Integrand size = 352, antiderivative size = 28 \[ \int \frac {-125 x-200 x^2-90 x^3-16 x^4-x^5+\left (-50 x-20 x^2-2 x^3\right ) \log (4)+e^x \left (-625-1000 x-450 x^2-80 x^3-5 x^4+\left (-250 x-100 x^2-10 x^3\right ) \log (4)\right )+\left (e^x \left (125+100 x+15 x^2\right ) \log (4)+\left (25 x+20 x^2+3 x^3\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x \left (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6\right )+\left (e^x \left (121500-194400 x+29160 x^2+38880 x^3+4860 x^4\right ) \log (4)+\left (24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (e^x \left (-24300+19440 x+4860 x^2\right ) \log ^2(4)+\left (-4860 x+3888 x^2+972 x^3\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (1620 e^x \log ^3(4)+324 x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (5 e^x+x\right )\right )} \, dx=\frac {x}{324 \left (-1+x+\frac {\log (4) \log \left (e^x+\frac {x}{5}\right )}{5+x}\right )^2} \] Output:
1/2*x/(9*x+18*ln(1/5*x+exp(x))*ln(2)/(5+x)-9)/(18*x+36*ln(1/5*x+exp(x))*ln (2)/(5+x)-18)
Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(28)=56\).
Time = 0.22 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.89 \[ \int \frac {-125 x-200 x^2-90 x^3-16 x^4-x^5+\left (-50 x-20 x^2-2 x^3\right ) \log (4)+e^x \left (-625-1000 x-450 x^2-80 x^3-5 x^4+\left (-250 x-100 x^2-10 x^3\right ) \log (4)\right )+\left (e^x \left (125+100 x+15 x^2\right ) \log (4)+\left (25 x+20 x^2+3 x^3\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x \left (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6\right )+\left (e^x \left (121500-194400 x+29160 x^2+38880 x^3+4860 x^4\right ) \log (4)+\left (24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (e^x \left (-24300+19440 x+4860 x^2\right ) \log ^2(4)+\left (-4860 x+3888 x^2+972 x^3\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (1620 e^x \log ^3(4)+324 x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (5 e^x+x\right )\right )} \, dx=\frac {x (5+x)^2 \left (8 x+4 x^2+\log (16)+5 e^x (8+4 x+\log (16))\right )}{648 \left (4 x+2 x^2+\log (4)+5 e^x (4+2 x+\log (4))\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2} \] Input:
Integrate[(-125*x - 200*x^2 - 90*x^3 - 16*x^4 - x^5 + (-50*x - 20*x^2 - 2* x^3)*Log[4] + E^x*(-625 - 1000*x - 450*x^2 - 80*x^3 - 5*x^4 + (-250*x - 10 0*x^2 - 10*x^3)*Log[4]) + (E^x*(125 + 100*x + 15*x^2)*Log[4] + (25*x + 20* x^2 + 3*x^3)*Log[4])*Log[(5*E^x + x)/5])/(-40500*x + 97200*x^2 - 53460*x^3 - 18144*x^4 + 10692*x^5 + 3888*x^6 + 324*x^7 + E^x*(-202500 + 486000*x - 267300*x^2 - 90720*x^3 + 53460*x^4 + 19440*x^5 + 1620*x^6) + (E^x*(121500 - 194400*x + 29160*x^2 + 38880*x^3 + 4860*x^4)*Log[4] + (24300*x - 38880*x ^2 + 5832*x^3 + 7776*x^4 + 972*x^5)*Log[4])*Log[(5*E^x + x)/5] + (E^x*(-24 300 + 19440*x + 4860*x^2)*Log[4]^2 + (-4860*x + 3888*x^2 + 972*x^3)*Log[4] ^2)*Log[(5*E^x + x)/5]^2 + (1620*E^x*Log[4]^3 + 324*x*Log[4]^3)*Log[(5*E^x + x)/5]^3),x]
Output:
(x*(5 + x)^2*(8*x + 4*x^2 + Log[16] + 5*E^x*(8 + 4*x + Log[16])))/(648*(4* x + 2*x^2 + Log[4] + 5*E^x*(4 + 2*x + Log[4]))*(-5 + 4*x + x^2 + Log[4]*Lo g[E^x + x/5])^2)
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-x^5-16 x^4-90 x^3-200 x^2+\left (e^x \left (15 x^2+100 x+125\right ) \log (4)+\left (3 x^3+20 x^2+25 x\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (x+5 e^x\right )\right )+\left (-2 x^3-20 x^2-50 x\right ) \log (4)+e^x \left (-5 x^4-80 x^3-450 x^2+\left (-10 x^3-100 x^2-250 x\right ) \log (4)-1000 x-625\right )-125 x}{324 x^7+3888 x^6+10692 x^5-18144 x^4-53460 x^3+97200 x^2+\left (e^x \left (4860 x^2+19440 x-24300\right ) \log ^2(4)+\left (972 x^3+3888 x^2-4860 x\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (x+5 e^x\right )\right )+\left (e^x \left (4860 x^4+38880 x^3+29160 x^2-194400 x+121500\right ) \log (4)+\left (972 x^5+7776 x^4+5832 x^3-38880 x^2+24300 x\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (x+5 e^x\right )\right )+e^x \left (1620 x^6+19440 x^5+53460 x^4-90720 x^3-267300 x^2+486000 x-202500\right )-40500 x+\left (324 x \log ^3(4)+1620 e^x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (x+5 e^x\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+5) \left ((x+5) \left (x \left (x^2+6 x+5+\log (16)\right )+5 e^x \left (x^2+x (6+\log (16))+5\right )\right )-\left (x+5 e^x\right ) (3 x+5) \log (4) \log \left (\frac {x}{5}+e^x\right )\right )}{324 \left (x+5 e^x\right ) \left (-x^2-4 x-\log (4) \log \left (\frac {x}{5}+e^x\right )+5\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{324} \int \frac {(x+5) \left ((x+5) \left (x \left (x^2+6 x+\log (16)+5\right )+5 e^x \left (x^2+(6+\log (16)) x+5\right )\right )-\left (x+5 e^x\right ) (3 x+5) \log (4) \log \left (\frac {x}{5}+e^x\right )\right )}{\left (x+5 e^x\right ) \left (-x^2-4 x-\log (4) \log \left (\frac {x}{5}+e^x\right )+5\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{324} \int \left (\frac {(x-1) x \log (16) (x+5)^2}{\left (x+5 e^x\right ) \left (x^2+4 x+\log (4) \log \left (\frac {x}{5}+e^x\right )-5\right )^3}+\frac {\left (x^3+11 \left (1+\frac {4 \log (2)}{11}\right ) x^2-3 \log (4) \log \left (\frac {x}{5}+e^x\right ) x+35 \left (1+\frac {4 \log (2)}{7}\right ) x-5 \log (4) \log \left (\frac {x}{5}+e^x\right )+25\right ) (x+5)}{\left (-x^2-4 x-\log (4) \log \left (\frac {x}{5}+e^x\right )+5\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{324} \left (125 \text {Subst}\left (\int \frac {1}{\left (25 x^2+20 x+\log (4) \log \left (x+e^{5 x}\right )-5\right )^2}dx,x,\frac {x}{5}\right )-100 (2+\log (2)) \int \frac {x}{\left (x^2+4 x+\log (4) \log \left (\frac {x}{5}+e^x\right )-5\right )^3}dx-20 (9+\log (4)) \int \frac {x^2}{\left (x^2+4 x+\log (4) \log \left (\frac {x}{5}+e^x\right )-5\right )^3}dx-25 \log (16) \int \frac {x}{\left (x+5 e^x\right ) \left (x^2+4 x+\log (4) \log \left (\frac {x}{5}+e^x\right )-5\right )^3}dx+15 \log (16) \int \frac {x^2}{\left (x+5 e^x\right ) \left (x^2+4 x+\log (4) \log \left (\frac {x}{5}+e^x\right )-5\right )^3}dx+20 \int \frac {x}{\left (x^2+4 x+\log (4) \log \left (\frac {x}{5}+e^x\right )-5\right )^2}dx+3 \int \frac {x^2}{\left (x^2+4 x+\log (4) \log \left (\frac {x}{5}+e^x\right )-5\right )^2}dx-4 \int \frac {x^4}{\left (x^2+4 x+\log (4) \log \left (\frac {x}{5}+e^x\right )-5\right )^3}dx+\log (16) \int \frac {x^4}{\left (x+5 e^x\right ) \left (x^2+4 x+\log (4) \log \left (\frac {x}{5}+e^x\right )-5\right )^3}dx-4 (12+\log (2)) \int \frac {x^3}{\left (x^2+4 x+\log (4) \log \left (\frac {x}{5}+e^x\right )-5\right )^3}dx+9 \log (16) \int \frac {x^3}{\left (x+5 e^x\right ) \left (x^2+4 x+\log (4) \log \left (\frac {x}{5}+e^x\right )-5\right )^3}dx\right )\) |
Input:
Int[(-125*x - 200*x^2 - 90*x^3 - 16*x^4 - x^5 + (-50*x - 20*x^2 - 2*x^3)*L og[4] + E^x*(-625 - 1000*x - 450*x^2 - 80*x^3 - 5*x^4 + (-250*x - 100*x^2 - 10*x^3)*Log[4]) + (E^x*(125 + 100*x + 15*x^2)*Log[4] + (25*x + 20*x^2 + 3*x^3)*Log[4])*Log[(5*E^x + x)/5])/(-40500*x + 97200*x^2 - 53460*x^3 - 181 44*x^4 + 10692*x^5 + 3888*x^6 + 324*x^7 + E^x*(-202500 + 486000*x - 267300 *x^2 - 90720*x^3 + 53460*x^4 + 19440*x^5 + 1620*x^6) + (E^x*(121500 - 1944 00*x + 29160*x^2 + 38880*x^3 + 4860*x^4)*Log[4] + (24300*x - 38880*x^2 + 5 832*x^3 + 7776*x^4 + 972*x^5)*Log[4])*Log[(5*E^x + x)/5] + (E^x*(-24300 + 19440*x + 4860*x^2)*Log[4]^2 + (-4860*x + 3888*x^2 + 972*x^3)*Log[4]^2)*Lo g[(5*E^x + x)/5]^2 + (1620*E^x*Log[4]^3 + 324*x*Log[4]^3)*Log[(5*E^x + x)/ 5]^3),x]
Output:
$Aborted
Time = 3.45 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(\frac {\left (x^{2}+10 x +25\right ) x}{324 {\left (2 \ln \left (2\right ) \ln \left (\frac {x}{5}+{\mathrm e}^{x}\right )+x^{2}+4 x -5\right )}^{2}}\) | \(33\) |
| parallelrisch | \(\frac {25 x^{3}+250 x^{2}+625 x}{32400 \ln \left (2\right )^{2} \ln \left (\frac {x}{5}+{\mathrm e}^{x}\right )^{2}+32400 \ln \left (2\right ) \ln \left (\frac {x}{5}+{\mathrm e}^{x}\right ) x^{2}+8100 x^{4}+129600 \ln \left (2\right ) \ln \left (\frac {x}{5}+{\mathrm e}^{x}\right ) x +64800 x^{3}-162000 \ln \left (2\right ) \ln \left (\frac {x}{5}+{\mathrm e}^{x}\right )+48600 x^{2}-324000 x +202500}\) | \(89\) |
Input:
int(((2*(15*x^2+100*x+125)*ln(2)*exp(x)+2*(3*x^3+20*x^2+25*x)*ln(2))*ln(1/ 5*x+exp(x))+(2*(-10*x^3-100*x^2-250*x)*ln(2)-5*x^4-80*x^3-450*x^2-1000*x-6 25)*exp(x)+2*(-2*x^3-20*x^2-50*x)*ln(2)-x^5-16*x^4-90*x^3-200*x^2-125*x)/( (12960*ln(2)^3*exp(x)+2592*x*ln(2)^3)*ln(1/5*x+exp(x))^3+(4*(4860*x^2+1944 0*x-24300)*ln(2)^2*exp(x)+4*(972*x^3+3888*x^2-4860*x)*ln(2)^2)*ln(1/5*x+ex p(x))^2+(2*(4860*x^4+38880*x^3+29160*x^2-194400*x+121500)*ln(2)*exp(x)+2*( 972*x^5+7776*x^4+5832*x^3-38880*x^2+24300*x)*ln(2))*ln(1/5*x+exp(x))+(1620 *x^6+19440*x^5+53460*x^4-90720*x^3-267300*x^2+486000*x-202500)*exp(x)+324* x^7+3888*x^6+10692*x^5-18144*x^4-53460*x^3+97200*x^2-40500*x),x,method=_RE TURNVERBOSE)
Output:
1/324*(x^2+10*x+25)*x/(2*ln(2)*ln(1/5*x+exp(x))+x^2+4*x-5)^2
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (24) = 48\).
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.43 \[ \int \frac {-125 x-200 x^2-90 x^3-16 x^4-x^5+\left (-50 x-20 x^2-2 x^3\right ) \log (4)+e^x \left (-625-1000 x-450 x^2-80 x^3-5 x^4+\left (-250 x-100 x^2-10 x^3\right ) \log (4)\right )+\left (e^x \left (125+100 x+15 x^2\right ) \log (4)+\left (25 x+20 x^2+3 x^3\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x \left (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6\right )+\left (e^x \left (121500-194400 x+29160 x^2+38880 x^3+4860 x^4\right ) \log (4)+\left (24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (e^x \left (-24300+19440 x+4860 x^2\right ) \log ^2(4)+\left (-4860 x+3888 x^2+972 x^3\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (1620 e^x \log ^3(4)+324 x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (5 e^x+x\right )\right )} \, dx=\frac {x^{3} + 10 \, x^{2} + 25 \, x}{324 \, {\left (x^{4} + 4 \, \log \left (2\right )^{2} \log \left (\frac {1}{5} \, x + e^{x}\right )^{2} + 8 \, x^{3} + 4 \, {\left (x^{2} + 4 \, x - 5\right )} \log \left (2\right ) \log \left (\frac {1}{5} \, x + e^{x}\right ) + 6 \, x^{2} - 40 \, x + 25\right )}} \] Input:
integrate(((2*(15*x^2+100*x+125)*log(2)*exp(x)+2*(3*x^3+20*x^2+25*x)*log(2 ))*log(1/5*x+exp(x))+(2*(-10*x^3-100*x^2-250*x)*log(2)-5*x^4-80*x^3-450*x^ 2-1000*x-625)*exp(x)+2*(-2*x^3-20*x^2-50*x)*log(2)-x^5-16*x^4-90*x^3-200*x ^2-125*x)/((12960*log(2)^3*exp(x)+2592*x*log(2)^3)*log(1/5*x+exp(x))^3+(4* (4860*x^2+19440*x-24300)*log(2)^2*exp(x)+4*(972*x^3+3888*x^2-4860*x)*log(2 )^2)*log(1/5*x+exp(x))^2+(2*(4860*x^4+38880*x^3+29160*x^2-194400*x+121500) *log(2)*exp(x)+2*(972*x^5+7776*x^4+5832*x^3-38880*x^2+24300*x)*log(2))*log (1/5*x+exp(x))+(1620*x^6+19440*x^5+53460*x^4-90720*x^3-267300*x^2+486000*x -202500)*exp(x)+324*x^7+3888*x^6+10692*x^5-18144*x^4-53460*x^3+97200*x^2-4 0500*x),x, algorithm="fricas")
Output:
1/324*(x^3 + 10*x^2 + 25*x)/(x^4 + 4*log(2)^2*log(1/5*x + e^x)^2 + 8*x^3 + 4*(x^2 + 4*x - 5)*log(2)*log(1/5*x + e^x) + 6*x^2 - 40*x + 25)
Time = 0.32 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.71 \[ \int \frac {-125 x-200 x^2-90 x^3-16 x^4-x^5+\left (-50 x-20 x^2-2 x^3\right ) \log (4)+e^x \left (-625-1000 x-450 x^2-80 x^3-5 x^4+\left (-250 x-100 x^2-10 x^3\right ) \log (4)\right )+\left (e^x \left (125+100 x+15 x^2\right ) \log (4)+\left (25 x+20 x^2+3 x^3\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x \left (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6\right )+\left (e^x \left (121500-194400 x+29160 x^2+38880 x^3+4860 x^4\right ) \log (4)+\left (24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (e^x \left (-24300+19440 x+4860 x^2\right ) \log ^2(4)+\left (-4860 x+3888 x^2+972 x^3\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (1620 e^x \log ^3(4)+324 x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (5 e^x+x\right )\right )} \, dx=\frac {x^{3} + 10 x^{2} + 25 x}{324 x^{4} + 2592 x^{3} + 1944 x^{2} - 12960 x + \left (1296 x^{2} \log {\left (2 \right )} + 5184 x \log {\left (2 \right )} - 6480 \log {\left (2 \right )}\right ) \log {\left (\frac {x}{5} + e^{x} \right )} + 1296 \log {\left (2 \right )}^{2} \log {\left (\frac {x}{5} + e^{x} \right )}^{2} + 8100} \] Input:
integrate(((2*(15*x**2+100*x+125)*ln(2)*exp(x)+2*(3*x**3+20*x**2+25*x)*ln( 2))*ln(1/5*x+exp(x))+(2*(-10*x**3-100*x**2-250*x)*ln(2)-5*x**4-80*x**3-450 *x**2-1000*x-625)*exp(x)+2*(-2*x**3-20*x**2-50*x)*ln(2)-x**5-16*x**4-90*x* *3-200*x**2-125*x)/((12960*ln(2)**3*exp(x)+2592*x*ln(2)**3)*ln(1/5*x+exp(x ))**3+(4*(4860*x**2+19440*x-24300)*ln(2)**2*exp(x)+4*(972*x**3+3888*x**2-4 860*x)*ln(2)**2)*ln(1/5*x+exp(x))**2+(2*(4860*x**4+38880*x**3+29160*x**2-1 94400*x+121500)*ln(2)*exp(x)+2*(972*x**5+7776*x**4+5832*x**3-38880*x**2+24 300*x)*ln(2))*ln(1/5*x+exp(x))+(1620*x**6+19440*x**5+53460*x**4-90720*x**3 -267300*x**2+486000*x-202500)*exp(x)+324*x**7+3888*x**6+10692*x**5-18144*x **4-53460*x**3+97200*x**2-40500*x),x)
Output:
(x**3 + 10*x**2 + 25*x)/(324*x**4 + 2592*x**3 + 1944*x**2 - 12960*x + (129 6*x**2*log(2) + 5184*x*log(2) - 6480*log(2))*log(x/5 + exp(x)) + 1296*log( 2)**2*log(x/5 + exp(x))**2 + 8100)
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (24) = 48\).
Time = 0.73 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.07 \[ \int \frac {-125 x-200 x^2-90 x^3-16 x^4-x^5+\left (-50 x-20 x^2-2 x^3\right ) \log (4)+e^x \left (-625-1000 x-450 x^2-80 x^3-5 x^4+\left (-250 x-100 x^2-10 x^3\right ) \log (4)\right )+\left (e^x \left (125+100 x+15 x^2\right ) \log (4)+\left (25 x+20 x^2+3 x^3\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x \left (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6\right )+\left (e^x \left (121500-194400 x+29160 x^2+38880 x^3+4860 x^4\right ) \log (4)+\left (24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (e^x \left (-24300+19440 x+4860 x^2\right ) \log ^2(4)+\left (-4860 x+3888 x^2+972 x^3\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (1620 e^x \log ^3(4)+324 x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (5 e^x+x\right )\right )} \, dx=\frac {x^{3} + 10 \, x^{2} + 25 \, x}{324 \, {\left (x^{4} + 4 \, \log \left (5\right )^{2} \log \left (2\right )^{2} + 4 \, \log \left (2\right )^{2} \log \left (x + 5 \, e^{x}\right )^{2} - 2 \, {\left (2 \, \log \left (5\right ) \log \left (2\right ) - 3\right )} x^{2} + 8 \, x^{3} - 8 \, {\left (2 \, \log \left (5\right ) \log \left (2\right ) + 5\right )} x + 20 \, \log \left (5\right ) \log \left (2\right ) + 4 \, {\left (x^{2} \log \left (2\right ) - 2 \, \log \left (5\right ) \log \left (2\right )^{2} + 4 \, x \log \left (2\right ) - 5 \, \log \left (2\right )\right )} \log \left (x + 5 \, e^{x}\right ) + 25\right )}} \] Input:
integrate(((2*(15*x^2+100*x+125)*log(2)*exp(x)+2*(3*x^3+20*x^2+25*x)*log(2 ))*log(1/5*x+exp(x))+(2*(-10*x^3-100*x^2-250*x)*log(2)-5*x^4-80*x^3-450*x^ 2-1000*x-625)*exp(x)+2*(-2*x^3-20*x^2-50*x)*log(2)-x^5-16*x^4-90*x^3-200*x ^2-125*x)/((12960*log(2)^3*exp(x)+2592*x*log(2)^3)*log(1/5*x+exp(x))^3+(4* (4860*x^2+19440*x-24300)*log(2)^2*exp(x)+4*(972*x^3+3888*x^2-4860*x)*log(2 )^2)*log(1/5*x+exp(x))^2+(2*(4860*x^4+38880*x^3+29160*x^2-194400*x+121500) *log(2)*exp(x)+2*(972*x^5+7776*x^4+5832*x^3-38880*x^2+24300*x)*log(2))*log (1/5*x+exp(x))+(1620*x^6+19440*x^5+53460*x^4-90720*x^3-267300*x^2+486000*x -202500)*exp(x)+324*x^7+3888*x^6+10692*x^5-18144*x^4-53460*x^3+97200*x^2-4 0500*x),x, algorithm="maxima")
Output:
1/324*(x^3 + 10*x^2 + 25*x)/(x^4 + 4*log(5)^2*log(2)^2 + 4*log(2)^2*log(x + 5*e^x)^2 - 2*(2*log(5)*log(2) - 3)*x^2 + 8*x^3 - 8*(2*log(5)*log(2) + 5) *x + 20*log(5)*log(2) + 4*(x^2*log(2) - 2*log(5)*log(2)^2 + 4*x*log(2) - 5 *log(2))*log(x + 5*e^x) + 25)
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (24) = 48\).
Time = 0.66 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.75 \[ \int \frac {-125 x-200 x^2-90 x^3-16 x^4-x^5+\left (-50 x-20 x^2-2 x^3\right ) \log (4)+e^x \left (-625-1000 x-450 x^2-80 x^3-5 x^4+\left (-250 x-100 x^2-10 x^3\right ) \log (4)\right )+\left (e^x \left (125+100 x+15 x^2\right ) \log (4)+\left (25 x+20 x^2+3 x^3\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x \left (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6\right )+\left (e^x \left (121500-194400 x+29160 x^2+38880 x^3+4860 x^4\right ) \log (4)+\left (24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (e^x \left (-24300+19440 x+4860 x^2\right ) \log ^2(4)+\left (-4860 x+3888 x^2+972 x^3\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (1620 e^x \log ^3(4)+324 x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (5 e^x+x\right )\right )} \, dx=\frac {x^{3} + 10 \, x^{2} + 25 \, x}{324 \, {\left (x^{4} - 4 \, x^{2} \log \left (5\right ) \log \left (2\right ) + 4 \, \log \left (5\right )^{2} \log \left (2\right )^{2} + 4 \, x^{2} \log \left (2\right ) \log \left (x + 5 \, e^{x}\right ) - 8 \, \log \left (5\right ) \log \left (2\right )^{2} \log \left (x + 5 \, e^{x}\right ) + 4 \, \log \left (2\right )^{2} \log \left (x + 5 \, e^{x}\right )^{2} + 8 \, x^{3} - 16 \, x \log \left (5\right ) \log \left (2\right ) + 16 \, x \log \left (2\right ) \log \left (x + 5 \, e^{x}\right ) + 6 \, x^{2} + 20 \, \log \left (5\right ) \log \left (2\right ) - 20 \, \log \left (2\right ) \log \left (x + 5 \, e^{x}\right ) - 40 \, x + 25\right )}} \] Input:
integrate(((2*(15*x^2+100*x+125)*log(2)*exp(x)+2*(3*x^3+20*x^2+25*x)*log(2 ))*log(1/5*x+exp(x))+(2*(-10*x^3-100*x^2-250*x)*log(2)-5*x^4-80*x^3-450*x^ 2-1000*x-625)*exp(x)+2*(-2*x^3-20*x^2-50*x)*log(2)-x^5-16*x^4-90*x^3-200*x ^2-125*x)/((12960*log(2)^3*exp(x)+2592*x*log(2)^3)*log(1/5*x+exp(x))^3+(4* (4860*x^2+19440*x-24300)*log(2)^2*exp(x)+4*(972*x^3+3888*x^2-4860*x)*log(2 )^2)*log(1/5*x+exp(x))^2+(2*(4860*x^4+38880*x^3+29160*x^2-194400*x+121500) *log(2)*exp(x)+2*(972*x^5+7776*x^4+5832*x^3-38880*x^2+24300*x)*log(2))*log (1/5*x+exp(x))+(1620*x^6+19440*x^5+53460*x^4-90720*x^3-267300*x^2+486000*x -202500)*exp(x)+324*x^7+3888*x^6+10692*x^5-18144*x^4-53460*x^3+97200*x^2-4 0500*x),x, algorithm="giac")
Output:
1/324*(x^3 + 10*x^2 + 25*x)/(x^4 - 4*x^2*log(5)*log(2) + 4*log(5)^2*log(2) ^2 + 4*x^2*log(2)*log(x + 5*e^x) - 8*log(5)*log(2)^2*log(x + 5*e^x) + 4*lo g(2)^2*log(x + 5*e^x)^2 + 8*x^3 - 16*x*log(5)*log(2) + 16*x*log(2)*log(x + 5*e^x) + 6*x^2 + 20*log(5)*log(2) - 20*log(2)*log(x + 5*e^x) - 40*x + 25)
Timed out. \[ \int \frac {-125 x-200 x^2-90 x^3-16 x^4-x^5+\left (-50 x-20 x^2-2 x^3\right ) \log (4)+e^x \left (-625-1000 x-450 x^2-80 x^3-5 x^4+\left (-250 x-100 x^2-10 x^3\right ) \log (4)\right )+\left (e^x \left (125+100 x+15 x^2\right ) \log (4)+\left (25 x+20 x^2+3 x^3\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x \left (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6\right )+\left (e^x \left (121500-194400 x+29160 x^2+38880 x^3+4860 x^4\right ) \log (4)+\left (24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (e^x \left (-24300+19440 x+4860 x^2\right ) \log ^2(4)+\left (-4860 x+3888 x^2+972 x^3\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (1620 e^x \log ^3(4)+324 x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (5 e^x+x\right )\right )} \, dx=\int -\frac {125\,x+2\,\ln \left (2\right )\,\left (2\,x^3+20\,x^2+50\,x\right )+{\mathrm {e}}^x\,\left (1000\,x+2\,\ln \left (2\right )\,\left (10\,x^3+100\,x^2+250\,x\right )+450\,x^2+80\,x^3+5\,x^4+625\right )-\ln \left (\frac {x}{5}+{\mathrm {e}}^x\right )\,\left (2\,\ln \left (2\right )\,\left (3\,x^3+20\,x^2+25\,x\right )+2\,{\mathrm {e}}^x\,\ln \left (2\right )\,\left (15\,x^2+100\,x+125\right )\right )+200\,x^2+90\,x^3+16\,x^4+x^5}{{\ln \left (\frac {x}{5}+{\mathrm {e}}^x\right )}^3\,\left (12960\,{\mathrm {e}}^x\,{\ln \left (2\right )}^3+2592\,x\,{\ln \left (2\right )}^3\right )-40500\,x+{\mathrm {e}}^x\,\left (1620\,x^6+19440\,x^5+53460\,x^4-90720\,x^3-267300\,x^2+486000\,x-202500\right )+\ln \left (\frac {x}{5}+{\mathrm {e}}^x\right )\,\left (2\,\ln \left (2\right )\,\left (972\,x^5+7776\,x^4+5832\,x^3-38880\,x^2+24300\,x\right )+2\,{\mathrm {e}}^x\,\ln \left (2\right )\,\left (4860\,x^4+38880\,x^3+29160\,x^2-194400\,x+121500\right )\right )+97200\,x^2-53460\,x^3-18144\,x^4+10692\,x^5+3888\,x^6+324\,x^7+{\ln \left (\frac {x}{5}+{\mathrm {e}}^x\right )}^2\,\left (4\,{\ln \left (2\right )}^2\,\left (972\,x^3+3888\,x^2-4860\,x\right )+4\,{\mathrm {e}}^x\,{\ln \left (2\right )}^2\,\left (4860\,x^2+19440\,x-24300\right )\right )} \,d x \] Input:
int(-(125*x + 2*log(2)*(50*x + 20*x^2 + 2*x^3) + exp(x)*(1000*x + 2*log(2) *(250*x + 100*x^2 + 10*x^3) + 450*x^2 + 80*x^3 + 5*x^4 + 625) - log(x/5 + exp(x))*(2*log(2)*(25*x + 20*x^2 + 3*x^3) + 2*exp(x)*log(2)*(100*x + 15*x^ 2 + 125)) + 200*x^2 + 90*x^3 + 16*x^4 + x^5)/(log(x/5 + exp(x))^3*(12960*e xp(x)*log(2)^3 + 2592*x*log(2)^3) - 40500*x + exp(x)*(486000*x - 267300*x^ 2 - 90720*x^3 + 53460*x^4 + 19440*x^5 + 1620*x^6 - 202500) + log(x/5 + exp (x))*(2*log(2)*(24300*x - 38880*x^2 + 5832*x^3 + 7776*x^4 + 972*x^5) + 2*e xp(x)*log(2)*(29160*x^2 - 194400*x + 38880*x^3 + 4860*x^4 + 121500)) + 972 00*x^2 - 53460*x^3 - 18144*x^4 + 10692*x^5 + 3888*x^6 + 324*x^7 + log(x/5 + exp(x))^2*(4*log(2)^2*(3888*x^2 - 4860*x + 972*x^3) + 4*exp(x)*log(2)^2* (19440*x + 4860*x^2 - 24300))),x)
Output:
int(-(125*x + 2*log(2)*(50*x + 20*x^2 + 2*x^3) + exp(x)*(1000*x + 2*log(2) *(250*x + 100*x^2 + 10*x^3) + 450*x^2 + 80*x^3 + 5*x^4 + 625) - log(x/5 + exp(x))*(2*log(2)*(25*x + 20*x^2 + 3*x^3) + 2*exp(x)*log(2)*(100*x + 15*x^ 2 + 125)) + 200*x^2 + 90*x^3 + 16*x^4 + x^5)/(log(x/5 + exp(x))^3*(12960*e xp(x)*log(2)^3 + 2592*x*log(2)^3) - 40500*x + exp(x)*(486000*x - 267300*x^ 2 - 90720*x^3 + 53460*x^4 + 19440*x^5 + 1620*x^6 - 202500) + log(x/5 + exp (x))*(2*log(2)*(24300*x - 38880*x^2 + 5832*x^3 + 7776*x^4 + 972*x^5) + 2*e xp(x)*log(2)*(29160*x^2 - 194400*x + 38880*x^3 + 4860*x^4 + 121500)) + 972 00*x^2 - 53460*x^3 - 18144*x^4 + 10692*x^5 + 3888*x^6 + 324*x^7 + log(x/5 + exp(x))^2*(4*log(2)^2*(3888*x^2 - 4860*x + 972*x^3) + 4*exp(x)*log(2)^2* (19440*x + 4860*x^2 - 24300))), x)
Time = 0.21 (sec) , antiderivative size = 384, normalized size of antiderivative = 13.71 \[ \int \frac {-125 x-200 x^2-90 x^3-16 x^4-x^5+\left (-50 x-20 x^2-2 x^3\right ) \log (4)+e^x \left (-625-1000 x-450 x^2-80 x^3-5 x^4+\left (-250 x-100 x^2-10 x^3\right ) \log (4)\right )+\left (e^x \left (125+100 x+15 x^2\right ) \log (4)+\left (25 x+20 x^2+3 x^3\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x \left (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6\right )+\left (e^x \left (121500-194400 x+29160 x^2+38880 x^3+4860 x^4\right ) \log (4)+\left (24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (e^x \left (-24300+19440 x+4860 x^2\right ) \log ^2(4)+\left (-4860 x+3888 x^2+972 x^3\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (1620 e^x \log ^3(4)+324 x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (5 e^x+x\right )\right )} \, dx=\frac {16 \,\mathrm {log}\left (5 e^{x}+x \right ) \mathrm {log}\left (e^{x}+\frac {x}{5}\right )^{2} \mathrm {log}\left (2\right )^{3}+16 \,\mathrm {log}\left (5 e^{x}+x \right ) \mathrm {log}\left (e^{x}+\frac {x}{5}\right ) \mathrm {log}\left (2\right )^{2} x^{2}+64 \,\mathrm {log}\left (5 e^{x}+x \right ) \mathrm {log}\left (e^{x}+\frac {x}{5}\right ) \mathrm {log}\left (2\right )^{2} x -80 \,\mathrm {log}\left (5 e^{x}+x \right ) \mathrm {log}\left (e^{x}+\frac {x}{5}\right ) \mathrm {log}\left (2\right )^{2}+4 \,\mathrm {log}\left (5 e^{x}+x \right ) \mathrm {log}\left (2\right ) x^{4}+32 \,\mathrm {log}\left (5 e^{x}+x \right ) \mathrm {log}\left (2\right ) x^{3}+24 \,\mathrm {log}\left (5 e^{x}+x \right ) \mathrm {log}\left (2\right ) x^{2}-160 \,\mathrm {log}\left (5 e^{x}+x \right ) \mathrm {log}\left (2\right ) x +100 \,\mathrm {log}\left (5 e^{x}+x \right ) \mathrm {log}\left (2\right )-16 \mathrm {log}\left (e^{x}+\frac {x}{5}\right )^{3} \mathrm {log}\left (2\right )^{3}-16 \mathrm {log}\left (e^{x}+\frac {x}{5}\right )^{2} \mathrm {log}\left (2\right )^{2} x^{2}-64 \mathrm {log}\left (e^{x}+\frac {x}{5}\right )^{2} \mathrm {log}\left (2\right )^{2} x +60 \mathrm {log}\left (e^{x}+\frac {x}{5}\right )^{2} \mathrm {log}\left (2\right )^{2}-4 \,\mathrm {log}\left (e^{x}+\frac {x}{5}\right ) \mathrm {log}\left (2\right ) x^{4}-32 \,\mathrm {log}\left (e^{x}+\frac {x}{5}\right ) \mathrm {log}\left (2\right ) x^{3}-44 \,\mathrm {log}\left (e^{x}+\frac {x}{5}\right ) \mathrm {log}\left (2\right ) x^{2}+80 \,\mathrm {log}\left (e^{x}+\frac {x}{5}\right ) \mathrm {log}\left (2\right ) x -5 x^{4}-37 x^{3}+275 x -125}{3888 \mathrm {log}\left (e^{x}+\frac {x}{5}\right )^{2} \mathrm {log}\left (2\right )^{2}+3888 \,\mathrm {log}\left (e^{x}+\frac {x}{5}\right ) \mathrm {log}\left (2\right ) x^{2}+15552 \,\mathrm {log}\left (e^{x}+\frac {x}{5}\right ) \mathrm {log}\left (2\right ) x -19440 \,\mathrm {log}\left (e^{x}+\frac {x}{5}\right ) \mathrm {log}\left (2\right )+972 x^{4}+7776 x^{3}+5832 x^{2}-38880 x +24300} \] Input:
int(((2*(15*x^2+100*x+125)*log(2)*exp(x)+2*(3*x^3+20*x^2+25*x)*log(2))*log (1/5*x+exp(x))+(2*(-10*x^3-100*x^2-250*x)*log(2)-5*x^4-80*x^3-450*x^2-1000 *x-625)*exp(x)+2*(-2*x^3-20*x^2-50*x)*log(2)-x^5-16*x^4-90*x^3-200*x^2-125 *x)/((12960*log(2)^3*exp(x)+2592*x*log(2)^3)*log(1/5*x+exp(x))^3+(4*(4860* x^2+19440*x-24300)*log(2)^2*exp(x)+4*(972*x^3+3888*x^2-4860*x)*log(2)^2)*l og(1/5*x+exp(x))^2+(2*(4860*x^4+38880*x^3+29160*x^2-194400*x+121500)*log(2 )*exp(x)+2*(972*x^5+7776*x^4+5832*x^3-38880*x^2+24300*x)*log(2))*log(1/5*x +exp(x))+(1620*x^6+19440*x^5+53460*x^4-90720*x^3-267300*x^2+486000*x-20250 0)*exp(x)+324*x^7+3888*x^6+10692*x^5-18144*x^4-53460*x^3+97200*x^2-40500*x ),x)
Output:
(16*log(5*e**x + x)*log((5*e**x + x)/5)**2*log(2)**3 + 16*log(5*e**x + x)* log((5*e**x + x)/5)*log(2)**2*x**2 + 64*log(5*e**x + x)*log((5*e**x + x)/5 )*log(2)**2*x - 80*log(5*e**x + x)*log((5*e**x + x)/5)*log(2)**2 + 4*log(5 *e**x + x)*log(2)*x**4 + 32*log(5*e**x + x)*log(2)*x**3 + 24*log(5*e**x + x)*log(2)*x**2 - 160*log(5*e**x + x)*log(2)*x + 100*log(5*e**x + x)*log(2) - 16*log((5*e**x + x)/5)**3*log(2)**3 - 16*log((5*e**x + x)/5)**2*log(2)* *2*x**2 - 64*log((5*e**x + x)/5)**2*log(2)**2*x + 60*log((5*e**x + x)/5)** 2*log(2)**2 - 4*log((5*e**x + x)/5)*log(2)*x**4 - 32*log((5*e**x + x)/5)*l og(2)*x**3 - 44*log((5*e**x + x)/5)*log(2)*x**2 + 80*log((5*e**x + x)/5)*l og(2)*x - 5*x**4 - 37*x**3 + 275*x - 125)/(972*(4*log((5*e**x + x)/5)**2*l og(2)**2 + 4*log((5*e**x + x)/5)*log(2)*x**2 + 16*log((5*e**x + x)/5)*log( 2)*x - 20*log((5*e**x + x)/5)*log(2) + x**4 + 8*x**3 + 6*x**2 - 40*x + 25) )