csgn(I*(-5+(x^2-4*x+4)*ln(2)))*csgn(I*(x+16))*csgn(I*
(-5+(x^2-4*x+4)*ln(2))*(x+16))+I*Pi*csgn(I*(-5+(x^2-4*x+4)*ln(2)))*csgn(I*(-5+(x^2-4*x+4)*ln(2))*(x+16))^2+I*P
i*csgn(I*(x+16))*csgn(I*(-5+(x^2-4*x+4)*ln(2))*(x+16))^2-I*Pi*csgn(I*(-5+(x^2-4*x+4)*ln(2))*(x+16))^3+2*ln(x+1
6))/x

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maxima [A]  time = 0.76, size = 38, normalized size = 1.41 \begin {gather*} \frac {16 \, {\left (e^{\left (x + 5\right )} \log \left (x^{2} \log \relax (2) - 4 \, x \log \relax (2) + 4 \, \log \relax (2) - 5\right ) + e^{\left (x + 5\right )} \log \left (x + 16\right )\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x^4+176*x^3-1152*x^2+1984*x-1024)*log(2)-80*x^2-1200*x+1280)*exp(5+x)*log((x^3+12*x^2-60*x+64)
*log(2)-5*x-80)+((48*x^3+384*x^2-960*x)*log(2)-80*x)*exp(5+x))/((x^5+12*x^4-60*x^3+64*x^2)*log(2)-5*x^3-80*x^2
),x, algorithm="maxima")

[Out]

16*(e^(x + 5)*log(x^2*log(2) - 4*x*log(2) + 4*log(2) - 5) + e^(x + 5)*log(x + 16))/x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{x+5}\,\left (80\,x-\ln \relax (2)\,\left (48\,x^3+384\,x^2-960\,x\right )\right )+{\mathrm {e}}^{x+5}\,\ln \left (\ln \relax (2)\,\left (x^3+12\,x^2-60\,x+64\right )-5\,x-80\right )\,\left (1200\,x-\ln \relax (2)\,\left (16\,x^4+176\,x^3-1152\,x^2+1984\,x-1024\right )+80\,x^2-1280\right )}{80\,x^2-\ln \relax (2)\,\left (x^5+12\,x^4-60\,x^3+64\,x^2\right )+5\,x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + 5)*(80*x - log(2)*(384*x^2 - 960*x + 48*x^3)) + exp(x + 5)*log(log(2)*(12*x^2 - 60*x + x^3 + 64)
- 5*x - 80)*(1200*x - log(2)*(1984*x - 1152*x^2 + 176*x^3 + 16*x^4 - 1024) + 80*x^2 - 1280))/(80*x^2 - log(2)*
(64*x^2 - 60*x^3 + 12*x^4 + x^5) + 5*x^3),x)

[Out]

int((exp(x + 5)*(80*x - log(2)*(384*x^2 - 960*x + 48*x^3)) + exp(x + 5)*log(log(2)*(12*x^2 - 60*x + x^3 + 64)
- 5*x - 80)*(1200*x - log(2)*(1984*x - 1152*x^2 + 176*x^3 + 16*x^4 - 1024) + 80*x^2 - 1280))/(80*x^2 - log(2)*
(64*x^2 - 60*x^3 + 12*x^4 + x^5) + 5*x^3), x)

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sympy [A]  time = 0.69, size = 31, normalized size = 1.15 \begin {gather*} \frac {16 e^{x + 5} \log {\left (- 5 x + \left (x^{3} + 12 x^{2} - 60 x + 64\right ) \log {\relax (2 )} - 80 \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((16*x**4+176*x**3-1152*x**2+1984*x-1024)*ln(2)-80*x**2-1200*x+1280)*exp(5+x)*ln((x**3+12*x**2-60*x
+64)*ln(2)-5*x-80)+((48*x**3+384*x**2-960*x)*ln(2)-80*x)*exp(5+x))/((x**5+12*x**4-60*x**3+64*x**2)*ln(2)-5*x**
3-80*x**2),x)

[Out]

16*exp(x + 5)*log(-5*x + (x**3 + 12*x**2 - 60*x + 64)*log(2) - 80)/x

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