\(\int \frac {26244 x^8+324 e^{20} x^8-43740 x^9-26244 x^{10}+54432 x^{11}+12312 x^{12}-23328 x^{13}-4860 x^{14}+3564 x^{15}+972 x^{16}+e^{-4+4 x} (-4+4 x)+e^{15} (-3888 x^8+1620 x^9+1944 x^{10})+e^{10} (17496 x^8-14580 x^9-14580 x^{10}+6804 x^{11}+3888 x^{12})+e^5 (-34992 x^8+43740 x^9+34992 x^{10}-38556 x^{11}-15552 x^{12}+8748 x^{13}+3240 x^{14})+e^{-2+2 x} (216 x^5+36 e^{10} x^5-396 x^6-72 x^7+144 x^8+36 x^9+e^5 (-180 x^5+144 x^6+72 x^7))}{x^5} \, dx\) [566]

Optimal result

Integrand size = 223, antiderivative size = 30 \[ \int \frac {26244 x^8+324 e^{20} x^8-43740 x^9-26244 x^{10}+54432 x^{11}+12312 x^{12}-23328 x^{13}-4860 x^{14}+3564 x^{15}+972 x^{16}+e^{-4+4 x} (-4+4 x)+e^{15} \left (-3888 x^8+1620 x^9+1944 x^{10}\right )+e^{10} \left (17496 x^8-14580 x^9-14580 x^{10}+6804 x^{11}+3888 x^{12}\right )+e^5 \left (-34992 x^8+43740 x^9+34992 x^{10}-38556 x^{11}-15552 x^{12}+8748 x^{13}+3240 x^{14}\right )+e^{-2+2 x} \left (216 x^5+36 e^{10} x^5-396 x^6-72 x^7+144 x^8+36 x^9+e^5 \left (-180 x^5+144 x^6+72 x^7\right )\right )}{x^5} \, dx=\left (\frac {e^{-2+2 x}}{x^2}+9 x^2 \left (-3+e^5+x+x^2\right )^2\right )^2 \]

[Out]

(exp(-1+x)^2/x^2+9*(exp(5)-3+x^2+x)^2*x^2)^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(193\) vs. \(2(30)=60\).

Time = 0.25 (sec) , antiderivative size = 193, normalized size of antiderivative = 6.43, number of steps used = 22, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {6, 14, 2228, 2227, 2225, 2207, 1602} \[ \int \frac {26244 x^8+324 e^{20} x^8-43740 x^9-26244 x^{10}+54432 x^{11}+12312 x^{12}-23328 x^{13}-4860 x^{14}+3564 x^{15}+972 x^{16}+e^{-4+4 x} (-4+4 x)+e^{15} \left (-3888 x^8+1620 x^9+1944 x^{10}\right )+e^{10} \left (17496 x^8-14580 x^9-14580 x^{10}+6804 x^{11}+3888 x^{12}\right )+e^5 \left (-34992 x^8+43740 x^9+34992 x^{10}-38556 x^{11}-15552 x^{12}+8748 x^{13}+3240 x^{14}\right )+e^{-2+2 x} \left (216 x^5+36 e^{10} x^5-396 x^6-72 x^7+144 x^8+36 x^9+e^5 \left (-180 x^5+144 x^6+72 x^7\right )\right )}{x^5} \, dx=18 e^{2 x-2} x^4+\frac {e^{4 x-4}}{x^4}+36 e^{2 x-2} x^3-54 e^{2 x-2} x^2-36 \left (1-e^5\right ) e^{2 x-2} x^2+81 \left (-x^2-x-e^5+3\right )^4 x^4+54 e^{2 x-2} x+36 \left (1-e^5\right ) e^{2 x-2} x-18 \left (11-4 e^5\right ) e^{2 x-2} x-27 e^{2 x-2}+18 \left (6-5 e^5+e^{10}\right ) e^{2 x-2}-18 \left (1-e^5\right ) e^{2 x-2}+9 \left (11-4 e^5\right ) e^{2 x-2} \]

[In]

Int[(26244*x^8 + 324*E^20*x^8 - 43740*x^9 - 26244*x^10 + 54432*x^11 + 12312*x^12 - 23328*x^13 - 4860*x^14 + 35
64*x^15 + 972*x^16 + E^(-4 + 4*x)*(-4 + 4*x) + E^15*(-3888*x^8 + 1620*x^9 + 1944*x^10) + E^10*(17496*x^8 - 145
80*x^9 - 14580*x^10 + 6804*x^11 + 3888*x^12) + E^5*(-34992*x^8 + 43740*x^9 + 34992*x^10 - 38556*x^11 - 15552*x
^12 + 8748*x^13 + 3240*x^14) + E^(-2 + 2*x)*(216*x^5 + 36*E^10*x^5 - 396*x^6 - 72*x^7 + 144*x^8 + 36*x^9 + E^5
*(-180*x^5 + 144*x^6 + 72*x^7)))/x^5,x]

[Out]

-27*E^(-2 + 2*x) + 9*E^(-2 + 2*x)*(11 - 4*E^5) - 18*E^(-2 + 2*x)*(1 - E^5) + 18*E^(-2 + 2*x)*(6 - 5*E^5 + E^10
) + E^(-4 + 4*x)/x^4 + 54*E^(-2 + 2*x)*x - 18*E^(-2 + 2*x)*(11 - 4*E^5)*x + 36*E^(-2 + 2*x)*(1 - E^5)*x - 54*E
^(-2 + 2*x)*x^2 - 36*E^(-2 + 2*x)*(1 - E^5)*x^2 + 36*E^(-2 + 2*x)*x^3 + 18*E^(-2 + 2*x)*x^4 + 81*x^4*(3 - E^5
- x - x^2)^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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 1602

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*x^(p - q +
1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rule 2207

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^m*
((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Dist[d*(m/(f*g*n*Log[F])), 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] && GtQ[m, 0] && IntegerQ[2*m] &&  !TrueQ[$UseGamma]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2227

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

Rule 2228

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
]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (26244+324 e^{20}\right ) x^8-43740 x^9-26244 x^{10}+54432 x^{11}+12312 x^{12}-23328 x^{13}-4860 x^{14}+3564 x^{15}+972 x^{16}+e^{-4+4 x} (-4+4 x)+e^{15} \left (-3888 x^8+1620 x^9+1944 x^{10}\right )+e^{10} \left (17496 x^8-14580 x^9-14580 x^{10}+6804 x^{11}+3888 x^{12}\right )+e^5 \left (-34992 x^8+43740 x^9+34992 x^{10}-38556 x^{11}-15552 x^{12}+8748 x^{13}+3240 x^{14}\right )+e^{-2+2 x} \left (216 x^5+36 e^{10} x^5-396 x^6-72 x^7+144 x^8+36 x^9+e^5 \left (-180 x^5+144 x^6+72 x^7\right )\right )}{x^5} \, dx \\ & = \int \left (\frac {4 e^{-4+4 x} (-1+x)}{x^5}+36 e^{-2+2 x} \left (-3+e^5+x+x^2\right ) \left (-2+e^5+3 x+x^2\right )+324 x^3 \left (-3+e^5+x+x^2\right )^3 \left (-3+e^5+2 x+3 x^2\right )\right ) \, dx \\ & = 4 \int \frac {e^{-4+4 x} (-1+x)}{x^5} \, dx+36 \int e^{-2+2 x} \left (-3+e^5+x+x^2\right ) \left (-2+e^5+3 x+x^2\right ) \, dx+324 \int x^3 \left (-3+e^5+x+x^2\right )^3 \left (-3+e^5+2 x+3 x^2\right ) \, dx \\ & = \frac {e^{-4+4 x}}{x^4}+81 x^4 \left (3-e^5-x-x^2\right )^4+36 \int \left (6 e^{-2+2 x} \left (1+\frac {1}{6} e^5 \left (-5+e^5\right )\right )+e^{-2+2 x} \left (-11+4 e^5\right ) x+2 e^{-2+2 x} \left (-1+e^5\right ) x^2+4 e^{-2+2 x} x^3+e^{-2+2 x} x^4\right ) \, dx \\ & = \frac {e^{-4+4 x}}{x^4}+81 x^4 \left (3-e^5-x-x^2\right )^4+36 \int e^{-2+2 x} x^4 \, dx+144 \int e^{-2+2 x} x^3 \, dx-\left (36 \left (11-4 e^5\right )\right ) \int e^{-2+2 x} x \, dx-\left (72 \left (1-e^5\right )\right ) \int e^{-2+2 x} x^2 \, dx+\left (36 \left (6-5 e^5+e^{10}\right )\right ) \int e^{-2+2 x} \, dx \\ & = 18 e^{-2+2 x} \left (6-5 e^5+e^{10}\right )+\frac {e^{-4+4 x}}{x^4}-18 e^{-2+2 x} \left (11-4 e^5\right ) x-36 e^{-2+2 x} \left (1-e^5\right ) x^2+72 e^{-2+2 x} x^3+18 e^{-2+2 x} x^4+81 x^4 \left (3-e^5-x-x^2\right )^4-72 \int e^{-2+2 x} x^3 \, dx-216 \int e^{-2+2 x} x^2 \, dx+\left (18 \left (11-4 e^5\right )\right ) \int e^{-2+2 x} \, dx+\left (72 \left (1-e^5\right )\right ) \int e^{-2+2 x} x \, dx \\ & = 9 e^{-2+2 x} \left (11-4 e^5\right )+18 e^{-2+2 x} \left (6-5 e^5+e^{10}\right )+\frac {e^{-4+4 x}}{x^4}-18 e^{-2+2 x} \left (11-4 e^5\right ) x+36 e^{-2+2 x} \left (1-e^5\right ) x-108 e^{-2+2 x} x^2-36 e^{-2+2 x} \left (1-e^5\right ) x^2+36 e^{-2+2 x} x^3+18 e^{-2+2 x} x^4+81 x^4 \left (3-e^5-x-x^2\right )^4+108 \int e^{-2+2 x} x^2 \, dx+216 \int e^{-2+2 x} x \, dx-\left (36 \left (1-e^5\right )\right ) \int e^{-2+2 x} \, dx \\ & = 9 e^{-2+2 x} \left (11-4 e^5\right )-18 e^{-2+2 x} \left (1-e^5\right )+18 e^{-2+2 x} \left (6-5 e^5+e^{10}\right )+\frac {e^{-4+4 x}}{x^4}+108 e^{-2+2 x} x-18 e^{-2+2 x} \left (11-4 e^5\right ) x+36 e^{-2+2 x} \left (1-e^5\right ) x-54 e^{-2+2 x} x^2-36 e^{-2+2 x} \left (1-e^5\right ) x^2+36 e^{-2+2 x} x^3+18 e^{-2+2 x} x^4+81 x^4 \left (3-e^5-x-x^2\right )^4-108 \int e^{-2+2 x} \, dx-108 \int e^{-2+2 x} x \, dx \\ & = -54 e^{-2+2 x}+9 e^{-2+2 x} \left (11-4 e^5\right )-18 e^{-2+2 x} \left (1-e^5\right )+18 e^{-2+2 x} \left (6-5 e^5+e^{10}\right )+\frac {e^{-4+4 x}}{x^4}+54 e^{-2+2 x} x-18 e^{-2+2 x} \left (11-4 e^5\right ) x+36 e^{-2+2 x} \left (1-e^5\right ) x-54 e^{-2+2 x} x^2-36 e^{-2+2 x} \left (1-e^5\right ) x^2+36 e^{-2+2 x} x^3+18 e^{-2+2 x} x^4+81 x^4 \left (3-e^5-x-x^2\right )^4+54 \int e^{-2+2 x} \, dx \\ & = -27 e^{-2+2 x}+9 e^{-2+2 x} \left (11-4 e^5\right )-18 e^{-2+2 x} \left (1-e^5\right )+18 e^{-2+2 x} \left (6-5 e^5+e^{10}\right )+\frac {e^{-4+4 x}}{x^4}+54 e^{-2+2 x} x-18 e^{-2+2 x} \left (11-4 e^5\right ) x+36 e^{-2+2 x} \left (1-e^5\right ) x-54 e^{-2+2 x} x^2-36 e^{-2+2 x} \left (1-e^5\right ) x^2+36 e^{-2+2 x} x^3+18 e^{-2+2 x} x^4+81 x^4 \left (3-e^5-x-x^2\right )^4 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.77 \[ \int \frac {26244 x^8+324 e^{20} x^8-43740 x^9-26244 x^{10}+54432 x^{11}+12312 x^{12}-23328 x^{13}-4860 x^{14}+3564 x^{15}+972 x^{16}+e^{-4+4 x} (-4+4 x)+e^{15} \left (-3888 x^8+1620 x^9+1944 x^{10}\right )+e^{10} \left (17496 x^8-14580 x^9-14580 x^{10}+6804 x^{11}+3888 x^{12}\right )+e^5 \left (-34992 x^8+43740 x^9+34992 x^{10}-38556 x^{11}-15552 x^{12}+8748 x^{13}+3240 x^{14}\right )+e^{-2+2 x} \left (216 x^5+36 e^{10} x^5-396 x^6-72 x^7+144 x^8+36 x^9+e^5 \left (-180 x^5+144 x^6+72 x^7\right )\right )}{x^5} \, dx=\frac {\left (e^{2 x}+9 e^{12} x^4+18 e^7 x^4 \left (-3+x+x^2\right )+9 e^2 x^4 \left (-3+x+x^2\right )^2\right )^2}{e^4 x^4} \]

[In]

Integrate[(26244*x^8 + 324*E^20*x^8 - 43740*x^9 - 26244*x^10 + 54432*x^11 + 12312*x^12 - 23328*x^13 - 4860*x^1
4 + 3564*x^15 + 972*x^16 + E^(-4 + 4*x)*(-4 + 4*x) + E^15*(-3888*x^8 + 1620*x^9 + 1944*x^10) + E^10*(17496*x^8
- 14580*x^9 - 14580*x^10 + 6804*x^11 + 3888*x^12) + E^5*(-34992*x^8 + 43740*x^9 + 34992*x^10 - 38556*x^11 - 1
5552*x^12 + 8748*x^13 + 3240*x^14) + E^(-2 + 2*x)*(216*x^5 + 36*E^10*x^5 - 396*x^6 - 72*x^7 + 144*x^8 + 36*x^9
+ E^5*(-180*x^5 + 144*x^6 + 72*x^7)))/x^5,x]

[Out]

(E^(2*x) + 9*E^12*x^4 + 18*E^7*x^4*(-3 + x + x^2) + 9*E^2*x^4*(-3 + x + x^2)^2)^2/(E^4*x^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(215\) vs. \(2(28)=56\).

Time = 0.35 (sec) , antiderivative size = 216, normalized size of antiderivative = 7.20

method	result	size
risch	\(-5508 x^{7} {\mathrm e}^{5}+8748 x^{5} {\mathrm e}^{5}-1944 x^{8} {\mathrm e}^{5}-8748 x^{4} {\mathrm e}^{5}+5832 x^{6} {\mathrm e}^{5}+324 x^{11}+81 x^{12}-486 x^{10}-2592 x^{9}+7776 x^{7}+1539 x^{8}+6561 x^{4}-4374 x^{6}-8748 x^{5}+486 x^{8} {\mathrm e}^{10}-2916 x^{5} {\mathrm e}^{10}+324 x^{5} {\mathrm e}^{15}+4374 x^{4} {\mathrm e}^{10}-972 x^{4} {\mathrm e}^{15}+81 x^{4} {\mathrm e}^{20}+\frac {{\mathrm e}^{-4+4 x}}{x^{4}}+\left (18 x^{4}+36 x^{2} {\mathrm e}^{5}+36 x^{3}+18 \,{\mathrm e}^{10}+36 x \,{\mathrm e}^{5}-90 x^{2}-108 \,{\mathrm e}^{5}-108 x +162\right ) {\mathrm e}^{-2+2 x}+972 x^{7} {\mathrm e}^{10}-2430 x^{6} {\mathrm e}^{10}+324 x^{6} {\mathrm e}^{15}+324 \,{\mathrm e}^{5} x^{10}+972 \,{\mathrm e}^{5} x^{9}\)	\(216\)
parts	\(-108 \,{\mathrm e}^{-2+2 x} \left (-1+x \right )+18 \,{\mathrm e}^{-2+2 x}+18 \,{\mathrm e}^{-2+2 x} \left (-1+x \right )^{4}+108 \,{\mathrm e}^{-2+2 x} \left (-1+x \right )^{3}+126 \,{\mathrm e}^{-2+2 x} \left (-1+x \right )^{2}+18 \,{\mathrm e}^{5} {\mathrm e}^{-2+2 x}+18 \,{\mathrm e}^{10} {\mathrm e}^{-2+2 x}+288 \,{\mathrm e}^{5} \left (\frac {{\mathrm e}^{-2+2 x} \left (-1+x \right )}{2}-\frac {{\mathrm e}^{-2+2 x}}{4}\right )+72 \,{\mathrm e}^{5} \left (\frac {{\mathrm e}^{-2+2 x} \left (-1+x \right )^{2}}{2}-\frac {{\mathrm e}^{-2+2 x} \left (-1+x \right )}{2}+\frac {{\mathrm e}^{-2+2 x}}{4}\right )+81 x^{12}+324 x^{11}+\frac {162 \left (10 \,{\mathrm e}^{5}-15\right ) x^{10}}{5}+36 \left (27 \,{\mathrm e}^{5}-72\right ) x^{9}+\frac {81 \left (12 \,{\mathrm e}^{10}-48 \,{\mathrm e}^{5}+38\right ) x^{8}}{2}+\frac {324 \left (21 \,{\mathrm e}^{10}-119 \,{\mathrm e}^{5}+168\right ) x^{7}}{7}+54 \left (6 \,{\mathrm e}^{15}-45 \,{\mathrm e}^{10}+108 \,{\mathrm e}^{5}-81\right ) x^{6}+\frac {324 \left (5 \,{\mathrm e}^{15}-45 \,{\mathrm e}^{10}+135 \,{\mathrm e}^{5}-135\right ) x^{5}}{5}+81 \left ({\mathrm e}^{20}-12 \,{\mathrm e}^{15}+54 \,{\mathrm e}^{10}-108 \,{\mathrm e}^{5}+81\right ) x^{4}+\frac {{\mathrm e}^{-4+4 x}}{x^{4}}\)	\(292\)
parallelrisch	\(\frac {-8748 x^{8} {\mathrm e}^{5}+324 x^{14} {\mathrm e}^{5}+7776 x^{11}+1539 x^{12}-2592 x^{13}-486 x^{14}+324 x^{15}+81 x^{16}-4374 x^{10}-8748 x^{9}+6561 x^{8}+486 x^{12} {\mathrm e}^{10}+972 x^{11} {\mathrm e}^{10}-2430 x^{10} {\mathrm e}^{10}+324 x^{10} {\mathrm e}^{15}-2916 x^{9} {\mathrm e}^{10}+324 x^{9} {\mathrm e}^{15}+4374 x^{8} {\mathrm e}^{10}-972 x^{8} {\mathrm e}^{15}+18 \,{\mathrm e}^{-2+2 x} x^{8}+81 x^{8} {\mathrm e}^{20}+36 \,{\mathrm e}^{-2+2 x} x^{7}-90 \,{\mathrm e}^{-2+2 x} x^{6}-108 \,{\mathrm e}^{-2+2 x} x^{5}+162 \,{\mathrm e}^{-2+2 x} x^{4}+{\mathrm e}^{-4+4 x}+36 \,{\mathrm e}^{5} {\mathrm e}^{-2+2 x} x^{6}+36 \,{\mathrm e}^{5} {\mathrm e}^{-2+2 x} x^{5}-108 \,{\mathrm e}^{5} {\mathrm e}^{-2+2 x} x^{4}+18 \,{\mathrm e}^{10} {\mathrm e}^{-2+2 x} x^{4}+972 \,{\mathrm e}^{5} x^{13}-1944 \,{\mathrm e}^{5} x^{12}-5508 \,{\mathrm e}^{5} x^{11}+5832 \,{\mathrm e}^{5} x^{10}+8748 \,{\mathrm e}^{5} x^{9}}{x^{4}}\)	\(296\)
derivativedivides	\(\text {Expression too large to display}\)	\(2996\)
default	\(\text {Expression too large to display}\)	\(2996\)

[In]

int(((-4+4*x)*exp(-1+x)^4+(36*x^5*exp(5)^2+(72*x^7+144*x^6-180*x^5)*exp(5)+36*x^9+144*x^8-72*x^7-396*x^6+216*x
^5)*exp(-1+x)^2+324*x^8*exp(5)^4+(1944*x^10+1620*x^9-3888*x^8)*exp(5)^3+(3888*x^12+6804*x^11-14580*x^10-14580*
x^9+17496*x^8)*exp(5)^2+(3240*x^14+8748*x^13-15552*x^12-38556*x^11+34992*x^10+43740*x^9-34992*x^8)*exp(5)+972*
x^16+3564*x^15-4860*x^14-23328*x^13+12312*x^12+54432*x^11-26244*x^10-43740*x^9+26244*x^8)/x^5,x,method=_RETURN
VERBOSE)

[Out]

-5508*x^7*exp(5)+8748*x^5*exp(5)-1944*x^8*exp(5)-8748*x^4*exp(5)+5832*x^6*exp(5)+324*x^11+81*x^12-486*x^10-259
2*x^9+7776*x^7+1539*x^8+6561*x^4-4374*x^6-8748*x^5+486*x^8*exp(10)-2916*x^5*exp(10)+324*x^5*exp(15)+4374*x^4*e
xp(10)-972*x^4*exp(15)+81*x^4*exp(20)+exp(-4+4*x)/x^4+(18*x^4+36*x^2*exp(5)+36*x^3+18*exp(10)+36*x*exp(5)-90*x
^2-108*exp(5)-108*x+162)*exp(-2+2*x)+972*x^7*exp(10)-2430*x^6*exp(10)+324*x^6*exp(15)+324*exp(5)*x^10+972*exp(
5)*x^9

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (28) = 56\).

Time = 0.26 (sec) , antiderivative size = 199, normalized size of antiderivative = 6.63 \[ \int \frac {26244 x^8+324 e^{20} x^8-43740 x^9-26244 x^{10}+54432 x^{11}+12312 x^{12}-23328 x^{13}-4860 x^{14}+3564 x^{15}+972 x^{16}+e^{-4+4 x} (-4+4 x)+e^{15} \left (-3888 x^8+1620 x^9+1944 x^{10}\right )+e^{10} \left (17496 x^8-14580 x^9-14580 x^{10}+6804 x^{11}+3888 x^{12}\right )+e^5 \left (-34992 x^8+43740 x^9+34992 x^{10}-38556 x^{11}-15552 x^{12}+8748 x^{13}+3240 x^{14}\right )+e^{-2+2 x} \left (216 x^5+36 e^{10} x^5-396 x^6-72 x^7+144 x^8+36 x^9+e^5 \left (-180 x^5+144 x^6+72 x^7\right )\right )}{x^5} \, dx=\frac {81 \, x^{16} + 324 \, x^{15} - 486 \, x^{14} - 2592 \, x^{13} + 1539 \, x^{12} + 7776 \, x^{11} - 4374 \, x^{10} - 8748 \, x^{9} + 81 \, x^{8} e^{20} + 6561 \, x^{8} + 324 \, {\left (x^{10} + x^{9} - 3 \, x^{8}\right )} e^{15} + 486 \, {\left (x^{12} + 2 \, x^{11} - 5 \, x^{10} - 6 \, x^{9} + 9 \, x^{8}\right )} e^{10} + 324 \, {\left (x^{14} + 3 \, x^{13} - 6 \, x^{12} - 17 \, x^{11} + 18 \, x^{10} + 27 \, x^{9} - 27 \, x^{8}\right )} e^{5} + 18 \, {\left (x^{8} + 2 \, x^{7} - 5 \, x^{6} - 6 \, x^{5} + x^{4} e^{10} + 9 \, x^{4} + 2 \, {\left (x^{6} + x^{5} - 3 \, x^{4}\right )} e^{5}\right )} e^{\left (2 \, x - 2\right )} + e^{\left (4 \, x - 4\right )}}{x^{4}} \]

[In]

integrate(((-4+4*x)*exp(-1+x)^4+(36*x^5*exp(5)^2+(72*x^7+144*x^6-180*x^5)*exp(5)+36*x^9+144*x^8-72*x^7-396*x^6
+216*x^5)*exp(-1+x)^2+324*x^8*exp(5)^4+(1944*x^10+1620*x^9-3888*x^8)*exp(5)^3+(3888*x^12+6804*x^11-14580*x^10-
14580*x^9+17496*x^8)*exp(5)^2+(3240*x^14+8748*x^13-15552*x^12-38556*x^11+34992*x^10+43740*x^9-34992*x^8)*exp(5
)+972*x^16+3564*x^15-4860*x^14-23328*x^13+12312*x^12+54432*x^11-26244*x^10-43740*x^9+26244*x^8)/x^5,x, algorit
hm="fricas")

[Out]

(81*x^16 + 324*x^15 - 486*x^14 - 2592*x^13 + 1539*x^12 + 7776*x^11 - 4374*x^10 - 8748*x^9 + 81*x^8*e^20 + 6561
*x^8 + 324*(x^10 + x^9 - 3*x^8)*e^15 + 486*(x^12 + 2*x^11 - 5*x^10 - 6*x^9 + 9*x^8)*e^10 + 324*(x^14 + 3*x^13
- 6*x^12 - 17*x^11 + 18*x^10 + 27*x^9 - 27*x^8)*e^5 + 18*(x^8 + 2*x^7 - 5*x^6 - 6*x^5 + x^4*e^10 + 9*x^4 + 2*(
x^6 + x^5 - 3*x^4)*e^5)*e^(2*x - 2) + e^(4*x - 4))/x^4

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (27) = 54\).

Time = 0.14 (sec) , antiderivative size = 202, normalized size of antiderivative = 6.73 \[ \int \frac {26244 x^8+324 e^{20} x^8-43740 x^9-26244 x^{10}+54432 x^{11}+12312 x^{12}-23328 x^{13}-4860 x^{14}+3564 x^{15}+972 x^{16}+e^{-4+4 x} (-4+4 x)+e^{15} \left (-3888 x^8+1620 x^9+1944 x^{10}\right )+e^{10} \left (17496 x^8-14580 x^9-14580 x^{10}+6804 x^{11}+3888 x^{12}\right )+e^5 \left (-34992 x^8+43740 x^9+34992 x^{10}-38556 x^{11}-15552 x^{12}+8748 x^{13}+3240 x^{14}\right )+e^{-2+2 x} \left (216 x^5+36 e^{10} x^5-396 x^6-72 x^7+144 x^8+36 x^9+e^5 \left (-180 x^5+144 x^6+72 x^7\right )\right )}{x^5} \, dx=81 x^{12} + 324 x^{11} + x^{10} \left (-486 + 324 e^{5}\right ) + x^{9} \left (-2592 + 972 e^{5}\right ) + x^{8} \left (- 1944 e^{5} + 1539 + 486 e^{10}\right ) + x^{7} \left (- 5508 e^{5} + 7776 + 972 e^{10}\right ) + x^{6} \left (- 2430 e^{10} - 4374 + 5832 e^{5} + 324 e^{15}\right ) + x^{5} \left (- 2916 e^{10} - 8748 + 8748 e^{5} + 324 e^{15}\right ) + x^{4} \left (- 972 e^{15} - 8748 e^{5} + 6561 + 4374 e^{10} + 81 e^{20}\right ) + \frac {\left (18 x^{8} + 36 x^{7} - 90 x^{6} + 36 x^{6} e^{5} - 108 x^{5} + 36 x^{5} e^{5} - 108 x^{4} e^{5} + 162 x^{4} + 18 x^{4} e^{10}\right ) e^{2 x - 2} + e^{4 x - 4}}{x^{4}} \]

[In]

integrate(((-4+4*x)*exp(-1+x)**4+(36*x**5*exp(5)**2+(72*x**7+144*x**6-180*x**5)*exp(5)+36*x**9+144*x**8-72*x**
7-396*x**6+216*x**5)*exp(-1+x)**2+324*x**8*exp(5)**4+(1944*x**10+1620*x**9-3888*x**8)*exp(5)**3+(3888*x**12+68
04*x**11-14580*x**10-14580*x**9+17496*x**8)*exp(5)**2+(3240*x**14+8748*x**13-15552*x**12-38556*x**11+34992*x**
10+43740*x**9-34992*x**8)*exp(5)+972*x**16+3564*x**15-4860*x**14-23328*x**13+12312*x**12+54432*x**11-26244*x**
10-43740*x**9+26244*x**8)/x**5,x)

[Out]

81*x**12 + 324*x**11 + x**10*(-486 + 324*exp(5)) + x**9*(-2592 + 972*exp(5)) + x**8*(-1944*exp(5) + 1539 + 486
*exp(10)) + x**7*(-5508*exp(5) + 7776 + 972*exp(10)) + x**6*(-2430*exp(10) - 4374 + 5832*exp(5) + 324*exp(15))
+ x**5*(-2916*exp(10) - 8748 + 8748*exp(5) + 324*exp(15)) + x**4*(-972*exp(15) - 8748*exp(5) + 6561 + 4374*ex
p(10) + 81*exp(20)) + ((18*x**8 + 36*x**7 - 90*x**6 + 36*x**6*exp(5) - 108*x**5 + 36*x**5*exp(5) - 108*x**4*ex
p(5) + 162*x**4 + 18*x**4*exp(10))*exp(2*x - 2) + exp(4*x - 4))/x**4

Maxima [C] (verification not implemented)

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

Time = 0.23 (sec) , antiderivative size = 319, normalized size of antiderivative = 10.63 \[ \int \frac {26244 x^8+324 e^{20} x^8-43740 x^9-26244 x^{10}+54432 x^{11}+12312 x^{12}-23328 x^{13}-4860 x^{14}+3564 x^{15}+972 x^{16}+e^{-4+4 x} (-4+4 x)+e^{15} \left (-3888 x^8+1620 x^9+1944 x^{10}\right )+e^{10} \left (17496 x^8-14580 x^9-14580 x^{10}+6804 x^{11}+3888 x^{12}\right )+e^5 \left (-34992 x^8+43740 x^9+34992 x^{10}-38556 x^{11}-15552 x^{12}+8748 x^{13}+3240 x^{14}\right )+e^{-2+2 x} \left (216 x^5+36 e^{10} x^5-396 x^6-72 x^7+144 x^8+36 x^9+e^5 \left (-180 x^5+144 x^6+72 x^7\right )\right )}{x^5} \, dx=81 \, x^{12} + 324 \, x^{11} + 324 \, x^{10} e^{5} - 486 \, x^{10} + 972 \, x^{9} e^{5} - 2592 \, x^{9} + 486 \, x^{8} e^{10} - 1944 \, x^{8} e^{5} + 1539 \, x^{8} + 972 \, x^{7} e^{10} - 5508 \, x^{7} e^{5} + 7776 \, x^{7} + 324 \, x^{6} e^{15} - 2430 \, x^{6} e^{10} + 5832 \, x^{6} e^{5} - 4374 \, x^{6} + 324 \, x^{5} e^{15} - 2916 \, x^{5} e^{10} + 8748 \, x^{5} e^{5} - 8748 \, x^{5} + 81 \, x^{4} e^{20} - 972 \, x^{4} e^{15} + 4374 \, x^{4} e^{10} - 8748 \, x^{4} e^{5} + 6561 \, x^{4} + 18 \, {\left (2 \, x^{2} e^{3} - 2 \, x e^{3} + e^{3}\right )} e^{\left (2 \, x\right )} + 36 \, {\left (2 \, x e^{3} - e^{3}\right )} e^{\left (2 \, x\right )} + 9 \, {\left (2 \, x^{4} - 4 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} e^{\left (2 \, x - 2\right )} + 18 \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x - 2\right )} - 18 \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x - 2\right )} - 99 \, {\left (2 \, x - 1\right )} e^{\left (2 \, x - 2\right )} + 256 \, e^{\left (-4\right )} \Gamma \left (-3, -4 \, x\right ) + 1024 \, e^{\left (-4\right )} \Gamma \left (-4, -4 \, x\right ) + 18 \, e^{\left (2 \, x + 8\right )} - 90 \, e^{\left (2 \, x + 3\right )} + 108 \, e^{\left (2 \, x - 2\right )} \]

[In]

integrate(((-4+4*x)*exp(-1+x)^4+(36*x^5*exp(5)^2+(72*x^7+144*x^6-180*x^5)*exp(5)+36*x^9+144*x^8-72*x^7-396*x^6
+216*x^5)*exp(-1+x)^2+324*x^8*exp(5)^4+(1944*x^10+1620*x^9-3888*x^8)*exp(5)^3+(3888*x^12+6804*x^11-14580*x^10-
14580*x^9+17496*x^8)*exp(5)^2+(3240*x^14+8748*x^13-15552*x^12-38556*x^11+34992*x^10+43740*x^9-34992*x^8)*exp(5
)+972*x^16+3564*x^15-4860*x^14-23328*x^13+12312*x^12+54432*x^11-26244*x^10-43740*x^9+26244*x^8)/x^5,x, algorit
hm="maxima")

[Out]

81*x^12 + 324*x^11 + 324*x^10*e^5 - 486*x^10 + 972*x^9*e^5 - 2592*x^9 + 486*x^8*e^10 - 1944*x^8*e^5 + 1539*x^8
+ 972*x^7*e^10 - 5508*x^7*e^5 + 7776*x^7 + 324*x^6*e^15 - 2430*x^6*e^10 + 5832*x^6*e^5 - 4374*x^6 + 324*x^5*e
^15 - 2916*x^5*e^10 + 8748*x^5*e^5 - 8748*x^5 + 81*x^4*e^20 - 972*x^4*e^15 + 4374*x^4*e^10 - 8748*x^4*e^5 + 65
61*x^4 + 18*(2*x^2*e^3 - 2*x*e^3 + e^3)*e^(2*x) + 36*(2*x*e^3 - e^3)*e^(2*x) + 9*(2*x^4 - 4*x^3 + 6*x^2 - 6*x
+ 3)*e^(2*x - 2) + 18*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x - 2) - 18*(2*x^2 - 2*x + 1)*e^(2*x - 2) - 99*(2*x - 1)*
e^(2*x - 2) + 256*e^(-4)*gamma(-3, -4*x) + 1024*e^(-4)*gamma(-4, -4*x) + 18*e^(2*x + 8) - 90*e^(2*x + 3) + 108
*e^(2*x - 2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 783 vs. \(2 (28) = 56\).

Time = 0.33 (sec) , antiderivative size = 783, normalized size of antiderivative = 26.10 \[ \int \frac {26244 x^8+324 e^{20} x^8-43740 x^9-26244 x^{10}+54432 x^{11}+12312 x^{12}-23328 x^{13}-4860 x^{14}+3564 x^{15}+972 x^{16}+e^{-4+4 x} (-4+4 x)+e^{15} \left (-3888 x^8+1620 x^9+1944 x^{10}\right )+e^{10} \left (17496 x^8-14580 x^9-14580 x^{10}+6804 x^{11}+3888 x^{12}\right )+e^5 \left (-34992 x^8+43740 x^9+34992 x^{10}-38556 x^{11}-15552 x^{12}+8748 x^{13}+3240 x^{14}\right )+e^{-2+2 x} \left (216 x^5+36 e^{10} x^5-396 x^6-72 x^7+144 x^8+36 x^9+e^5 \left (-180 x^5+144 x^6+72 x^7\right )\right )}{x^5} \, dx=\text {Too large to display} \]

[In]

integrate(((-4+4*x)*exp(-1+x)^4+(36*x^5*exp(5)^2+(72*x^7+144*x^6-180*x^5)*exp(5)+36*x^9+144*x^8-72*x^7-396*x^6
+216*x^5)*exp(-1+x)^2+324*x^8*exp(5)^4+(1944*x^10+1620*x^9-3888*x^8)*exp(5)^3+(3888*x^12+6804*x^11-14580*x^10-
14580*x^9+17496*x^8)*exp(5)^2+(3240*x^14+8748*x^13-15552*x^12-38556*x^11+34992*x^10+43740*x^9-34992*x^8)*exp(5
)+972*x^16+3564*x^15-4860*x^14-23328*x^13+12312*x^12+54432*x^11-26244*x^10-43740*x^9+26244*x^8)/x^5,x, algorit
hm="giac")

[Out]

(81*(x - 1)^16 + 1620*(x - 1)^15 + 324*(x - 1)^14*e^5 + 14094*(x - 1)^14 + 5508*(x - 1)^13*e^5 + 69984*(x - 1)
^13 + 486*(x - 1)^12*e^10 + 40176*(x - 1)^12*e^5 + 218457*(x - 1)^12 + 6804*(x - 1)^11*e^10 + 164916*(x - 1)^1
1*e^5 + 443232*(x - 1)^11 + 324*(x - 1)^10*e^15 + 40338*(x - 1)^10*e^10 + 419256*(x - 1)^10*e^5 + 576558*(x -
1)^10 + 3564*(x - 1)^9*e^15 + 133164*(x - 1)^9*e^10 + 680076*(x - 1)^9*e^5 + 435780*(x - 1)^9 + 81*(x - 1)^8*e
^20 + 16524*(x - 1)^8*e^15 + 269730*(x - 1)^8*e^10 + 685260*(x - 1)^8*e^5 + 18*(x - 1)^8*e^(2*x - 2) + 107892*
(x - 1)^8 + 648*(x - 1)^7*e^20 + 42768*(x - 1)^7*e^15 + 344088*(x - 1)^7*e^10 + 367416*(x - 1)^7*e^5 + 180*(x
- 1)^7*e^(2*x - 2) - 106596*(x - 1)^7 + 2268*(x - 1)^6*e^20 + 68040*(x - 1)^6*e^15 + 265356*(x - 1)^6*e^10 + 1
4580*(x - 1)^6*e^5 + 36*(x - 1)^6*e^(2*x + 3) + 666*(x - 1)^6*e^(2*x - 2) - 92178*(x - 1)^6 + 4536*(x - 1)^5*e
^20 + 68040*(x - 1)^5*e^15 + 99144*(x - 1)^5*e^10 - 102708*(x - 1)^5*e^5 + 252*(x - 1)^5*e^(2*x + 3) + 1116*(x
- 1)^5*e^(2*x - 2) - 7776*(x - 1)^5 + 5589*(x - 1)^4*e^20 + 41148*(x - 1)^4*e^15 - 10692*(x - 1)^4*e^10 - 456
84*(x - 1)^4*e^5 + 18*(x - 1)^4*e^(2*x + 8) + 612*(x - 1)^4*e^(2*x + 3) + 792*(x - 1)^4*e^(2*x - 2) + 16200*(x
- 1)^4 + 4212*(x - 1)^3*e^20 + 12960*(x - 1)^3*e^15 - 26244*(x - 1)^3*e^10 + 5508*(x - 1)^3*e^5 + 72*(x - 1)^
3*e^(2*x + 8) + 648*(x - 1)^3*e^(2*x + 3) + 36*(x - 1)^3*e^(2*x - 2) + 3564*(x - 1)^3 + 1782*(x - 1)^2*e^20 +
972*(x - 1)^2*e^15 - 9234*(x - 1)^2*e^10 + 8424*(x - 1)^2*e^5 + 108*(x - 1)^2*e^(2*x + 8) + 252*(x - 1)^2*e^(2
*x + 3) - 198*(x - 1)^2*e^(2*x - 2) - 1944*(x - 1)^2 + 324*(x - 1)*e^20 - 324*(x - 1)*e^15 - 972*(x - 1)*e^10
+ 1620*(x - 1)*e^5 + 72*(x - 1)*e^(2*x + 8) - 36*(x - 1)*e^(2*x + 3) - 36*(x - 1)*e^(2*x - 2) - 648*x + e^(4*x
- 4) + 18*e^(2*x + 8) - 36*e^(2*x + 3) + 18*e^(2*x - 2) + 648)/((x - 1)^4 + 4*(x - 1)^3 + 6*(x - 1)^2 + 4*x -
3)

Mupad [B] (verification not implemented)

Time = 8.13 (sec) , antiderivative size = 152, normalized size of antiderivative = 5.07 \[ \int \frac {26244 x^8+324 e^{20} x^8-43740 x^9-26244 x^{10}+54432 x^{11}+12312 x^{12}-23328 x^{13}-4860 x^{14}+3564 x^{15}+972 x^{16}+e^{-4+4 x} (-4+4 x)+e^{15} \left (-3888 x^8+1620 x^9+1944 x^{10}\right )+e^{10} \left (17496 x^8-14580 x^9-14580 x^{10}+6804 x^{11}+3888 x^{12}\right )+e^5 \left (-34992 x^8+43740 x^9+34992 x^{10}-38556 x^{11}-15552 x^{12}+8748 x^{13}+3240 x^{14}\right )+e^{-2+2 x} \left (216 x^5+36 e^{10} x^5-396 x^6-72 x^7+144 x^8+36 x^9+e^5 \left (-180 x^5+144 x^6+72 x^7\right )\right )}{x^5} \, dx=x^8\,\left (486\,{\mathrm {e}}^{10}-1944\,{\mathrm {e}}^5+1539\right )+x^7\,\left (972\,{\mathrm {e}}^{10}-5508\,{\mathrm {e}}^5+7776\right )+x^{10}\,\left (324\,{\mathrm {e}}^5-486\right )+x^9\,\left (972\,{\mathrm {e}}^5-2592\right )+81\,x^4\,{\left ({\mathrm {e}}^5-3\right )}^4+324\,x^5\,{\left ({\mathrm {e}}^5-3\right )}^3+{\mathrm {e}}^{2\,x-2}\,\left (18\,x^4+36\,x^3+\left (36\,{\mathrm {e}}^5-90\right )\,x^2+\left (36\,{\mathrm {e}}^5-108\right )\,x+18\,{\left ({\mathrm {e}}^5-3\right )}^2\right )+\frac {{\mathrm {e}}^{4\,x-4}}{x^4}+324\,x^{11}+81\,x^{12}+162\,x^6\,\left (2\,{\mathrm {e}}^5-3\right )\,{\left ({\mathrm {e}}^5-3\right )}^2 \]

[In]

int((exp(4*x - 4)*(4*x - 4) + exp(10)*(17496*x^8 - 14580*x^9 - 14580*x^10 + 6804*x^11 + 3888*x^12) + 324*x^8*e
xp(20) + exp(5)*(43740*x^9 - 34992*x^8 + 34992*x^10 - 38556*x^11 - 15552*x^12 + 8748*x^13 + 3240*x^14) + exp(1
5)*(1620*x^9 - 3888*x^8 + 1944*x^10) + exp(2*x - 2)*(36*x^5*exp(10) + exp(5)*(144*x^6 - 180*x^5 + 72*x^7) + 21
6*x^5 - 396*x^6 - 72*x^7 + 144*x^8 + 36*x^9) + 26244*x^8 - 43740*x^9 - 26244*x^10 + 54432*x^11 + 12312*x^12 -
23328*x^13 - 4860*x^14 + 3564*x^15 + 972*x^16)/x^5,x)

[Out]

x^8*(486*exp(10) - 1944*exp(5) + 1539) + x^7*(972*exp(10) - 5508*exp(5) + 7776) + x^10*(324*exp(5) - 486) + x^
9*(972*exp(5) - 2592) + 81*x^4*(exp(5) - 3)^4 + 324*x^5*(exp(5) - 3)^3 + exp(2*x - 2)*(18*(exp(5) - 3)^2 + x^2
*(36*exp(5) - 90) + 36*x^3 + 18*x^4 + x*(36*exp(5) - 108)) + exp(4*x - 4)/x^4 + 324*x^11 + 81*x^12 + 162*x^6*(
2*exp(5) - 3)*(exp(5) - 3)^2