\(\int \frac {-45 x^2-3 x^3+(30 x+6 x^2) \log (2)+e^{2 x} (-15-39 x+6 x^2+(42-6 x) \log (2))+e^x (-60 x-42 x^2+6 x^3+(30+48 x-6 x^2) \log (2))}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx\) [9920]

Optimal result

Integrand size = 104, antiderivative size = 22 \[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=\frac {3 \left (e^x+x\right )^2 (x-\log (2))}{(5-x)^4} \]

[Out]

(exp(x)+x)^2*(3*x-3*ln(2))/(5-x)^4

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.85 (sec) , antiderivative size = 553, normalized size of antiderivative = 25.14, number of steps used = 39, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6820, 12, 6874, 37, 45, 2230, 2208, 2209} \[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=12 e^{10} \operatorname {ExpIntegralEi}(-2 (5-x))+6 e^5 \operatorname {ExpIntegralEi}(x-5)-\frac {1}{8} e^5 (200-\log (1099511627776)) \operatorname {ExpIntegralEi}(x-5)-\frac {1}{2} e^5 (10+\log (16)) \operatorname {ExpIntegralEi}(x-5)-2 e^{10} (20-\log (16)) \operatorname {ExpIntegralEi}(-2 (5-x))+\frac {3}{2} e^5 (16-\log (4)) \operatorname {ExpIntegralEi}(x-5)+4 e^{10} (7-\log (4)) \operatorname {ExpIntegralEi}(-2 (5-x))+\frac {3 x^4}{20 (5-x)^4}+\frac {6 e^x}{5-x}+\frac {6 e^{2 x}}{5-x}-\frac {3 e^{2 x}}{(5-x)^2}-\frac {e^x (200-\log (1099511627776))}{8 (5-x)}+\frac {e^x (200-\log (1099511627776))}{8 (5-x)^2}-\frac {e^x (200-\log (1099511627776))}{4 (5-x)^3}+\frac {3 e^x (200-\log (1099511627776))}{4 (5-x)^4}+\frac {\log (1024)}{(5-x)^3}-\frac {15 \log (1024)}{4 (5-x)^4}-\frac {e^x (10+\log (16))}{2 (5-x)}+\frac {e^x (10+\log (16))}{2 (5-x)^2}-\frac {e^x (10+\log (16))}{(5-x)^3}-\frac {e^{2 x} (20-\log (16))}{5-x}+\frac {e^{2 x} (20-\log (16))}{2 (5-x)^2}-\frac {e^{2 x} (20-\log (16))}{2 (5-x)^3}+\frac {3 e^{2 x} (20-\log (16))}{4 (5-x)^4}+\frac {3 e^x (16-\log (4))}{2 (5-x)}-\frac {3 e^x (16-\log (4))}{2 (5-x)^2}+\frac {3 (15-\log (4))}{2 (5-x)^2}-\frac {10 (15-\log (4))}{(5-x)^3}+\frac {75 (15-\log (4))}{4 (5-x)^4}+\frac {2 e^{2 x} (7-\log (4))}{5-x}-\frac {e^{2 x} (7-\log (4))}{(5-x)^2}+\frac {e^{2 x} (7-\log (4))}{(5-x)^3} \]

[In]

Int[(-45*x^2 - 3*x^3 + (30*x + 6*x^2)*Log[2] + E^(2*x)*(-15 - 39*x + 6*x^2 + (42 - 6*x)*Log[2]) + E^x*(-60*x -
42*x^2 + 6*x^3 + (30 + 48*x - 6*x^2)*Log[2]))/(-3125 + 3125*x - 1250*x^2 + 250*x^3 - 25*x^4 + x^5),x]

[Out]

(-3*E^(2*x))/(5 - x)^2 + (6*E^x)/(5 - x) + (6*E^(2*x))/(5 - x) + (3*x^4)/(20*(5 - x)^4) + 12*E^10*ExpIntegralE
i[-2*(5 - x)] + 6*E^5*ExpIntegralEi[-5 + x] + (E^(2*x)*(7 - Log[4]))/(5 - x)^3 - (E^(2*x)*(7 - Log[4]))/(5 - x
)^2 + (2*E^(2*x)*(7 - Log[4]))/(5 - x) + 4*E^10*ExpIntegralEi[-2*(5 - x)]*(7 - Log[4]) + (75*(15 - Log[4]))/(4
*(5 - x)^4) - (10*(15 - Log[4]))/(5 - x)^3 + (3*(15 - Log[4]))/(2*(5 - x)^2) - (3*E^x*(16 - Log[4]))/(2*(5 - x
)^2) + (3*E^x*(16 - Log[4]))/(2*(5 - x)) + (3*E^5*ExpIntegralEi[-5 + x]*(16 - Log[4]))/2 + (3*E^(2*x)*(20 - Lo
g[16]))/(4*(5 - x)^4) - (E^(2*x)*(20 - Log[16]))/(2*(5 - x)^3) + (E^(2*x)*(20 - Log[16]))/(2*(5 - x)^2) - (E^(
2*x)*(20 - Log[16]))/(5 - x) - 2*E^10*ExpIntegralEi[-2*(5 - x)]*(20 - Log[16]) - (E^x*(10 + Log[16]))/(5 - x)^
3 + (E^x*(10 + Log[16]))/(2*(5 - x)^2) - (E^x*(10 + Log[16]))/(2*(5 - x)) - (E^5*ExpIntegralEi[-5 + x]*(10 + L
og[16]))/2 - (15*Log[1024])/(4*(5 - x)^4) + Log[1024]/(5 - x)^3 + (3*E^x*(200 - Log[1099511627776]))/(4*(5 - x
)^4) - (E^x*(200 - Log[1099511627776]))/(4*(5 - x)^3) + (E^x*(200 - Log[1099511627776]))/(8*(5 - x)^2) - (E^x*
(200 - Log[1099511627776]))/(8*(5 - x)) - (E^5*ExpIntegralEi[-5 + x]*(200 - Log[1099511627776]))/8

Rule 12

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2208

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*((b*F^(g*(e + f*x)))^n/(d*(m + 1))), x] - Dist[f*g*n*(Log[F]/(d*(m + 1))), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !TrueQ[$UseGamm
a]

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 2230

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !TrueQ[$UseGamma]

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 {3 \left (e^x+x\right ) \left (x^2-x (-15+\log (4))-e^x \left (-5+2 x^2+14 \log (2)-x (13+\log (4))\right )-\log (1024)\right )}{(5-x)^5} \, dx \\ & = 3 \int \frac {\left (e^x+x\right ) \left (x^2-x (-15+\log (4))-e^x \left (-5+2 x^2+14 \log (2)-x (13+\log (4))\right )-\log (1024)\right )}{(5-x)^5} \, dx \\ & = 3 \int \left (-\frac {x^3}{(-5+x)^5}+\frac {x^2 (-15+\log (4))}{(-5+x)^5}+\frac {e^{2 x} \left (5-2 x^2-14 \log (2)+x (13+\log (4))\right )}{(5-x)^5}+\frac {x \log (1024)}{(-5+x)^5}+\frac {e^x \left (-2 x^3+x^2 (14+\log (4))-\log (1024)+x (20-\log (65536))\right )}{(5-x)^5}\right ) \, dx \\ & = -\left (3 \int \frac {x^3}{(-5+x)^5} \, dx\right )+3 \int \frac {e^{2 x} \left (5-2 x^2-14 \log (2)+x (13+\log (4))\right )}{(5-x)^5} \, dx+3 \int \frac {e^x \left (-2 x^3+x^2 (14+\log (4))-\log (1024)+x (20-\log (65536))\right )}{(5-x)^5} \, dx+(3 (-15+\log (4))) \int \frac {x^2}{(-5+x)^5} \, dx+(3 \log (1024)) \int \frac {x}{(-5+x)^5} \, dx \\ & = \frac {3 x^4}{20 (5-x)^4}+3 \int \left (\frac {2 e^{2 x}}{(-5+x)^3}+\frac {e^{2 x} (7-\log (4))}{(-5+x)^4}+\frac {e^{2 x} (-20+\log (16))}{(-5+x)^5}\right ) \, dx+3 \int \left (\frac {2 e^x}{(-5+x)^2}+\frac {e^x (16-\log (4))}{(-5+x)^3}+\frac {e^x (-10-\log (16))}{(-5+x)^4}+\frac {e^x (-200+\log (1099511627776))}{(-5+x)^5}\right ) \, dx+(3 (-15+\log (4))) \int \left (\frac {25}{(-5+x)^5}+\frac {10}{(-5+x)^4}+\frac {1}{(-5+x)^3}\right ) \, dx+(3 \log (1024)) \int \left (\frac {5}{(-5+x)^5}+\frac {1}{(-5+x)^4}\right ) \, dx \\ & = \frac {3 x^4}{20 (5-x)^4}+\frac {75 (15-\log (4))}{4 (5-x)^4}-\frac {10 (15-\log (4))}{(5-x)^3}+\frac {3 (15-\log (4))}{2 (5-x)^2}-\frac {15 \log (1024)}{4 (5-x)^4}+\frac {\log (1024)}{(5-x)^3}+6 \int \frac {e^{2 x}}{(-5+x)^3} \, dx+6 \int \frac {e^x}{(-5+x)^2} \, dx+(3 (7-\log (4))) \int \frac {e^{2 x}}{(-5+x)^4} \, dx+(3 (16-\log (4))) \int \frac {e^x}{(-5+x)^3} \, dx+(3 (-20+\log (16))) \int \frac {e^{2 x}}{(-5+x)^5} \, dx-(3 (10+\log (16))) \int \frac {e^x}{(-5+x)^4} \, dx-(3 (200-\log (1099511627776))) \int \frac {e^x}{(-5+x)^5} \, dx \\ & = -\frac {3 e^{2 x}}{(5-x)^2}+\frac {6 e^x}{5-x}+\frac {3 x^4}{20 (5-x)^4}+\frac {e^{2 x} (7-\log (4))}{(5-x)^3}+\frac {75 (15-\log (4))}{4 (5-x)^4}-\frac {10 (15-\log (4))}{(5-x)^3}+\frac {3 (15-\log (4))}{2 (5-x)^2}-\frac {3 e^x (16-\log (4))}{2 (5-x)^2}+\frac {3 e^{2 x} (20-\log (16))}{4 (5-x)^4}-\frac {e^x (10+\log (16))}{(5-x)^3}-\frac {15 \log (1024)}{4 (5-x)^4}+\frac {\log (1024)}{(5-x)^3}+\frac {3 e^x (200-\log (1099511627776))}{4 (5-x)^4}+6 \int \frac {e^{2 x}}{(-5+x)^2} \, dx+6 \int \frac {e^x}{-5+x} \, dx+(2 (7-\log (4))) \int \frac {e^{2 x}}{(-5+x)^3} \, dx+\frac {1}{2} (3 (16-\log (4))) \int \frac {e^x}{(-5+x)^2} \, dx+\frac {1}{2} (3 (-20+\log (16))) \int \frac {e^{2 x}}{(-5+x)^4} \, dx-(10+\log (16)) \int \frac {e^x}{(-5+x)^3} \, dx-\frac {1}{4} (3 (200-\log (1099511627776))) \int \frac {e^x}{(-5+x)^4} \, dx \\ & = -\frac {3 e^{2 x}}{(5-x)^2}+\frac {6 e^x}{5-x}+\frac {6 e^{2 x}}{5-x}+\frac {3 x^4}{20 (5-x)^4}+6 e^5 \operatorname {ExpIntegralEi}(-5+x)+\frac {e^{2 x} (7-\log (4))}{(5-x)^3}-\frac {e^{2 x} (7-\log (4))}{(5-x)^2}+\frac {75 (15-\log (4))}{4 (5-x)^4}-\frac {10 (15-\log (4))}{(5-x)^3}+\frac {3 (15-\log (4))}{2 (5-x)^2}-\frac {3 e^x (16-\log (4))}{2 (5-x)^2}+\frac {3 e^x (16-\log (4))}{2 (5-x)}+\frac {3 e^{2 x} (20-\log (16))}{4 (5-x)^4}-\frac {e^{2 x} (20-\log (16))}{2 (5-x)^3}-\frac {e^x (10+\log (16))}{(5-x)^3}+\frac {e^x (10+\log (16))}{2 (5-x)^2}-\frac {15 \log (1024)}{4 (5-x)^4}+\frac {\log (1024)}{(5-x)^3}+\frac {3 e^x (200-\log (1099511627776))}{4 (5-x)^4}-\frac {e^x (200-\log (1099511627776))}{4 (5-x)^3}+12 \int \frac {e^{2 x}}{-5+x} \, dx+(2 (7-\log (4))) \int \frac {e^{2 x}}{(-5+x)^2} \, dx+\frac {1}{2} (3 (16-\log (4))) \int \frac {e^x}{-5+x} \, dx+(-20+\log (16)) \int \frac {e^{2 x}}{(-5+x)^3} \, dx-\frac {1}{2} (10+\log (16)) \int \frac {e^x}{(-5+x)^2} \, dx-\frac {1}{4} (200-\log (1099511627776)) \int \frac {e^x}{(-5+x)^3} \, dx \\ & = -\frac {3 e^{2 x}}{(5-x)^2}+\frac {6 e^x}{5-x}+\frac {6 e^{2 x}}{5-x}+\frac {3 x^4}{20 (5-x)^4}+12 e^{10} \operatorname {ExpIntegralEi}(-2 (5-x))+6 e^5 \operatorname {ExpIntegralEi}(-5+x)+\frac {e^{2 x} (7-\log (4))}{(5-x)^3}-\frac {e^{2 x} (7-\log (4))}{(5-x)^2}+\frac {2 e^{2 x} (7-\log (4))}{5-x}+\frac {75 (15-\log (4))}{4 (5-x)^4}-\frac {10 (15-\log (4))}{(5-x)^3}+\frac {3 (15-\log (4))}{2 (5-x)^2}-\frac {3 e^x (16-\log (4))}{2 (5-x)^2}+\frac {3 e^x (16-\log (4))}{2 (5-x)}+\frac {3}{2} e^5 \operatorname {ExpIntegralEi}(-5+x) (16-\log (4))+\frac {3 e^{2 x} (20-\log (16))}{4 (5-x)^4}-\frac {e^{2 x} (20-\log (16))}{2 (5-x)^3}+\frac {e^{2 x} (20-\log (16))}{2 (5-x)^2}-\frac {e^x (10+\log (16))}{(5-x)^3}+\frac {e^x (10+\log (16))}{2 (5-x)^2}-\frac {e^x (10+\log (16))}{2 (5-x)}-\frac {15 \log (1024)}{4 (5-x)^4}+\frac {\log (1024)}{(5-x)^3}+\frac {3 e^x (200-\log (1099511627776))}{4 (5-x)^4}-\frac {e^x (200-\log (1099511627776))}{4 (5-x)^3}+\frac {e^x (200-\log (1099511627776))}{8 (5-x)^2}+(4 (7-\log (4))) \int \frac {e^{2 x}}{-5+x} \, dx+(-20+\log (16)) \int \frac {e^{2 x}}{(-5+x)^2} \, dx-\frac {1}{2} (10+\log (16)) \int \frac {e^x}{-5+x} \, dx-\frac {1}{8} (200-\log (1099511627776)) \int \frac {e^x}{(-5+x)^2} \, dx \\ & = -\frac {3 e^{2 x}}{(5-x)^2}+\frac {6 e^x}{5-x}+\frac {6 e^{2 x}}{5-x}+\frac {3 x^4}{20 (5-x)^4}+12 e^{10} \operatorname {ExpIntegralEi}(-2 (5-x))+6 e^5 \operatorname {ExpIntegralEi}(-5+x)+\frac {e^{2 x} (7-\log (4))}{(5-x)^3}-\frac {e^{2 x} (7-\log (4))}{(5-x)^2}+\frac {2 e^{2 x} (7-\log (4))}{5-x}+4 e^{10} \operatorname {ExpIntegralEi}(-2 (5-x)) (7-\log (4))+\frac {75 (15-\log (4))}{4 (5-x)^4}-\frac {10 (15-\log (4))}{(5-x)^3}+\frac {3 (15-\log (4))}{2 (5-x)^2}-\frac {3 e^x (16-\log (4))}{2 (5-x)^2}+\frac {3 e^x (16-\log (4))}{2 (5-x)}+\frac {3}{2} e^5 \operatorname {ExpIntegralEi}(-5+x) (16-\log (4))+\frac {3 e^{2 x} (20-\log (16))}{4 (5-x)^4}-\frac {e^{2 x} (20-\log (16))}{2 (5-x)^3}+\frac {e^{2 x} (20-\log (16))}{2 (5-x)^2}-\frac {e^{2 x} (20-\log (16))}{5-x}-\frac {e^x (10+\log (16))}{(5-x)^3}+\frac {e^x (10+\log (16))}{2 (5-x)^2}-\frac {e^x (10+\log (16))}{2 (5-x)}-\frac {1}{2} e^5 \operatorname {ExpIntegralEi}(-5+x) (10+\log (16))-\frac {15 \log (1024)}{4 (5-x)^4}+\frac {\log (1024)}{(5-x)^3}+\frac {3 e^x (200-\log (1099511627776))}{4 (5-x)^4}-\frac {e^x (200-\log (1099511627776))}{4 (5-x)^3}+\frac {e^x (200-\log (1099511627776))}{8 (5-x)^2}-\frac {e^x (200-\log (1099511627776))}{8 (5-x)}+(2 (-20+\log (16))) \int \frac {e^{2 x}}{-5+x} \, dx-\frac {1}{8} (200-\log (1099511627776)) \int \frac {e^x}{-5+x} \, dx \\ & = -\frac {3 e^{2 x}}{(5-x)^2}+\frac {6 e^x}{5-x}+\frac {6 e^{2 x}}{5-x}+\frac {3 x^4}{20 (5-x)^4}+12 e^{10} \operatorname {ExpIntegralEi}(-2 (5-x))+6 e^5 \operatorname {ExpIntegralEi}(-5+x)+\frac {e^{2 x} (7-\log (4))}{(5-x)^3}-\frac {e^{2 x} (7-\log (4))}{(5-x)^2}+\frac {2 e^{2 x} (7-\log (4))}{5-x}+4 e^{10} \operatorname {ExpIntegralEi}(-2 (5-x)) (7-\log (4))+\frac {75 (15-\log (4))}{4 (5-x)^4}-\frac {10 (15-\log (4))}{(5-x)^3}+\frac {3 (15-\log (4))}{2 (5-x)^2}-\frac {3 e^x (16-\log (4))}{2 (5-x)^2}+\frac {3 e^x (16-\log (4))}{2 (5-x)}+\frac {3}{2} e^5 \operatorname {ExpIntegralEi}(-5+x) (16-\log (4))+\frac {3 e^{2 x} (20-\log (16))}{4 (5-x)^4}-\frac {e^{2 x} (20-\log (16))}{2 (5-x)^3}+\frac {e^{2 x} (20-\log (16))}{2 (5-x)^2}-\frac {e^{2 x} (20-\log (16))}{5-x}-2 e^{10} \operatorname {ExpIntegralEi}(-2 (5-x)) (20-\log (16))-\frac {e^x (10+\log (16))}{(5-x)^3}+\frac {e^x (10+\log (16))}{2 (5-x)^2}-\frac {e^x (10+\log (16))}{2 (5-x)}-\frac {1}{2} e^5 \operatorname {ExpIntegralEi}(-5+x) (10+\log (16))-\frac {15 \log (1024)}{4 (5-x)^4}+\frac {\log (1024)}{(5-x)^3}+\frac {3 e^x (200-\log (1099511627776))}{4 (5-x)^4}-\frac {e^x (200-\log (1099511627776))}{4 (5-x)^3}+\frac {e^x (200-\log (1099511627776))}{8 (5-x)^2}-\frac {e^x (200-\log (1099511627776))}{8 (5-x)}-\frac {1}{8} e^5 \operatorname {ExpIntegralEi}(-5+x) (200-\log (1099511627776)) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(22)=44\).

Time = 9.58 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=\frac {8 e^x x (6 x-\log (64))+2 x^2 (12 x-\log (4096))+e^{2 x} (24 x-\log (16777216))}{8 (-5+x)^4} \]

[In]

Integrate[(-45*x^2 - 3*x^3 + (30*x + 6*x^2)*Log[2] + E^(2*x)*(-15 - 39*x + 6*x^2 + (42 - 6*x)*Log[2]) + E^x*(-
60*x - 42*x^2 + 6*x^3 + (30 + 48*x - 6*x^2)*Log[2]))/(-3125 + 3125*x - 1250*x^2 + 250*x^3 - 25*x^4 + x^5),x]

[Out]

(8*E^x*x*(6*x - Log[64]) + 2*x^2*(12*x - Log[4096]) + E^(2*x)*(24*x - Log[16777216]))/(8*(-5 + x)^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(22)=44\).

Time = 0.34 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.23

method	result	size
norman	\(\frac {3 x^{3}-3 x^{2} \ln \left (2\right )+3 x \,{\mathrm e}^{2 x}+6 \,{\mathrm e}^{x} x^{2}-3 \ln \left (2\right ) {\mathrm e}^{2 x}-6 x \ln \left (2\right ) {\mathrm e}^{x}}{\left (-5+x \right )^{4}}\)	\(49\)
parallelrisch	\(-\frac {3 x^{2} \ln \left (2\right )+6 x \ln \left (2\right ) {\mathrm e}^{x}+3 \ln \left (2\right ) {\mathrm e}^{2 x}-3 x^{3}-6 \,{\mathrm e}^{x} x^{2}-3 x \,{\mathrm e}^{2 x}}{x^{4}-20 x^{3}+150 x^{2}-500 x +625}\)	\(65\)
risch	\(\frac {-3 x^{2} \ln \left (2\right )+3 x^{3}}{x^{4}-20 x^{3}+150 x^{2}-500 x +625}-\frac {3 \left (\ln \left (2\right )-x \right ) {\mathrm e}^{2 x}}{\left (-5+x \right )^{4}}-\frac {6 x \left (\ln \left (2\right )-x \right ) {\mathrm e}^{x}}{\left (-5+x \right )^{4}}\)	\(69\)
parts	\(-\frac {30 \ln \left (2\right )-225}{\left (-5+x \right )^{3}}-\frac {3 \left (2 \ln \left (2\right )-30\right )}{2 \left (-5+x \right )^{2}}+\frac {3}{-5+x}-\frac {3 \left (100 \ln \left (2\right )-500\right )}{4 \left (-5+x \right )^{4}}+\frac {15 \,{\mathrm e}^{2 x}}{\left (-5+x \right )^{4}}+\frac {3 \,{\mathrm e}^{2 x}}{\left (-5+x \right )^{3}}-\frac {3 \ln \left (2\right ) {\mathrm e}^{2 x}}{\left (-5+x \right )^{4}}-\frac {6 \ln \left (2\right ) {\mathrm e}^{x}}{\left (-5+x \right )^{3}}-\frac {30 \ln \left (2\right ) {\mathrm e}^{x}}{\left (-5+x \right )^{4}}+\frac {60 \,{\mathrm e}^{x}}{\left (-5+x \right )^{3}}+\frac {6 \,{\mathrm e}^{x}}{\left (-5+x \right )^{2}}+\frac {150 \,{\mathrm e}^{x}}{\left (-5+x \right )^{4}}\)	\(132\)
default	\(\frac {225}{\left (-5+x \right )^{3}}+\frac {45}{\left (-5+x \right )^{2}}+\frac {375}{\left (-5+x \right )^{4}}+\frac {3}{-5+x}+\frac {15 \,{\mathrm e}^{2 x}}{\left (-5+x \right )^{4}}+\frac {3 \,{\mathrm e}^{2 x}}{\left (-5+x \right )^{3}}-\frac {10 \ln \left (2\right )}{\left (-5+x \right )^{3}}-\frac {75 \ln \left (2\right )}{2 \left (-5+x \right )^{4}}+6 \ln \left (2\right ) \left (-\frac {10}{3 \left (-5+x \right )^{3}}-\frac {1}{2 \left (-5+x \right )^{2}}-\frac {25}{4 \left (-5+x \right )^{4}}\right )+\frac {60 \,{\mathrm e}^{x}}{\left (-5+x \right )^{3}}+\frac {6 \,{\mathrm e}^{x}}{\left (-5+x \right )^{2}}+\frac {150 \,{\mathrm e}^{x}}{\left (-5+x \right )^{4}}+30 \ln \left (2\right ) \left (-\frac {{\mathrm e}^{x}}{4 \left (-5+x \right )^{4}}-\frac {{\mathrm e}^{x}}{12 \left (-5+x \right )^{3}}-\frac {{\mathrm e}^{x}}{24 \left (-5+x \right )^{2}}-\frac {{\mathrm e}^{x}}{24 \left (-5+x \right )}-\frac {{\mathrm e}^{5} \operatorname {Ei}_{1}\left (5-x \right )}{24}\right )+42 \ln \left (2\right ) \left (-\frac {{\mathrm e}^{2 x}}{4 \left (-5+x \right )^{4}}-\frac {{\mathrm e}^{2 x}}{6 \left (-5+x \right )^{3}}-\frac {{\mathrm e}^{2 x}}{6 \left (-5+x \right )^{2}}-\frac {{\mathrm e}^{2 x}}{3 \left (-5+x \right )}-\frac {2 \,{\mathrm e}^{10} \operatorname {Ei}_{1}\left (-2 x +10\right )}{3}\right )+48 \ln \left (2\right ) \left (-\frac {3 \,{\mathrm e}^{x}}{4 \left (-5+x \right )^{3}}-\frac {3 \,{\mathrm e}^{x}}{8 \left (-5+x \right )^{2}}-\frac {3 \,{\mathrm e}^{x}}{8 \left (-5+x \right )}-\frac {3 \,{\mathrm e}^{5} \operatorname {Ei}_{1}\left (5-x \right )}{8}-\frac {5 \,{\mathrm e}^{x}}{4 \left (-5+x \right )^{4}}\right )-6 \ln \left (2\right ) \left (-\frac {7 \,{\mathrm e}^{2 x}}{6 \left (-5+x \right )^{3}}-\frac {7 \,{\mathrm e}^{2 x}}{6 \left (-5+x \right )^{2}}-\frac {7 \,{\mathrm e}^{2 x}}{3 \left (-5+x \right )}-\frac {14 \,{\mathrm e}^{10} \operatorname {Ei}_{1}\left (-2 x +10\right )}{3}-\frac {5 \,{\mathrm e}^{2 x}}{4 \left (-5+x \right )^{4}}\right )-6 \ln \left (2\right ) \left (-\frac {65 \,{\mathrm e}^{x}}{12 \left (-5+x \right )^{3}}-\frac {77 \,{\mathrm e}^{x}}{24 \left (-5+x \right )^{2}}-\frac {77 \,{\mathrm e}^{x}}{24 \left (-5+x \right )}-\frac {77 \,{\mathrm e}^{5} \operatorname {Ei}_{1}\left (5-x \right )}{24}-\frac {25 \,{\mathrm e}^{x}}{4 \left (-5+x \right )^{4}}\right )\)	\(399\)

[In]

int((((-6*x+42)*ln(2)+6*x^2-39*x-15)*exp(x)^2+((-6*x^2+48*x+30)*ln(2)+6*x^3-42*x^2-60*x)*exp(x)+(6*x^2+30*x)*l
n(2)-3*x^3-45*x^2)/(x^5-25*x^4+250*x^3-1250*x^2+3125*x-3125),x,method=_RETURNVERBOSE)

[Out]

(3*x^3-3*x^2*ln(2)+3*x*exp(x)^2+6*exp(x)*x^2-3*ln(2)*exp(x)^2-6*x*ln(2)*exp(x))/(-5+x)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (19) = 38\).

Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.59 \[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=\frac {3 \, {\left (x^{3} - x^{2} \log \left (2\right ) + {\left (x - \log \left (2\right )\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{2} - x \log \left (2\right )\right )} e^{x}\right )}}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625} \]

[In]

integrate((((-6*x+42)*log(2)+6*x^2-39*x-15)*exp(x)^2+((-6*x^2+48*x+30)*log(2)+6*x^3-42*x^2-60*x)*exp(x)+(6*x^2
+30*x)*log(2)-3*x^3-45*x^2)/(x^5-25*x^4+250*x^3-1250*x^2+3125*x-3125),x, algorithm="fricas")

[Out]

3*(x^3 - x^2*log(2) + (x - log(2))*e^(2*x) + 2*(x^2 - x*log(2))*e^x)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (19) = 38\).

Time = 0.51 (sec) , antiderivative size = 206, normalized size of antiderivative = 9.36 \[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=- \frac {- 3 x^{3} + 3 x^{2} \log {\left (2 \right )}}{x^{4} - 20 x^{3} + 150 x^{2} - 500 x + 625} + \frac {\left (3 x^{5} - 60 x^{4} - 3 x^{4} \log {\left (2 \right )} + 60 x^{3} \log {\left (2 \right )} + 450 x^{3} - 1500 x^{2} - 450 x^{2} \log {\left (2 \right )} + 1500 x \log {\left (2 \right )} + 1875 x - 1875 \log {\left (2 \right )}\right ) e^{2 x} + \left (6 x^{6} - 120 x^{5} - 6 x^{5} \log {\left (2 \right )} + 120 x^{4} \log {\left (2 \right )} + 900 x^{4} - 3000 x^{3} - 900 x^{3} \log {\left (2 \right )} + 3000 x^{2} \log {\left (2 \right )} + 3750 x^{2} - 3750 x \log {\left (2 \right )}\right ) e^{x}}{x^{8} - 40 x^{7} + 700 x^{6} - 7000 x^{5} + 43750 x^{4} - 175000 x^{3} + 437500 x^{2} - 625000 x + 390625} \]

[In]

integrate((((-6*x+42)*ln(2)+6*x**2-39*x-15)*exp(x)**2+((-6*x**2+48*x+30)*ln(2)+6*x**3-42*x**2-60*x)*exp(x)+(6*
x**2+30*x)*ln(2)-3*x**3-45*x**2)/(x**5-25*x**4+250*x**3-1250*x**2+3125*x-3125),x)

[Out]

-(-3*x**3 + 3*x**2*log(2))/(x**4 - 20*x**3 + 150*x**2 - 500*x + 625) + ((3*x**5 - 60*x**4 - 3*x**4*log(2) + 60
*x**3*log(2) + 450*x**3 - 1500*x**2 - 450*x**2*log(2) + 1500*x*log(2) + 1875*x - 1875*log(2))*exp(2*x) + (6*x*
*6 - 120*x**5 - 6*x**5*log(2) + 120*x**4*log(2) + 900*x**4 - 3000*x**3 - 900*x**3*log(2) + 3000*x**2*log(2) +
3750*x**2 - 3750*x*log(2))*exp(x))/(x**8 - 40*x**7 + 700*x**6 - 7000*x**5 + 43750*x**4 - 175000*x**3 + 437500*
x**2 - 625000*x + 390625)

Maxima [F]

\[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=\int { -\frac {3 \, {\left (x^{3} + 15 \, x^{2} - {\left (2 \, x^{2} - 2 \, {\left (x - 7\right )} \log \left (2\right ) - 13 \, x - 5\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{3} - 7 \, x^{2} - {\left (x^{2} - 8 \, x - 5\right )} \log \left (2\right ) - 10 \, x\right )} e^{x} - 2 \, {\left (x^{2} + 5 \, x\right )} \log \left (2\right )\right )}}{x^{5} - 25 \, x^{4} + 250 \, x^{3} - 1250 \, x^{2} + 3125 \, x - 3125} \,d x } \]

[In]

integrate((((-6*x+42)*log(2)+6*x^2-39*x-15)*exp(x)^2+((-6*x^2+48*x+30)*log(2)+6*x^3-42*x^2-60*x)*exp(x)+(6*x^2
+30*x)*log(2)-3*x^3-45*x^2)/(x^5-25*x^4+250*x^3-1250*x^2+3125*x-3125),x, algorithm="maxima")

[Out]

-30*integrate(e^x/(x^5 - 25*x^4 + 250*x^3 - 1250*x^2 + 3125*x - 3125), x)*log(2) - 1/2*(6*x^2 - 20*x + 25)*log
(2)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) - 5/2*(4*x - 5)*log(2)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) + 3/4
*(4*x^3 - 30*x^2 + 100*x - 125)/(x^4 - 20*x^3 + 150*x^2 - 500*x + 625) + 15/4*(6*x^2 - 20*x + 25)/(x^4 - 20*x^
3 + 150*x^2 - 500*x + 625) + 3*((x - log(2))*e^(2*x) + 2*(x^2 - x*log(2))*e^x)/(x^4 - 20*x^3 + 150*x^2 - 500*x
+ 625) - 30*e^5*exp_integral_e(5, -x + 5)*log(2)/(x - 5)^4

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (19) = 38\).

Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.77 \[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=\frac {3 \, {\left (x^{3} + 2 \, x^{2} e^{x} - x^{2} \log \left (2\right ) - 2 \, x e^{x} \log \left (2\right ) + x e^{\left (2 \, x\right )} - e^{\left (2 \, x\right )} \log \left (2\right )\right )}}{x^{4} - 20 \, x^{3} + 150 \, x^{2} - 500 \, x + 625} \]

[In]

integrate((((-6*x+42)*log(2)+6*x^2-39*x-15)*exp(x)^2+((-6*x^2+48*x+30)*log(2)+6*x^3-42*x^2-60*x)*exp(x)+(6*x^2
+30*x)*log(2)-3*x^3-45*x^2)/(x^5-25*x^4+250*x^3-1250*x^2+3125*x-3125),x, algorithm="giac")

[Out]

3*(x^3 + 2*x^2*e^x - x^2*log(2) - 2*x*e^x*log(2) + x*e^(2*x) - e^(2*x)*log(2))/(x^4 - 20*x^3 + 150*x^2 - 500*x
+ 625)

Mupad [B] (verification not implemented)

Time = 13.66 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.82 \[ \int \frac {-45 x^2-3 x^3+\left (30 x+6 x^2\right ) \log (2)+e^{2 x} \left (-15-39 x+6 x^2+(42-6 x) \log (2)\right )+e^x \left (-60 x-42 x^2+6 x^3+\left (30+48 x-6 x^2\right ) \log (2)\right )}{-3125+3125 x-1250 x^2+250 x^3-25 x^4+x^5} \, dx=-\frac {{\mathrm {e}}^{2\,x}\,\ln \left (8\right )+x^2\,\left (\ln \left (8\right )-6\,{\mathrm {e}}^x\right )-x\,\left (3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\ln \left (64\right )\right )-3\,x^3}{x^4-20\,x^3+150\,x^2-500\,x+625} \]

[In]

int(-(exp(2*x)*(39*x + log(2)*(6*x - 42) - 6*x^2 + 15) + exp(x)*(60*x - log(2)*(48*x - 6*x^2 + 30) + 42*x^2 -
6*x^3) - log(2)*(30*x + 6*x^2) + 45*x^2 + 3*x^3)/(3125*x - 1250*x^2 + 250*x^3 - 25*x^4 + x^5 - 3125),x)

[Out]

-(exp(2*x)*log(8) + x^2*(log(8) - 6*exp(x)) - x*(3*exp(2*x) - exp(x)*log(64)) - 3*x^3)/(150*x^2 - 500*x - 20*x
^3 + x^4 + 625)