3.2.48 \(\int \frac {-\frac {22500 e^8}{x}-22500 e^{20} x^3+22500 e^{20} x^3 \log (x)-900 e^{10} x \log ^2(x)+900 e^{10} x \log ^3(x)+\log ^5(x)+\frac {e^6 (-90000 e^5 x^3+22500 e^5 x^3 \log (x))}{x^3}+\frac {e^4 (-135000 e^{10} x^3+67500 e^{10} x^3 \log (x)-900 x \log ^2(x))}{x^2}+\frac {e^2 (-90000 e^{15} x^3+67500 e^{15} x^3 \log (x)-1800 e^5 x \log ^2(x)+900 e^5 x \log ^3(x))}{x}}{\log ^5(x)} \, dx\) [148]

Optimal. Leaf size=30 \[ x+9 \left (1+\frac {25 \left (e^5+\frac {e^2}{x}\right )^2 x^2}{\log ^2(x)}\right )^2 \]

[Out]

3*(25*x^2*(exp(5)+exp(-ln(x)+2))^2/ln(x)^2+1)*(75*x^2*(exp(5)+exp(-ln(x)+2))^2/ln(x)^2+3)+x

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(377\) vs. \(2(30)=60\).
time = 0.97, antiderivative size = 377, normalized size of antiderivative = 12.57, number of steps used = 101, number of rules used = 12, integrand size = 161, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {6820, 6874, 2395, 2334, 2335, 2339, 30, 2343, 2346, 2209, 2357, 2367} \begin {gather*} \frac {5625 e^{20} x^4}{\log ^4(x)}+\frac {7500 e^{20} x^4}{\log ^3(x)}+\frac {15000 e^{20} x^4}{\log ^2(x)}+\frac {60000 e^{20} x^4}{\log (x)}+\frac {22500 e^{17} x^3}{\log ^4(x)}+\frac {22500 e^{17} x^3}{\log ^3(x)}+\frac {33750 e^{17} x^3}{\log ^2(x)}+\frac {101250 e^{17} x^3}{\log (x)}+\frac {33750 e^{14} x^2}{\log ^4(x)}+\frac {22500 e^{14} x^2}{\log ^3(x)}+\frac {22500 e^{14} x^2}{\log ^2(x)}+\frac {450 e^{10} x^2}{\log ^2(x)}+\frac {45000 e^{14} x^2}{\log (x)}+\frac {900 e^{10} x^2}{\log (x)}+x+\frac {22500 e^{11} x}{\log ^4(x)}+\frac {5625 e^8}{\log ^4(x)}-\frac {7500 e^{11} \left (e^3 x+1\right )^3 x}{\log ^3(x)}+\frac {7500 e^{11} x}{\log ^3(x)}-\frac {15000 e^{11} \left (e^3 x+1\right )^3 x}{\log ^2(x)}+\frac {11250 e^{11} \left (e^3 x+1\right )^2 x}{\log ^2(x)}+\frac {3750 e^{11} x}{\log ^2(x)}+\frac {900 e^7 x}{\log ^2(x)}+\frac {450 e^4}{\log ^2(x)}-\frac {60000 e^{11} \left (e^3 x+1\right )^3 x}{\log (x)}+\frac {78750 e^{11} \left (e^3 x+1\right )^2 x}{\log (x)}-\frac {22500 e^{11} \left (e^3 x+1\right ) x}{\log (x)}-\frac {900 e^7 \left (e^3 x+1\right ) x}{\log (x)}+\frac {3750 e^{11} x}{\log (x)}+\frac {900 e^7 x}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-22500*E^8)/x - 22500*E^20*x^3 + 22500*E^20*x^3*Log[x] - 900*E^10*x*Log[x]^2 + 900*E^10*x*Log[x]^3 + Log
[x]^5 + (E^6*(-90000*E^5*x^3 + 22500*E^5*x^3*Log[x]))/x^3 + (E^4*(-135000*E^10*x^3 + 67500*E^10*x^3*Log[x] - 9
00*x*Log[x]^2))/x^2 + (E^2*(-90000*E^15*x^3 + 67500*E^15*x^3*Log[x] - 1800*E^5*x*Log[x]^2 + 900*E^5*x*Log[x]^3
))/x)/Log[x]^5,x]

[Out]

x + (5625*E^8)/Log[x]^4 + (22500*E^11*x)/Log[x]^4 + (33750*E^14*x^2)/Log[x]^4 + (22500*E^17*x^3)/Log[x]^4 + (5
625*E^20*x^4)/Log[x]^4 + (7500*E^11*x)/Log[x]^3 + (22500*E^14*x^2)/Log[x]^3 + (22500*E^17*x^3)/Log[x]^3 + (750
0*E^20*x^4)/Log[x]^3 - (7500*E^11*x*(1 + E^3*x)^3)/Log[x]^3 + (450*E^4)/Log[x]^2 + (900*E^7*x)/Log[x]^2 + (375
0*E^11*x)/Log[x]^2 + (450*E^10*x^2)/Log[x]^2 + (22500*E^14*x^2)/Log[x]^2 + (33750*E^17*x^3)/Log[x]^2 + (15000*
E^20*x^4)/Log[x]^2 + (11250*E^11*x*(1 + E^3*x)^2)/Log[x]^2 - (15000*E^11*x*(1 + E^3*x)^3)/Log[x]^2 + (900*E^7*
x)/Log[x] + (3750*E^11*x)/Log[x] + (900*E^10*x^2)/Log[x] + (45000*E^14*x^2)/Log[x] + (101250*E^17*x^3)/Log[x]
+ (60000*E^20*x^4)/Log[x] - (900*E^7*x*(1 + E^3*x))/Log[x] - (22500*E^11*x*(1 + E^3*x))/Log[x] + (78750*E^11*x
*(1 + E^3*x)^2)/Log[x] - (60000*E^11*x*(1 + E^3*x)^3)/Log[x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x*((a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2335

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, 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 2357

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

Rule 2367

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

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 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 {gather*} \begin {aligned} \text {integral} &=\int \frac {-22500 e^8 \left (1+e^3 x\right )^4+22500 e^{11} x \left (1+e^3 x\right )^3 \log (x)-900 e^4 \left (1+e^3 x\right )^2 \log ^2(x)+900 e^7 x \left (1+e^3 x\right ) \log ^3(x)+x \log ^5(x)}{x \log ^5(x)} \, dx\\ &=\int \left (1-\frac {22500 e^8 \left (1+e^3 x\right )^4}{x \log ^5(x)}+\frac {22500 e^{11} \left (1+e^3 x\right )^3}{\log ^4(x)}-\frac {900 e^4 \left (1+e^3 x\right )^2}{x \log ^3(x)}+\frac {900 e^7 \left (1+e^3 x\right )}{\log ^2(x)}\right ) \, dx\\ &=x-\left (900 e^4\right ) \int \frac {\left (1+e^3 x\right )^2}{x \log ^3(x)} \, dx+\left (900 e^7\right ) \int \frac {1+e^3 x}{\log ^2(x)} \, dx-\left (22500 e^8\right ) \int \frac {\left (1+e^3 x\right )^4}{x \log ^5(x)} \, dx+\left (22500 e^{11}\right ) \int \frac {\left (1+e^3 x\right )^3}{\log ^4(x)} \, dx\\ &=x-\frac {7500 e^{11} x \left (1+e^3 x\right )^3}{\log ^3(x)}-\frac {900 e^7 x \left (1+e^3 x\right )}{\log (x)}-\left (900 e^4\right ) \int \left (\frac {2 e^3}{\log ^3(x)}+\frac {1}{x \log ^3(x)}+\frac {e^6 x}{\log ^3(x)}\right ) \, dx-\left (900 e^7\right ) \int \frac {1}{\log (x)} \, dx+\left (1800 e^7\right ) \int \frac {1+e^3 x}{\log (x)} \, dx-\left (22500 e^8\right ) \int \left (\frac {4 e^3}{\log ^5(x)}+\frac {1}{x \log ^5(x)}+\frac {6 e^6 x}{\log ^5(x)}+\frac {4 e^9 x^2}{\log ^5(x)}+\frac {e^{12} x^3}{\log ^5(x)}\right ) \, dx-\left (22500 e^{11}\right ) \int \frac {\left (1+e^3 x\right )^2}{\log ^3(x)} \, dx+\left (30000 e^{11}\right ) \int \frac {\left (1+e^3 x\right )^3}{\log ^3(x)} \, dx\\ &=x-\frac {7500 e^{11} x \left (1+e^3 x\right )^3}{\log ^3(x)}+\frac {11250 e^{11} x \left (1+e^3 x\right )^2}{\log ^2(x)}-\frac {15000 e^{11} x \left (1+e^3 x\right )^3}{\log ^2(x)}-\frac {900 e^7 x \left (1+e^3 x\right )}{\log (x)}-900 e^7 \text {li}(x)-\left (900 e^4\right ) \int \frac {1}{x \log ^3(x)} \, dx+\left (1800 e^7\right ) \int \left (\frac {1}{\log (x)}+\frac {e^3 x}{\log (x)}\right ) \, dx-\left (1800 e^7\right ) \int \frac {1}{\log ^3(x)} \, dx-\left (22500 e^8\right ) \int \frac {1}{x \log ^5(x)} \, dx-\left (900 e^{10}\right ) \int \frac {x}{\log ^3(x)} \, dx+\left (22500 e^{11}\right ) \int \frac {1+e^3 x}{\log ^2(x)} \, dx-\left (33750 e^{11}\right ) \int \frac {\left (1+e^3 x\right )^2}{\log ^2(x)} \, dx-\left (45000 e^{11}\right ) \int \frac {\left (1+e^3 x\right )^2}{\log ^2(x)} \, dx+\left (60000 e^{11}\right ) \int \frac {\left (1+e^3 x\right )^3}{\log ^2(x)} \, dx-\left (90000 e^{11}\right ) \int \frac {1}{\log ^5(x)} \, dx-\left (135000 e^{14}\right ) \int \frac {x}{\log ^5(x)} \, dx-\left (90000 e^{17}\right ) \int \frac {x^2}{\log ^5(x)} \, dx-\left (22500 e^{20}\right ) \int \frac {x^3}{\log ^5(x)} \, dx\\ &=x+\frac {22500 e^{11} x}{\log ^4(x)}+\frac {33750 e^{14} x^2}{\log ^4(x)}+\frac {22500 e^{17} x^3}{\log ^4(x)}+\frac {5625 e^{20} x^4}{\log ^4(x)}-\frac {7500 e^{11} x \left (1+e^3 x\right )^3}{\log ^3(x)}+\frac {900 e^7 x}{\log ^2(x)}+\frac {450 e^{10} x^2}{\log ^2(x)}+\frac {11250 e^{11} x \left (1+e^3 x\right )^2}{\log ^2(x)}-\frac {15000 e^{11} x \left (1+e^3 x\right )^3}{\log ^2(x)}-\frac {900 e^7 x \left (1+e^3 x\right )}{\log (x)}-\frac {22500 e^{11} x \left (1+e^3 x\right )}{\log (x)}+\frac {78750 e^{11} x \left (1+e^3 x\right )^2}{\log (x)}-\frac {60000 e^{11} x \left (1+e^3 x\right )^3}{\log (x)}-900 e^7 \text {li}(x)-\left (900 e^4\right ) \text {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log (x)\right )-\left (900 e^7\right ) \int \frac {1}{\log ^2(x)} \, dx+\left (1800 e^7\right ) \int \frac {1}{\log (x)} \, dx-\left (22500 e^8\right ) \text {Subst}\left (\int \frac {1}{x^5} \, dx,x,\log (x)\right )-\left (900 e^{10}\right ) \int \frac {x}{\log ^2(x)} \, dx+\left (1800 e^{10}\right ) \int \frac {x}{\log (x)} \, dx-\left (22500 e^{11}\right ) \int \frac {1}{\log ^4(x)} \, dx-\left (22500 e^{11}\right ) \int \frac {1}{\log (x)} \, dx+\left (45000 e^{11}\right ) \int \frac {1+e^3 x}{\log (x)} \, dx+\left (67500 e^{11}\right ) \int \frac {1+e^3 x}{\log (x)} \, dx+\left (90000 e^{11}\right ) \int \frac {1+e^3 x}{\log (x)} \, dx-\left (101250 e^{11}\right ) \int \frac {\left (1+e^3 x\right )^2}{\log (x)} \, dx-\left (135000 e^{11}\right ) \int \frac {\left (1+e^3 x\right )^2}{\log (x)} \, dx-\left (180000 e^{11}\right ) \int \frac {\left (1+e^3 x\right )^2}{\log (x)} \, dx+\left (240000 e^{11}\right ) \int \frac {\left (1+e^3 x\right )^3}{\log (x)} \, dx-\left (67500 e^{14}\right ) \int \frac {x}{\log ^4(x)} \, dx-\left (67500 e^{17}\right ) \int \frac {x^2}{\log ^4(x)} \, dx-\left (22500 e^{20}\right ) \int \frac {x^3}{\log ^4(x)} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.04, size = 38, normalized size = 1.27 \begin {gather*} x+\frac {5625 e^8 \left (1+e^3 x\right )^4}{\log ^4(x)}+\frac {450 e^4 \left (1+e^3 x\right )^2}{\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-22500*E^8)/x - 22500*E^20*x^3 + 22500*E^20*x^3*Log[x] - 900*E^10*x*Log[x]^2 + 900*E^10*x*Log[x]^3
+ Log[x]^5 + (E^6*(-90000*E^5*x^3 + 22500*E^5*x^3*Log[x]))/x^3 + (E^4*(-135000*E^10*x^3 + 67500*E^10*x^3*Log[
x] - 900*x*Log[x]^2))/x^2 + (E^2*(-90000*E^15*x^3 + 67500*E^15*x^3*Log[x] - 1800*E^5*x*Log[x]^2 + 900*E^5*x*Lo
g[x]^3))/x)/Log[x]^5,x]

[Out]

x + (5625*E^8*(1 + E^3*x)^4)/Log[x]^4 + (450*E^4*(1 + E^3*x)^2)/Log[x]^2

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Maple [A]
time = 0.06, size = 93, normalized size = 3.10

method	result	size
risch	\(x +\frac {225 \,{\mathrm e}^{4} \left (25 x^{4} {\mathrm e}^{16}+100 \,{\mathrm e}^{13} x^{3}+150 \,{\mathrm e}^{10} x^{2}+2 \,{\mathrm e}^{6} x^{2} \ln \left (x \right )^{2}+100 x \,{\mathrm e}^{7}+4 \,{\mathrm e}^{3} x \ln \left (x \right )^{2}+25 \,{\mathrm e}^{4}+2 \ln \left (x \right )^{2}\right )}{\ln \left (x \right )^{4}}\)	\(68\)
default	\(x +\frac {22500 \,{\mathrm e}^{6} {\mathrm e}^{5} x}{\ln \left (x \right )^{4}}+\frac {33750 \,{\mathrm e}^{4} {\mathrm e}^{10} x^{2}}{\ln \left (x \right )^{4}}+\frac {450 \,{\mathrm e}^{4}}{\ln \left (x \right )^{2}}+\frac {900 \,{\mathrm e}^{2} {\mathrm e}^{5} x}{\ln \left (x \right )^{2}}+\frac {22500 \,{\mathrm e}^{2} x^{3} {\mathrm e}^{15}}{\ln \left (x \right )^{4}}+\frac {450 \,{\mathrm e}^{10} x^{2}}{\ln \left (x \right )^{2}}+\frac {5625 \,{\mathrm e}^{20} x^{4}}{\ln \left (x \right )^{4}}+\frac {5625 \,{\mathrm e}^{8}}{\ln \left (x \right )^{4}}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-22500*x^3*exp(-ln(x)+2)^4+(22500*x^3*exp(5)*ln(x)-90000*x^3*exp(5))*exp(-ln(x)+2)^3+(-900*x*ln(x)^2+6750
0*x^3*exp(5)^2*ln(x)-135000*x^3*exp(5)^2)*exp(-ln(x)+2)^2+(900*x*exp(5)*ln(x)^3-1800*x*exp(5)*ln(x)^2+67500*x^
3*exp(5)^3*ln(x)-90000*x^3*exp(5)^3)*exp(-ln(x)+2)+ln(x)^5+900*x*exp(5)^2*ln(x)^3-900*x*exp(5)^2*ln(x)^2+22500
*x^3*exp(5)^4*ln(x)-22500*x^3*exp(5)^4)/ln(x)^5,x,method=_RETURNVERBOSE)

[Out]

x+22500*exp(6)*exp(5)*x/ln(x)^4+33750*exp(4)*exp(10)*x^2/ln(x)^4+450*exp(4)/ln(x)^2+900*exp(2)*exp(5)*x/ln(x)^
2+22500*exp(2)*x^3*exp(15)/ln(x)^4+450*exp(5)^2*x^2/ln(x)^2+5625*exp(5)^4*x^4/ln(x)^4+5625*exp(8)/ln(x)^4

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.33, size = 138, normalized size = 4.60 \begin {gather*} 900 \, e^{7} \Gamma \left (-1, -\log \left (x\right )\right ) + 1800 \, e^{10} \Gamma \left (-1, -2 \, \log \left (x\right )\right ) + 1800 \, e^{7} \Gamma \left (-2, -\log \left (x\right )\right ) + 3600 \, e^{10} \Gamma \left (-2, -2 \, \log \left (x\right )\right ) + 22500 \, e^{11} \Gamma \left (-3, -\log \left (x\right )\right ) + 540000 \, e^{14} \Gamma \left (-3, -2 \, \log \left (x\right )\right ) + 1822500 \, e^{17} \Gamma \left (-3, -3 \, \log \left (x\right )\right ) + 1440000 \, e^{20} \Gamma \left (-3, -4 \, \log \left (x\right )\right ) + 90000 \, e^{11} \Gamma \left (-4, -\log \left (x\right )\right ) + 2160000 \, e^{14} \Gamma \left (-4, -2 \, \log \left (x\right )\right ) + 7290000 \, e^{17} \Gamma \left (-4, -3 \, \log \left (x\right )\right ) + 5760000 \, e^{20} \Gamma \left (-4, -4 \, \log \left (x\right )\right ) + x + \frac {450 \, e^{4}}{\log \left (x\right )^{2}} + \frac {5625 \, e^{8}}{\log \left (x\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-22500*x^3*exp(-log(x)+2)^4+(22500*x^3*exp(5)*log(x)-90000*x^3*exp(5))*exp(-log(x)+2)^3+(-900*x*log
(x)^2+67500*x^3*exp(5)^2*log(x)-135000*x^3*exp(5)^2)*exp(-log(x)+2)^2+(900*x*exp(5)*log(x)^3-1800*x*exp(5)*log
(x)^2+67500*x^3*exp(5)^3*log(x)-90000*x^3*exp(5)^3)*exp(-log(x)+2)+log(x)^5+900*x*exp(5)^2*log(x)^3-900*x*exp(
5)^2*log(x)^2+22500*x^3*exp(5)^4*log(x)-22500*x^3*exp(5)^4)/log(x)^5,x, algorithm="maxima")

[Out]

900*e^7*gamma(-1, -log(x)) + 1800*e^10*gamma(-1, -2*log(x)) + 1800*e^7*gamma(-2, -log(x)) + 3600*e^10*gamma(-2
, -2*log(x)) + 22500*e^11*gamma(-3, -log(x)) + 540000*e^14*gamma(-3, -2*log(x)) + 1822500*e^17*gamma(-3, -3*lo
g(x)) + 1440000*e^20*gamma(-3, -4*log(x)) + 90000*e^11*gamma(-4, -log(x)) + 2160000*e^14*gamma(-4, -2*log(x))
+ 7290000*e^17*gamma(-4, -3*log(x)) + 5760000*e^20*gamma(-4, -4*log(x)) + x + 450*e^4/log(x)^2 + 5625*e^8/log(
x)^4

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (29) = 58\).
time = 0.32, size = 62, normalized size = 2.07 \begin {gather*} \frac {5625 \, x^{4} e^{20} + x \log \left (x\right )^{4} + 22500 \, x^{3} e^{17} + 33750 \, x^{2} e^{14} + 450 \, {\left (x^{2} e^{10} + 2 \, x e^{7} + e^{4}\right )} \log \left (x\right )^{2} + 22500 \, x e^{11} + 5625 \, e^{8}}{\log \left (x\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-22500*x^3*exp(-log(x)+2)^4+(22500*x^3*exp(5)*log(x)-90000*x^3*exp(5))*exp(-log(x)+2)^3+(-900*x*log
(x)^2+67500*x^3*exp(5)^2*log(x)-135000*x^3*exp(5)^2)*exp(-log(x)+2)^2+(900*x*exp(5)*log(x)^3-1800*x*exp(5)*log
(x)^2+67500*x^3*exp(5)^3*log(x)-90000*x^3*exp(5)^3)*exp(-log(x)+2)+log(x)^5+900*x*exp(5)^2*log(x)^3-900*x*exp(
5)^2*log(x)^2+22500*x^3*exp(5)^4*log(x)-22500*x^3*exp(5)^4)/log(x)^5,x, algorithm="fricas")

[Out]

(5625*x^4*e^20 + x*log(x)^4 + 22500*x^3*e^17 + 33750*x^2*e^14 + 450*(x^2*e^10 + 2*x*e^7 + e^4)*log(x)^2 + 2250
0*x*e^11 + 5625*e^8)/log(x)^4

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (24) = 48\).
time = 0.07, size = 68, normalized size = 2.27 \begin {gather*} x + \frac {5625 x^{4} e^{20} + 22500 x^{3} e^{17} + 33750 x^{2} e^{14} + 22500 x e^{11} + \left (450 x^{2} e^{10} + 900 x e^{7} + 450 e^{4}\right ) \log {\left (x \right )}^{2} + 5625 e^{8}}{\log {\left (x \right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-22500*x**3*exp(-ln(x)+2)**4+(22500*x**3*exp(5)*ln(x)-90000*x**3*exp(5))*exp(-ln(x)+2)**3+(-900*x*l
n(x)**2+67500*x**3*exp(5)**2*ln(x)-135000*x**3*exp(5)**2)*exp(-ln(x)+2)**2+(900*x*exp(5)*ln(x)**3-1800*x*exp(5
)*ln(x)**2+67500*x**3*exp(5)**3*ln(x)-90000*x**3*exp(5)**3)*exp(-ln(x)+2)+ln(x)**5+900*x*exp(5)**2*ln(x)**3-90
0*x*exp(5)**2*ln(x)**2+22500*x**3*exp(5)**4*ln(x)-22500*x**3*exp(5)**4)/ln(x)**5,x)

[Out]

x + (5625*x**4*exp(20) + 22500*x**3*exp(17) + 33750*x**2*exp(14) + 22500*x*exp(11) + (450*x**2*exp(10) + 900*x
*exp(7) + 450*exp(4))*log(x)**2 + 5625*exp(8))/log(x)**4

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (29) = 58\).
time = 0.41, size = 70, normalized size = 2.33 \begin {gather*} \frac {5625 \, x^{4} e^{20} + 450 \, x^{2} e^{10} \log \left (x\right )^{2} + x \log \left (x\right )^{4} + 22500 \, x^{3} e^{17} + 900 \, x e^{7} \log \left (x\right )^{2} + 33750 \, x^{2} e^{14} + 450 \, e^{4} \log \left (x\right )^{2} + 22500 \, x e^{11} + 5625 \, e^{8}}{\log \left (x\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-22500*x^3*exp(-log(x)+2)^4+(22500*x^3*exp(5)*log(x)-90000*x^3*exp(5))*exp(-log(x)+2)^3+(-900*x*log
(x)^2+67500*x^3*exp(5)^2*log(x)-135000*x^3*exp(5)^2)*exp(-log(x)+2)^2+(900*x*exp(5)*log(x)^3-1800*x*exp(5)*log
(x)^2+67500*x^3*exp(5)^3*log(x)-90000*x^3*exp(5)^3)*exp(-log(x)+2)+log(x)^5+900*x*exp(5)^2*log(x)^3-900*x*exp(
5)^2*log(x)^2+22500*x^3*exp(5)^4*log(x)-22500*x^3*exp(5)^4)/log(x)^5,x, algorithm="giac")

[Out]

(5625*x^4*e^20 + 450*x^2*e^10*log(x)^2 + x*log(x)^4 + 22500*x^3*e^17 + 900*x*e^7*log(x)^2 + 33750*x^2*e^14 + 4
50*e^4*log(x)^2 + 22500*x*e^11 + 5625*e^8)/log(x)^4

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Mupad [B]
time = 0.38, size = 60, normalized size = 2.00 \begin {gather*} x+\frac {5625\,{\mathrm {e}}^8+22500\,x\,{\mathrm {e}}^{11}+33750\,x^2\,{\mathrm {e}}^{14}+22500\,x^3\,{\mathrm {e}}^{17}+5625\,x^4\,{\mathrm {e}}^{20}+{\ln \left (x\right )}^2\,\left (450\,{\mathrm {e}}^{10}\,x^2+900\,{\mathrm {e}}^7\,x+450\,{\mathrm {e}}^4\right )}{{\ln \left (x\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(4 - 2*log(x))*(900*x*log(x)^2 + 135000*x^3*exp(10) - 67500*x^3*exp(10)*log(x)) + 22500*x^3*exp(8 - 4
*log(x)) - log(x)^5 + 22500*x^3*exp(20) + exp(6 - 3*log(x))*(90000*x^3*exp(5) - 22500*x^3*exp(5)*log(x)) + exp
(2 - log(x))*(90000*x^3*exp(15) + 1800*x*exp(5)*log(x)^2 - 900*x*exp(5)*log(x)^3 - 67500*x^3*exp(15)*log(x)) +
900*x*exp(10)*log(x)^2 - 900*x*exp(10)*log(x)^3 - 22500*x^3*exp(20)*log(x))/log(x)^5,x)

[Out]

x + (5625*exp(8) + 22500*x*exp(11) + 33750*x^2*exp(14) + 22500*x^3*exp(17) + 5625*x^4*exp(20) + log(x)^2*(450*
exp(4) + 900*x*exp(7) + 450*x^2*exp(10)))/log(x)^4

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