\(\int \frac {-64 x^8+(384 x^6+144 x^8) \log (x)+(-768 x^4-16 e^x x^4-896 x^6) \log ^2(x)+(512 x^2+1920 x^4+e^x (32 x^2+40 x^4+8 x^5)) \log ^3(x)+(-1536 x^2+e^x (-96 x^2-32 x^3)) \log ^4(x)+(256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)) \log ^5(x)}{\log ^5(x)} \, dx\) [2352]

Optimal result

Integrand size = 141, antiderivative size = 23 \[ \int \frac {-64 x^8+\left (384 x^6+144 x^8\right ) \log (x)+\left (-768 x^4-16 e^x x^4-896 x^6\right ) \log ^2(x)+\left (512 x^2+1920 x^4+e^x \left (32 x^2+40 x^4+8 x^5\right )\right ) \log ^3(x)+\left (-1536 x^2+e^x \left (-96 x^2-32 x^3\right )\right ) \log ^4(x)+\left (256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)\right ) \log ^5(x)}{\log ^5(x)} \, dx=x \left (x+\left (e^x+\left (4-\frac {2 x^2}{\log (x)}\right )^2\right )^2\right ) \]

[Out]

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(23)=46\).

Time = 0.84 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.26, number of steps used = 45, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6874, 2207, 2225, 2326, 2343, 2346, 2209, 2395} \[ \int \frac {-64 x^8+\left (384 x^6+144 x^8\right ) \log (x)+\left (-768 x^4-16 e^x x^4-896 x^6\right ) \log ^2(x)+\left (512 x^2+1920 x^4+e^x \left (32 x^2+40 x^4+8 x^5\right )\right ) \log ^3(x)+\left (-1536 x^2+e^x \left (-96 x^2-32 x^3\right )\right ) \log ^4(x)+\left (256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)\right ) \log ^5(x)}{\log ^5(x)} \, dx=\frac {16 x^9}{\log ^4(x)}-\frac {128 x^7}{\log ^3(x)}+\frac {384 x^5}{\log ^2(x)}-\frac {512 x^3}{\log (x)}+x^2+\frac {8 e^x \left (x^5 \log (x)-4 x^3 \log ^2(x)+4 x \log ^3(x)\right )}{\log ^3(x)}+256 x-\frac {e^{2 x}}{2}+\frac {1}{2} e^{2 x} (2 x+1) \]

[In]

Int[(-64*x^8 + (384*x^6 + 144*x^8)*Log[x] + (-768*x^4 - 16*E^x*x^4 - 896*x^6)*Log[x]^2 + (512*x^2 + 1920*x^4 +
E^x*(32*x^2 + 40*x^4 + 8*x^5))*Log[x]^3 + (-1536*x^2 + E^x*(-96*x^2 - 32*x^3))*Log[x]^4 + (256 + 2*x + E^(2*x
)*(1 + 2*x) + E^x*(32 + 32*x))*Log[x]^5)/Log[x]^5,x]

[Out]

-1/2*E^(2*x) + 256*x + x^2 + (E^(2*x)*(1 + 2*x))/2 + (16*x^9)/Log[x]^4 - (128*x^7)/Log[x]^3 + (384*x^5)/Log[x]
^2 - (512*x^3)/Log[x] + (8*E^x*(x^5*Log[x] - 4*x^3*Log[x]^2 + 4*x*Log[x]^3))/Log[x]^3

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 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 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 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 2343

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

Rule 2346

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2395

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (e^{2 x} (1+2 x)+\frac {8 e^x \left (-2 x^4+4 x^2 \log (x)+5 x^4 \log (x)+x^5 \log (x)-12 x^2 \log ^2(x)-4 x^3 \log ^2(x)+4 \log ^3(x)+4 x \log ^3(x)\right )}{\log ^3(x)}+\frac {2 \left (-32 x^8+192 x^6 \log (x)+72 x^8 \log (x)-384 x^4 \log ^2(x)-448 x^6 \log ^2(x)+256 x^2 \log ^3(x)+960 x^4 \log ^3(x)-768 x^2 \log ^4(x)+128 \log ^5(x)+x \log ^5(x)\right )}{\log ^5(x)}\right ) \, dx \\ & = 2 \int \frac {-32 x^8+192 x^6 \log (x)+72 x^8 \log (x)-384 x^4 \log ^2(x)-448 x^6 \log ^2(x)+256 x^2 \log ^3(x)+960 x^4 \log ^3(x)-768 x^2 \log ^4(x)+128 \log ^5(x)+x \log ^5(x)}{\log ^5(x)} \, dx+8 \int \frac {e^x \left (-2 x^4+4 x^2 \log (x)+5 x^4 \log (x)+x^5 \log (x)-12 x^2 \log ^2(x)-4 x^3 \log ^2(x)+4 \log ^3(x)+4 x \log ^3(x)\right )}{\log ^3(x)} \, dx+\int e^{2 x} (1+2 x) \, dx \\ & = \frac {1}{2} e^{2 x} (1+2 x)+\frac {8 e^x \left (x^5 \log (x)-4 x^3 \log ^2(x)+4 x \log ^3(x)\right )}{\log ^3(x)}+2 \int \left (128+x-\frac {32 x^8}{\log ^5(x)}+\frac {24 x^6 \left (8+3 x^2\right )}{\log ^4(x)}-\frac {64 x^4 \left (6+7 x^2\right )}{\log ^3(x)}+\frac {64 x^2 \left (4+15 x^2\right )}{\log ^2(x)}-\frac {768 x^2}{\log (x)}\right ) \, dx-\int e^{2 x} \, dx \\ & = -\frac {e^{2 x}}{2}+256 x+x^2+\frac {1}{2} e^{2 x} (1+2 x)+\frac {8 e^x \left (x^5 \log (x)-4 x^3 \log ^2(x)+4 x \log ^3(x)\right )}{\log ^3(x)}+48 \int \frac {x^6 \left (8+3 x^2\right )}{\log ^4(x)} \, dx-64 \int \frac {x^8}{\log ^5(x)} \, dx-128 \int \frac {x^4 \left (6+7 x^2\right )}{\log ^3(x)} \, dx+128 \int \frac {x^2 \left (4+15 x^2\right )}{\log ^2(x)} \, dx-1536 \int \frac {x^2}{\log (x)} \, dx \\ & = -\frac {e^{2 x}}{2}+256 x+x^2+\frac {1}{2} e^{2 x} (1+2 x)+\frac {16 x^9}{\log ^4(x)}+\frac {8 e^x \left (x^5 \log (x)-4 x^3 \log ^2(x)+4 x \log ^3(x)\right )}{\log ^3(x)}+48 \int \left (\frac {8 x^6}{\log ^4(x)}+\frac {3 x^8}{\log ^4(x)}\right ) \, dx-128 \int \left (\frac {6 x^4}{\log ^3(x)}+\frac {7 x^6}{\log ^3(x)}\right ) \, dx+128 \int \left (\frac {4 x^2}{\log ^2(x)}+\frac {15 x^4}{\log ^2(x)}\right ) \, dx-144 \int \frac {x^8}{\log ^4(x)} \, dx-1536 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right ) \\ & = -\frac {e^{2 x}}{2}+256 x+x^2+\frac {1}{2} e^{2 x} (1+2 x)-1536 \operatorname {ExpIntegralEi}(3 \log (x))+\frac {16 x^9}{\log ^4(x)}+\frac {48 x^9}{\log ^3(x)}+\frac {8 e^x \left (x^5 \log (x)-4 x^3 \log ^2(x)+4 x \log ^3(x)\right )}{\log ^3(x)}+144 \int \frac {x^8}{\log ^4(x)} \, dx+384 \int \frac {x^6}{\log ^4(x)} \, dx-432 \int \frac {x^8}{\log ^3(x)} \, dx+512 \int \frac {x^2}{\log ^2(x)} \, dx-768 \int \frac {x^4}{\log ^3(x)} \, dx-896 \int \frac {x^6}{\log ^3(x)} \, dx+1920 \int \frac {x^4}{\log ^2(x)} \, dx \\ & = -\frac {e^{2 x}}{2}+256 x+x^2+\frac {1}{2} e^{2 x} (1+2 x)-1536 \operatorname {ExpIntegralEi}(3 \log (x))+\frac {16 x^9}{\log ^4(x)}-\frac {128 x^7}{\log ^3(x)}+\frac {384 x^5}{\log ^2(x)}+\frac {448 x^7}{\log ^2(x)}+\frac {216 x^9}{\log ^2(x)}-\frac {512 x^3}{\log (x)}-\frac {1920 x^5}{\log (x)}+\frac {8 e^x \left (x^5 \log (x)-4 x^3 \log ^2(x)+4 x \log ^3(x)\right )}{\log ^3(x)}+432 \int \frac {x^8}{\log ^3(x)} \, dx+896 \int \frac {x^6}{\log ^3(x)} \, dx+1536 \int \frac {x^2}{\log (x)} \, dx-1920 \int \frac {x^4}{\log ^2(x)} \, dx-1944 \int \frac {x^8}{\log ^2(x)} \, dx-3136 \int \frac {x^6}{\log ^2(x)} \, dx+9600 \int \frac {x^4}{\log (x)} \, dx \\ & = -\frac {e^{2 x}}{2}+256 x+x^2+\frac {1}{2} e^{2 x} (1+2 x)-1536 \operatorname {ExpIntegralEi}(3 \log (x))+\frac {16 x^9}{\log ^4(x)}-\frac {128 x^7}{\log ^3(x)}+\frac {384 x^5}{\log ^2(x)}-\frac {512 x^3}{\log (x)}+\frac {3136 x^7}{\log (x)}+\frac {1944 x^9}{\log (x)}+\frac {8 e^x \left (x^5 \log (x)-4 x^3 \log ^2(x)+4 x \log ^3(x)\right )}{\log ^3(x)}+1536 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )+1944 \int \frac {x^8}{\log ^2(x)} \, dx+3136 \int \frac {x^6}{\log ^2(x)} \, dx-9600 \int \frac {x^4}{\log (x)} \, dx+9600 \text {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )-17496 \int \frac {x^8}{\log (x)} \, dx-21952 \int \frac {x^6}{\log (x)} \, dx \\ & = -\frac {e^{2 x}}{2}+256 x+x^2+\frac {1}{2} e^{2 x} (1+2 x)+9600 \operatorname {ExpIntegralEi}(5 \log (x))+\frac {16 x^9}{\log ^4(x)}-\frac {128 x^7}{\log ^3(x)}+\frac {384 x^5}{\log ^2(x)}-\frac {512 x^3}{\log (x)}+\frac {8 e^x \left (x^5 \log (x)-4 x^3 \log ^2(x)+4 x \log ^3(x)\right )}{\log ^3(x)}-9600 \text {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )+17496 \int \frac {x^8}{\log (x)} \, dx-17496 \text {Subst}\left (\int \frac {e^{9 x}}{x} \, dx,x,\log (x)\right )+21952 \int \frac {x^6}{\log (x)} \, dx-21952 \text {Subst}\left (\int \frac {e^{7 x}}{x} \, dx,x,\log (x)\right ) \\ & = -\frac {e^{2 x}}{2}+256 x+x^2+\frac {1}{2} e^{2 x} (1+2 x)-21952 \operatorname {ExpIntegralEi}(7 \log (x))-17496 \operatorname {ExpIntegralEi}(9 \log (x))+\frac {16 x^9}{\log ^4(x)}-\frac {128 x^7}{\log ^3(x)}+\frac {384 x^5}{\log ^2(x)}-\frac {512 x^3}{\log (x)}+\frac {8 e^x \left (x^5 \log (x)-4 x^3 \log ^2(x)+4 x \log ^3(x)\right )}{\log ^3(x)}+17496 \text {Subst}\left (\int \frac {e^{9 x}}{x} \, dx,x,\log (x)\right )+21952 \text {Subst}\left (\int \frac {e^{7 x}}{x} \, dx,x,\log (x)\right ) \\ & = -\frac {e^{2 x}}{2}+256 x+x^2+\frac {1}{2} e^{2 x} (1+2 x)+\frac {16 x^9}{\log ^4(x)}-\frac {128 x^7}{\log ^3(x)}+\frac {384 x^5}{\log ^2(x)}-\frac {512 x^3}{\log (x)}+\frac {8 e^x \left (x^5 \log (x)-4 x^3 \log ^2(x)+4 x \log ^3(x)\right )}{\log ^3(x)} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(66\) vs. \(2(23)=46\).

Time = 1.13 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.87 \[ \int \frac {-64 x^8+\left (384 x^6+144 x^8\right ) \log (x)+\left (-768 x^4-16 e^x x^4-896 x^6\right ) \log ^2(x)+\left (512 x^2+1920 x^4+e^x \left (32 x^2+40 x^4+8 x^5\right )\right ) \log ^3(x)+\left (-1536 x^2+e^x \left (-96 x^2-32 x^3\right )\right ) \log ^4(x)+\left (256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)\right ) \log ^5(x)}{\log ^5(x)} \, dx=256 x+32 e^x x+e^{2 x} x+x^2+\frac {16 x^9}{\log ^4(x)}-\frac {128 x^7}{\log ^3(x)}+\frac {8 \left (48+e^x\right ) x^5}{\log ^2(x)}-\frac {32 \left (16+e^x\right ) x^3}{\log (x)} \]

[In]

Integrate[(-64*x^8 + (384*x^6 + 144*x^8)*Log[x] + (-768*x^4 - 16*E^x*x^4 - 896*x^6)*Log[x]^2 + (512*x^2 + 1920
*x^4 + E^x*(32*x^2 + 40*x^4 + 8*x^5))*Log[x]^3 + (-1536*x^2 + E^x*(-96*x^2 - 32*x^3))*Log[x]^4 + (256 + 2*x +
E^(2*x)*(1 + 2*x) + E^x*(32 + 32*x))*Log[x]^5)/Log[x]^5,x]

[Out]

256*x + 32*E^x*x + E^(2*x)*x + x^2 + (16*x^9)/Log[x]^4 - (128*x^7)/Log[x]^3 + (8*(48 + E^x)*x^5)/Log[x]^2 - (3
2*(16 + E^x)*x^3)/Log[x]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(22)=44\).

Time = 0.13 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.22

method	result	size
risch	\(x \,{\mathrm e}^{2 x}+x^{2}+32 \,{\mathrm e}^{x} x +256 x +\frac {8 x^{3} \left (2 x^{6}-16 x^{4} \ln \left (x \right )+x^{2} {\mathrm e}^{x} \ln \left (x \right )^{2}+48 x^{2} \ln \left (x \right )^{2}-4 \,{\mathrm e}^{x} \ln \left (x \right )^{3}-64 \ln \left (x \right )^{3}\right )}{\ln \left (x \right )^{4}}\)	\(74\)
parallelrisch	\(\frac {16 x^{9}-128 x^{7} \ln \left (x \right )+8 \,{\mathrm e}^{x} x^{5} \ln \left (x \right )^{2}+384 x^{5} \ln \left (x \right )^{2}-32 \ln \left (x \right )^{3} {\mathrm e}^{x} x^{3}+x \,{\mathrm e}^{2 x} \ln \left (x \right )^{4}-512 x^{3} \ln \left (x \right )^{3}+x^{2} \ln \left (x \right )^{4}+32 \ln \left (x \right )^{4} {\mathrm e}^{x} x +256 x \ln \left (x \right )^{4}}{\ln \left (x \right )^{4}}\)	\(93\)

[In]

int((((1+2*x)*exp(x)^2+(32*x+32)*exp(x)+2*x+256)*ln(x)^5+((-32*x^3-96*x^2)*exp(x)-1536*x^2)*ln(x)^4+((8*x^5+40
*x^4+32*x^2)*exp(x)+1920*x^4+512*x^2)*ln(x)^3+(-16*exp(x)*x^4-896*x^6-768*x^4)*ln(x)^2+(144*x^8+384*x^6)*ln(x)
-64*x^8)/ln(x)^5,x,method=_RETURNVERBOSE)

[Out]

x*exp(x)^2+x^2+32*exp(x)*x+256*x+8*x^3*(2*x^6-16*x^4*ln(x)+x^2*exp(x)*ln(x)^2+48*x^2*ln(x)^2-4*exp(x)*ln(x)^3-
64*ln(x)^3)/ln(x)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (23) = 46\).

Time = 0.26 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.35 \[ \int \frac {-64 x^8+\left (384 x^6+144 x^8\right ) \log (x)+\left (-768 x^4-16 e^x x^4-896 x^6\right ) \log ^2(x)+\left (512 x^2+1920 x^4+e^x \left (32 x^2+40 x^4+8 x^5\right )\right ) \log ^3(x)+\left (-1536 x^2+e^x \left (-96 x^2-32 x^3\right )\right ) \log ^4(x)+\left (256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)\right ) \log ^5(x)}{\log ^5(x)} \, dx=\frac {16 \, x^{9} - 128 \, x^{7} \log \left (x\right ) + {\left (x^{2} + x e^{\left (2 \, x\right )} + 32 \, x e^{x} + 256 \, x\right )} \log \left (x\right )^{4} - 32 \, {\left (x^{3} e^{x} + 16 \, x^{3}\right )} \log \left (x\right )^{3} + 8 \, {\left (x^{5} e^{x} + 48 \, x^{5}\right )} \log \left (x\right )^{2}}{\log \left (x\right )^{4}} \]

[In]

integrate((((1+2*x)*exp(x)^2+(32*x+32)*exp(x)+2*x+256)*log(x)^5+((-32*x^3-96*x^2)*exp(x)-1536*x^2)*log(x)^4+((
8*x^5+40*x^4+32*x^2)*exp(x)+1920*x^4+512*x^2)*log(x)^3+(-16*exp(x)*x^4-896*x^6-768*x^4)*log(x)^2+(144*x^8+384*
x^6)*log(x)-64*x^8)/log(x)^5,x, algorithm="fricas")

[Out]

(16*x^9 - 128*x^7*log(x) + (x^2 + x*e^(2*x) + 32*x*e^x + 256*x)*log(x)^4 - 32*(x^3*e^x + 16*x^3)*log(x)^3 + 8*
(x^5*e^x + 48*x^5)*log(x)^2)/log(x)^4

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (19) = 38\).

Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.78 \[ \int \frac {-64 x^8+\left (384 x^6+144 x^8\right ) \log (x)+\left (-768 x^4-16 e^x x^4-896 x^6\right ) \log ^2(x)+\left (512 x^2+1920 x^4+e^x \left (32 x^2+40 x^4+8 x^5\right )\right ) \log ^3(x)+\left (-1536 x^2+e^x \left (-96 x^2-32 x^3\right )\right ) \log ^4(x)+\left (256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)\right ) \log ^5(x)}{\log ^5(x)} \, dx=x^{2} + 256 x + \frac {x e^{2 x} \log {\left (x \right )}^{2} + \left (8 x^{5} - 32 x^{3} \log {\left (x \right )} + 32 x \log {\left (x \right )}^{2}\right ) e^{x}}{\log {\left (x \right )}^{2}} + \frac {16 x^{9} - 128 x^{7} \log {\left (x \right )} + 384 x^{5} \log {\left (x \right )}^{2} - 512 x^{3} \log {\left (x \right )}^{3}}{\log {\left (x \right )}^{4}} \]

[In]

integrate((((1+2*x)*exp(x)**2+(32*x+32)*exp(x)+2*x+256)*ln(x)**5+((-32*x**3-96*x**2)*exp(x)-1536*x**2)*ln(x)**
4+((8*x**5+40*x**4+32*x**2)*exp(x)+1920*x**4+512*x**2)*ln(x)**3+(-16*exp(x)*x**4-896*x**6-768*x**4)*ln(x)**2+(
144*x**8+384*x**6)*ln(x)-64*x**8)/ln(x)**5,x)

[Out]

x**2 + 256*x + (x*exp(2*x)*log(x)**2 + (8*x**5 - 32*x**3*log(x) + 32*x*log(x)**2)*exp(x))/log(x)**2 + (16*x**9
- 128*x**7*log(x) + 384*x**5*log(x)**2 - 512*x**3*log(x)**3)/log(x)**4

Maxima [F]

\[ \int \frac {-64 x^8+\left (384 x^6+144 x^8\right ) \log (x)+\left (-768 x^4-16 e^x x^4-896 x^6\right ) \log ^2(x)+\left (512 x^2+1920 x^4+e^x \left (32 x^2+40 x^4+8 x^5\right )\right ) \log ^3(x)+\left (-1536 x^2+e^x \left (-96 x^2-32 x^3\right )\right ) \log ^4(x)+\left (256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)\right ) \log ^5(x)}{\log ^5(x)} \, dx=\int { -\frac {64 \, x^{8} - {\left ({\left (2 \, x + 1\right )} e^{\left (2 \, x\right )} + 32 \, {\left (x + 1\right )} e^{x} + 2 \, x + 256\right )} \log \left (x\right )^{5} + 32 \, {\left (48 \, x^{2} + {\left (x^{3} + 3 \, x^{2}\right )} e^{x}\right )} \log \left (x\right )^{4} - 8 \, {\left (240 \, x^{4} + 64 \, x^{2} + {\left (x^{5} + 5 \, x^{4} + 4 \, x^{2}\right )} e^{x}\right )} \log \left (x\right )^{3} + 16 \, {\left (56 \, x^{6} + x^{4} e^{x} + 48 \, x^{4}\right )} \log \left (x\right )^{2} - 48 \, {\left (3 \, x^{8} + 8 \, x^{6}\right )} \log \left (x\right )}{\log \left (x\right )^{5}} \,d x } \]

[In]

integrate((((1+2*x)*exp(x)^2+(32*x+32)*exp(x)+2*x+256)*log(x)^5+((-32*x^3-96*x^2)*exp(x)-1536*x^2)*log(x)^4+((
8*x^5+40*x^4+32*x^2)*exp(x)+1920*x^4+512*x^2)*log(x)^3+(-16*exp(x)*x^4-896*x^6-768*x^4)*log(x)^2+(144*x^8+384*
x^6)*log(x)-64*x^8)/log(x)^5,x, algorithm="maxima")

[Out]

x^2 + 1/2*(2*x - 1)*e^(2*x) + 32*(x - 1)*e^x + 256*x + 8*((x^5 - 4*x^3*log(x))*e^x - 16*(15*x^5 + 4*x^3)*log(x
))/log(x)^2 + 1/2*e^(2*x) + 32*e^x + 19200*gamma(-2, -5*log(x)) + 43904*gamma(-2, -7*log(x)) + 131712*gamma(-3
, -7*log(x)) + 104976*gamma(-3, -9*log(x)) + 419904*gamma(-4, -9*log(x)) + 9600*integrate(x^4/log(x), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (23) = 46\).

Time = 0.32 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.00 \[ \int \frac {-64 x^8+\left (384 x^6+144 x^8\right ) \log (x)+\left (-768 x^4-16 e^x x^4-896 x^6\right ) \log ^2(x)+\left (512 x^2+1920 x^4+e^x \left (32 x^2+40 x^4+8 x^5\right )\right ) \log ^3(x)+\left (-1536 x^2+e^x \left (-96 x^2-32 x^3\right )\right ) \log ^4(x)+\left (256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)\right ) \log ^5(x)}{\log ^5(x)} \, dx=\frac {16 \, x^{9} - 128 \, x^{7} \log \left (x\right ) + 8 \, x^{5} e^{x} \log \left (x\right )^{2} + 384 \, x^{5} \log \left (x\right )^{2} - 32 \, x^{3} e^{x} \log \left (x\right )^{3} - 512 \, x^{3} \log \left (x\right )^{3} + x^{2} \log \left (x\right )^{4} + x e^{\left (2 \, x\right )} \log \left (x\right )^{4} + 32 \, x e^{x} \log \left (x\right )^{4} + 256 \, x \log \left (x\right )^{4}}{\log \left (x\right )^{4}} \]

[In]

integrate((((1+2*x)*exp(x)^2+(32*x+32)*exp(x)+2*x+256)*log(x)^5+((-32*x^3-96*x^2)*exp(x)-1536*x^2)*log(x)^4+((
8*x^5+40*x^4+32*x^2)*exp(x)+1920*x^4+512*x^2)*log(x)^3+(-16*exp(x)*x^4-896*x^6-768*x^4)*log(x)^2+(144*x^8+384*
x^6)*log(x)-64*x^8)/log(x)^5,x, algorithm="giac")

[Out]

(16*x^9 - 128*x^7*log(x) + 8*x^5*e^x*log(x)^2 + 384*x^5*log(x)^2 - 32*x^3*e^x*log(x)^3 - 512*x^3*log(x)^3 + x^
2*log(x)^4 + x*e^(2*x)*log(x)^4 + 32*x*e^x*log(x)^4 + 256*x*log(x)^4)/log(x)^4

Mupad [B] (verification not implemented)

Time = 9.30 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.30 \[ \int \frac {-64 x^8+\left (384 x^6+144 x^8\right ) \log (x)+\left (-768 x^4-16 e^x x^4-896 x^6\right ) \log ^2(x)+\left (512 x^2+1920 x^4+e^x \left (32 x^2+40 x^4+8 x^5\right )\right ) \log ^3(x)+\left (-1536 x^2+e^x \left (-96 x^2-32 x^3\right )\right ) \log ^4(x)+\left (256+2 x+e^{2 x} (1+2 x)+e^x (32+32 x)\right ) \log ^5(x)}{\log ^5(x)} \, dx=256\,x+x\,{\mathrm {e}}^{2\,x}-\frac {512\,x^3}{\ln \left (x\right )}+\frac {384\,x^5}{{\ln \left (x\right )}^2}-\frac {128\,x^7}{{\ln \left (x\right )}^3}+\frac {16\,x^9}{{\ln \left (x\right )}^4}+32\,x\,{\mathrm {e}}^x+x^2-\frac {32\,x^3\,{\mathrm {e}}^x}{\ln \left (x\right )}+\frac {8\,x^5\,{\mathrm {e}}^x}{{\ln \left (x\right )}^2} \]

[In]

int((log(x)*(384*x^6 + 144*x^8) + log(x)^5*(2*x + exp(x)*(32*x + 32) + exp(2*x)*(2*x + 1) + 256) - log(x)^4*(e
xp(x)*(96*x^2 + 32*x^3) + 1536*x^2) - log(x)^2*(16*x^4*exp(x) + 768*x^4 + 896*x^6) + log(x)^3*(exp(x)*(32*x^2
+ 40*x^4 + 8*x^5) + 512*x^2 + 1920*x^4) - 64*x^8)/log(x)^5,x)

[Out]

256*x + x*exp(2*x) - (512*x^3)/log(x) + (384*x^5)/log(x)^2 - (128*x^7)/log(x)^3 + (16*x^9)/log(x)^4 + 32*x*exp
(x) + x^2 - (32*x^3*exp(x))/log(x) + (8*x^5*exp(x))/log(x)^2