∫ 2562890625e16+log2( 15x______20+5x+log (4)) x8(160+8 log(4)+(40+2 log(4)) log( 15x______ 20+5x+log(4))) ______________________________________________________________________________________________________________(20+5x+log (4))8 (100x+25x2 +5xlog (4)) dx

Optimal. Leaf size=25 \[ \frac {1}{5} e^{\left (4+\log \left (\frac {3 x}{4+x+\frac {\log (4)}{5}}\right )\right )^2} \]

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Rubi [B] time = 0.33, antiderivative size = 63, normalized size of antiderivative = 2.52, number of steps used = 5, number of rules used = 5, integrand size = 75, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {6, 12, 1584, 21, 2288} \begin {gather*} -\frac {512578125 x^8 (20+\log (4)) e^{\log ^2\left (\frac {15 x}{5 x+20+\log (4)}\right )+16}}{(5 x+20+\log (4))^{10} \left (\frac {5 x}{(5 x+20+\log (4))^2}-\frac {1}{5 x+20+\log (4)}\right )} \end {gather*}

Int[(2562890625*E^(16 + Log[(15*x)/(20 + 5*x + Log[4])]^2)*x^8*(160 + 8*Log[4] + (40 + 2*Log[4])*Log[(15*x)/(2

0 + 5*x + Log[4])]))/((20 + 5*x + Log[4])^8*(100*x + 25*x^2 + 5*x*Log[4])),x]

(-512578125*E^(16 + Log[(15*x)/(20 + 5*x + Log[4])]^2)*x^8*(20 + Log[4]))/((20 + 5*x + Log[4])^10*((5*x)/(20 +

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x

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

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +

n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -

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,

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2562890625 e^{16+\log ^2\left (\frac {15 x}{20+5 x+\log (4)}\right )} x^8 \left (160+8 \log (4)+(40+2 \log (4)) \log \left (\frac {15 x}{20+5 x+\log (4)}\right )\right )}{(20+5 x+\log (4))^8 \left (25 x^2+x (100+5 \log (4))\right )} \, dx\\ &=2562890625 \int \frac {e^{16+\log ^2\left (\frac {15 x}{20+5 x+\log (4)}\right )} x^8 \left (160+8 \log (4)+(40+2 \log (4)) \log \left (\frac {15 x}{20+5 x+\log (4)}\right )\right )}{(20+5 x+\log (4))^8 \left (25 x^2+x (100+5 \log (4))\right )} \, dx\\ &=2562890625 \int \frac {e^{16+\log ^2\left (\frac {15 x}{20+5 x+\log (4)}\right )} x^7 \left (160+8 \log (4)+(40+2 \log (4)) \log \left (\frac {15 x}{20+5 x+\log (4)}\right )\right )}{(20+5 x+\log (4))^8 (100+25 x+5 \log (4))} \, dx\\ &=512578125 \int \frac {e^{16+\log ^2\left (\frac {15 x}{20+5 x+\log (4)}\right )} x^7 \left (160+8 \log (4)+(40+2 \log (4)) \log \left (\frac {15 x}{20+5 x+\log (4)}\right )\right )}{(20+5 x+\log (4))^9} \, dx\\ &=-\frac {512578125 e^{16+\log ^2\left (\frac {15 x}{20+5 x+\log (4)}\right )} x^8 (20+\log (4))}{(20+5 x+\log (4))^{10} \left (\frac {5 x}{(20+5 x+\log (4))^2}-\frac {1}{20+5 x+\log (4)}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.03, size = 33, normalized size = 1.32 \begin {gather*} \frac {512578125 e^{16+\log ^2\left (\frac {15 x}{20+5 x+\log (4)}\right )} x^8}{(20+5 x+\log (4))^8} \end {gather*}

Integrate[(2562890625*E^(16 + Log[(15*x)/(20 + 5*x + Log[4])]^2)*x^8*(160 + 8*Log[4] + (40 + 2*Log[4])*Log[(15

*x)/(20 + 5*x + Log[4])]))/((20 + 5*x + Log[4])^8*(100*x + 25*x^2 + 5*x*Log[4])),x]

(512578125*E^(16 + Log[(15*x)/(20 + 5*x + Log[4])]^2)*x^8)/(20 + 5*x + Log[4])^8

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fricas [A] time = 0.68, size = 39, normalized size = 1.56 \begin {gather*} \frac {1}{5} \, e^{\left (\log \left (\frac {15 \, x}{5 \, x + 2 \, \log \relax (2) + 20}\right )^{2} + 8 \, \log \left (\frac {15 \, x}{5 \, x + 2 \, \log \relax (2) + 20}\right ) + 16\right )} \end {gather*}

integrate(((4*log(2)+40)*log(15*x/(2*log(2)+20+5*x))+16*log(2)+160)*exp(log(15*x/(2*log(2)+20+5*x))^2+8*log(15

*x/(2*log(2)+20+5*x))+16)/(10*x*log(2)+25*x^2+100*x),x, algorithm="fricas")

1/5*e^(log(15*x/(5*x + 2*log(2) + 20))^2 + 8*log(15*x/(5*x + 2*log(2) + 20)) + 16)

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giac [A] time = 0.44, size = 39, normalized size = 1.56 \begin {gather*} \frac {1}{5} \, e^{\left (\log \left (\frac {15 \, x}{5 \, x + 2 \, \log \relax (2) + 20}\right )^{2} + 8 \, \log \left (\frac {15 \, x}{5 \, x + 2 \, \log \relax (2) + 20}\right ) + 16\right )} \end {gather*}

integrate(((4*log(2)+40)*log(15*x/(2*log(2)+20+5*x))+16*log(2)+160)*exp(log(15*x/(2*log(2)+20+5*x))^2+8*log(15

*x/(2*log(2)+20+5*x))+16)/(10*x*log(2)+25*x^2+100*x),x, algorithm="giac")

1/5*e^(log(15*x/(5*x + 2*log(2) + 20))^2 + 8*log(15*x/(5*x + 2*log(2) + 20)) + 16)

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method	result	size

risch	\(\frac {512578125 x^{8} {\mathrm e}^{\ln \left (\frac {15 x}{2 \ln \relax (2)+20+5 x}\right )^{2}+16}}{\left (2 \ln \relax (2)+20+5 x \right )^{8}}\)	\(37\)
norman	\(\frac {{\mathrm e}^{\ln \left (\frac {15 x}{2 \ln \relax (2)+20+5 x}\right )^{2}+8 \ln \left (\frac {15 x}{2 \ln \relax (2)+20+5 x}\right )+16}}{5}\)	\(40\)

int(((4*ln(2)+40)*ln(15*x/(2*ln(2)+20+5*x))+16*ln(2)+160)*exp(ln(15*x/(2*ln(2)+20+5*x))^2+8*ln(15*x/(2*ln(2)+2

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maxima [B] time = 0.81, size = 336, normalized size = 13.44 \begin {gather*} \frac {78125 \cdot 3^{2 \, \log \relax (5) + 8} x^{8} e^{\left (\log \relax (5)^{2} + \log \relax (3)^{2} - 2 \, \log \relax (5) \log \left (5 \, x + 2 \, \log \relax (2) + 20\right ) - 2 \, \log \relax (3) \log \left (5 \, x + 2 \, \log \relax (2) + 20\right ) + \log \left (5 \, x + 2 \, \log \relax (2) + 20\right )^{2} + 2 \, \log \relax (5) \log \relax (x) + 2 \, \log \relax (3) \log \relax (x) - 2 \, \log \left (5 \, x + 2 \, \log \relax (2) + 20\right ) \log \relax (x) + \log \relax (x)^{2} + 16\right )}}{390625 \, x^{8} + 1250000 \, x^{7} {\left (\log \relax (2) + 10\right )} + 256 \, \log \relax (2)^{8} + 1750000 \, {\left (\log \relax (2)^{2} + 20 \, \log \relax (2) + 100\right )} x^{6} + 20480 \, \log \relax (2)^{7} + 1400000 \, {\left (\log \relax (2)^{3} + 30 \, \log \relax (2)^{2} + 300 \, \log \relax (2) + 1000\right )} x^{5} + 716800 \, \log \relax (2)^{6} + 700000 \, {\left (\log \relax (2)^{4} + 40 \, \log \relax (2)^{3} + 600 \, \log \relax (2)^{2} + 4000 \, \log \relax (2) + 10000\right )} x^{4} + 14336000 \, \log \relax (2)^{5} + 224000 \, {\left (\log \relax (2)^{5} + 50 \, \log \relax (2)^{4} + 1000 \, \log \relax (2)^{3} + 10000 \, \log \relax (2)^{2} + 50000 \, \log \relax (2) + 100000\right )} x^{3} + 179200000 \, \log \relax (2)^{4} + 44800 \, {\left (\log \relax (2)^{6} + 60 \, \log \relax (2)^{5} + 1500 \, \log \relax (2)^{4} + 20000 \, \log \relax (2)^{3} + 150000 \, \log \relax (2)^{2} + 600000 \, \log \relax (2) + 1000000\right )} x^{2} + 1433600000 \, \log \relax (2)^{3} + 5120 \, {\left (\log \relax (2)^{7} + 70 \, \log \relax (2)^{6} + 2100 \, \log \relax (2)^{5} + 35000 \, \log \relax (2)^{4} + 350000 \, \log \relax (2)^{3} + 2100000 \, \log \relax (2)^{2} + 7000000 \, \log \relax (2) + 10000000\right )} x + 7168000000 \, \log \relax (2)^{2} + 20480000000 \, \log \relax (2) + 25600000000} \end {gather*}

integrate(((4*log(2)+40)*log(15*x/(2*log(2)+20+5*x))+16*log(2)+160)*exp(log(15*x/(2*log(2)+20+5*x))^2+8*log(15

*x/(2*log(2)+20+5*x))+16)/(10*x*log(2)+25*x^2+100*x),x, algorithm="maxima")

78125*3^(2*log(5) + 8)*x^8*e^(log(5)^2 + log(3)^2 - 2*log(5)*log(5*x + 2*log(2) + 20) - 2*log(3)*log(5*x + 2*l

og(2) + 20) + log(5*x + 2*log(2) + 20)^2 + 2*log(5)*log(x) + 2*log(3)*log(x) - 2*log(5*x + 2*log(2) + 20)*log(

x) + log(x)^2 + 16)/(390625*x^8 + 1250000*x^7*(log(2) + 10) + 256*log(2)^8 + 1750000*(log(2)^2 + 20*log(2) + 1

00)*x^6 + 20480*log(2)^7 + 1400000*(log(2)^3 + 30*log(2)^2 + 300*log(2) + 1000)*x^5 + 716800*log(2)^6 + 700000

*(log(2)^4 + 40*log(2)^3 + 600*log(2)^2 + 4000*log(2) + 10000)*x^4 + 14336000*log(2)^5 + 224000*(log(2)^5 + 50

*log(2)^4 + 1000*log(2)^3 + 10000*log(2)^2 + 50000*log(2) + 100000)*x^3 + 179200000*log(2)^4 + 44800*(log(2)^6

+ 60*log(2)^5 + 1500*log(2)^4 + 20000*log(2)^3 + 150000*log(2)^2 + 600000*log(2) + 1000000)*x^2 + 1433600000*

log(2)^3 + 5120*(log(2)^7 + 70*log(2)^6 + 2100*log(2)^5 + 35000*log(2)^4 + 350000*log(2)^3 + 2100000*log(2)^2

+ 7000000*log(2) + 10000000)*x + 7168000000*log(2)^2 + 20480000000*log(2) + 25600000000)

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mupad [B] time = 9.41, size = 49, normalized size = 1.96 \begin {gather*} \frac {512578125\,x^8\,{\mathrm {e}}^{{\ln \left (\frac {x}{5\,x+\ln \relax (4)+20}\right )}^2+{\ln \left (15\right )}^2+16}\,{\left (\frac {x}{5\,x+\ln \relax (4)+20}\right )}^{\ln \left (225\right )}}{{\left (5\,x+\ln \relax (4)+20\right )}^8} \end {gather*}

int((exp(8*log((15*x)/(5*x + 2*log(2) + 20)) + log((15*x)/(5*x + 2*log(2) + 20))^2 + 16)*(16*log(2) + log((15*

x)/(5*x + 2*log(2) + 20))*(4*log(2) + 40) + 160))/(100*x + 10*x*log(2) + 25*x^2),x)

(512578125*x^8*exp(log(x/(5*x + log(4) + 20))^2 + log(15)^2 + 16)*(x/(5*x + log(4) + 20))^log(225))/(5*x + log

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sympy [B] time = 3.08, size = 379, normalized size = 15.16 \begin {gather*} \frac {512578125 x^{8} e^{\log {\left (\frac {15 x}{5 x + 2 \log {\relax (2 )} + 20} \right )}^{2} + 16}}{390625 x^{8} + 1250000 x^{7} \log {\relax (2 )} + 12500000 x^{7} + 1750000 x^{6} \log {\relax (2 )}^{2} + 35000000 x^{6} \log {\relax (2 )} + 175000000 x^{6} + 1400000 x^{5} \log {\relax (2 )}^{3} + 42000000 x^{5} \log {\relax (2 )}^{2} + 420000000 x^{5} \log {\relax (2 )} + 1400000000 x^{5} + 700000 x^{4} \log {\relax (2 )}^{4} + 28000000 x^{4} \log {\relax (2 )}^{3} + 420000000 x^{4} \log {\relax (2 )}^{2} + 2800000000 x^{4} \log {\relax (2 )} + 7000000000 x^{4} + 224000 x^{3} \log {\relax (2 )}^{5} + 11200000 x^{3} \log {\relax (2 )}^{4} + 224000000 x^{3} \log {\relax (2 )}^{3} + 2240000000 x^{3} \log {\relax (2 )}^{2} + 11200000000 x^{3} \log {\relax (2 )} + 22400000000 x^{3} + 44800 x^{2} \log {\relax (2 )}^{6} + 2688000 x^{2} \log {\relax (2 )}^{5} + 67200000 x^{2} \log {\relax (2 )}^{4} + 896000000 x^{2} \log {\relax (2 )}^{3} + 6720000000 x^{2} \log {\relax (2 )}^{2} + 26880000000 x^{2} \log {\relax (2 )} + 44800000000 x^{2} + 5120 x \log {\relax (2 )}^{7} + 358400 x \log {\relax (2 )}^{6} + 10752000 x \log {\relax (2 )}^{5} + 179200000 x \log {\relax (2 )}^{4} + 1792000000 x \log {\relax (2 )}^{3} + 10752000000 x \log {\relax (2 )}^{2} + 35840000000 x \log {\relax (2 )} + 51200000000 x + 256 \log {\relax (2 )}^{8} + 20480 \log {\relax (2 )}^{7} + 716800 \log {\relax (2 )}^{6} + 14336000 \log {\relax (2 )}^{5} + 179200000 \log {\relax (2 )}^{4} + 1433600000 \log {\relax (2 )}^{3} + 7168000000 \log {\relax (2 )}^{2} + 20480000000 \log {\relax (2 )} + 25600000000} \end {gather*}

integrate(((4*ln(2)+40)*ln(15*x/(2*ln(2)+20+5*x))+16*ln(2)+160)*exp(ln(15*x/(2*ln(2)+20+5*x))**2+8*ln(15*x/(2*

512578125*x**8*exp(log(15*x/(5*x + 2*log(2) + 20))**2 + 16)/(390625*x**8 + 1250000*x**7*log(2) + 12500000*x**7

+ 1750000*x**6*log(2)**2 + 35000000*x**6*log(2) + 175000000*x**6 + 1400000*x**5*log(2)**3 + 42000000*x**5*log

(2)**2 + 420000000*x**5*log(2) + 1400000000*x**5 + 700000*x**4*log(2)**4 + 28000000*x**4*log(2)**3 + 420000000

*x**4*log(2)**2 + 2800000000*x**4*log(2) + 7000000000*x**4 + 224000*x**3*log(2)**5 + 11200000*x**3*log(2)**4 +

224000000*x**3*log(2)**3 + 2240000000*x**3*log(2)**2 + 11200000000*x**3*log(2) + 22400000000*x**3 + 44800*x**

2*log(2)**6 + 2688000*x**2*log(2)**5 + 67200000*x**2*log(2)**4 + 896000000*x**2*log(2)**3 + 6720000000*x**2*lo

g(2)**2 + 26880000000*x**2*log(2) + 44800000000*x**2 + 5120*x*log(2)**7 + 358400*x*log(2)**6 + 10752000*x*log(

2)**5 + 179200000*x*log(2)**4 + 1792000000*x*log(2)**3 + 10752000000*x*log(2)**2 + 35840000000*x*log(2) + 5120

0000000*x + 256*log(2)**8 + 20480*log(2)**7 + 716800*log(2)**6 + 14336000*log(2)**5 + 179200000*log(2)**4 + 14

33600000*log(2)**3 + 7168000000*log(2)**2 + 20480000000*log(2) + 25600000000)

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3.70.13 \(\int \frac {2562890625 e^{16+\log ^2(\frac {15 x}{20+5 x+\log (4)})} x^8 (160+8 \log (4)+(40+2 \log (4)) \log (\frac {15 x}{20+5 x+\log (4)}))}{(20+5 x+\log (4))^8 (100 x+25 x^2+5 x \log (4))} \, dx\)