Integrand size = 352, antiderivative size = 28 \[ \int \frac {-125 x-200 x^2-90 x^3-16 x^4-x^5+\left (-50 x-20 x^2-2 x^3\right ) \log (4)+e^x \left (-625-1000 x-450 x^2-80 x^3-5 x^4+\left (-250 x-100 x^2-10 x^3\right ) \log (4)\right )+\left (e^x \left (125+100 x+15 x^2\right ) \log (4)+\left (25 x+20 x^2+3 x^3\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x \left (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6\right )+\left (e^x \left (121500-194400 x+29160 x^2+38880 x^3+4860 x^4\right ) \log (4)+\left (24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (e^x \left (-24300+19440 x+4860 x^2\right ) \log ^2(4)+\left (-4860 x+3888 x^2+972 x^3\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (1620 e^x \log ^3(4)+324 x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (5 e^x+x\right )\right )} \, dx=\frac {x}{324 \left (-1+x+\frac {\log (4) \log \left (e^x+\frac {x}{5}\right )}{5+x}\right )^2} \]
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\[ \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-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 \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {(5+x) \left ((5+x) \left (x \left (5+6 x+x^2+\log (16)\right )+5 e^x \left (5+x^2+x (6+\log (16))\right )\right )-\left (5 e^x+x\right ) (5+3 x) \log (4) \log \left (e^x+\frac {x}{5}\right )\right )}{324 \left (5 e^x+x\right ) \left (5-4 x-x^2-\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx \\ & = \frac {1}{324} \int \frac {(5+x) \left ((5+x) \left (x \left (5+6 x+x^2+\log (16)\right )+5 e^x \left (5+x^2+x (6+\log (16))\right )\right )-\left (5 e^x+x\right ) (5+3 x) \log (4) \log \left (e^x+\frac {x}{5}\right )\right )}{\left (5 e^x+x\right ) \left (5-4 x-x^2-\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx \\ & = \frac {1}{324} \int \left (\frac {(-1+x) x (5+x)^2 \log (16)}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {(5+x) \left (25+x^3+11 x^2 \left (1+\frac {4 \log (2)}{11}\right )+35 x \left (1+\frac {4 \log (2)}{7}\right )-5 \log (4) \log \left (e^x+\frac {x}{5}\right )-3 x \log (4) \log \left (e^x+\frac {x}{5}\right )\right )}{\left (5-4 x-x^2-\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}\right ) \, dx \\ & = \frac {1}{324} \int \frac {(5+x) \left (25+x^3+11 x^2 \left (1+\frac {4 \log (2)}{11}\right )+35 x \left (1+\frac {4 \log (2)}{7}\right )-5 \log (4) \log \left (e^x+\frac {x}{5}\right )-3 x \log (4) \log \left (e^x+\frac {x}{5}\right )\right )}{\left (5-4 x-x^2-\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{324} \log (16) \int \frac {(-1+x) x (5+x)^2}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx \\ & = \frac {1}{324} \int \frac {(5+x) \left ((5+x) \left (5+x^2+x (6+\log (16))\right )-(5+3 x) \log (4) \log \left (e^x+\frac {x}{5}\right )\right )}{\left (5-4 x-x^2-\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{324} \log (16) \int \left (-\frac {25 x}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {15 x^2}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {9 x^3}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {x^4}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}\right ) \, dx \\ & = \frac {1}{324} \int \left (-\frac {x (5+x)^2 (8+4 x+\log (16))}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {25+20 x+3 x^2}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2}\right ) \, dx+\frac {1}{324} \log (16) \int \frac {x^4}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{36} \log (16) \int \frac {x^3}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{108} (5 \log (16)) \int \frac {x^2}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (25 \log (16)) \int \frac {x}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx \\ & = -\left (\frac {1}{324} \int \frac {x (5+x)^2 (8+4 x+\log (16))}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx\right )+\frac {1}{324} \int \frac {25+20 x+3 x^2}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2} \, dx+\frac {1}{324} \log (16) \int \frac {x^4}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{36} \log (16) \int \frac {x^3}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{108} (5 \log (16)) \int \frac {x^2}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (25 \log (16)) \int \frac {x}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx \\ & = -\left (\frac {1}{324} \int \left (\frac {4 x^4}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {25 x (8+\log (16))}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {10 x^2 (18+\log (16))}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {x^3 (48+\log (16))}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}\right ) \, dx\right )+\frac {1}{324} \int \left (\frac {25}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2}+\frac {20 x}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2}+\frac {3 x^2}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2}\right ) \, dx+\frac {1}{324} \log (16) \int \frac {x^4}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{36} \log (16) \int \frac {x^3}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{108} (5 \log (16)) \int \frac {x^2}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (25 \log (16)) \int \frac {x}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx \\ & = \frac {1}{108} \int \frac {x^2}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2} \, dx-\frac {1}{81} \int \frac {x^4}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {5}{81} \int \frac {x}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2} \, dx+\frac {25}{324} \int \frac {1}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2} \, dx+\frac {1}{324} \log (16) \int \frac {x^4}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{36} \log (16) \int \frac {x^3}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{108} (5 \log (16)) \int \frac {x^2}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (25 \log (16)) \int \frac {x}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (25 (8+\log (16))) \int \frac {x}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{162} (5 (18+\log (16))) \int \frac {x^2}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (48+\log (16)) \int \frac {x^3}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx \\ & = \frac {1}{108} \int \frac {x^2}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2} \, dx-\frac {1}{81} \int \frac {x^4}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {5}{81} \int \frac {x}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2} \, dx+\frac {125}{324} \text {Subst}\left (\int \frac {1}{\left (-5+20 x+25 x^2+\log (4) \log \left (e^{5 x}+x\right )\right )^2} \, dx,x,\frac {x}{5}\right )+\frac {1}{324} \log (16) \int \frac {x^4}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{36} \log (16) \int \frac {x^3}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{108} (5 \log (16)) \int \frac {x^2}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (25 \log (16)) \int \frac {x}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (25 (8+\log (16))) \int \frac {x}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{162} (5 (18+\log (16))) \int \frac {x^2}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (48+\log (16)) \int \frac {x^3}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(28)=56\).
Time = 0.35 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.89 \[ \int \frac {-125 x-200 x^2-90 x^3-16 x^4-x^5+\left (-50 x-20 x^2-2 x^3\right ) \log (4)+e^x \left (-625-1000 x-450 x^2-80 x^3-5 x^4+\left (-250 x-100 x^2-10 x^3\right ) \log (4)\right )+\left (e^x \left (125+100 x+15 x^2\right ) \log (4)+\left (25 x+20 x^2+3 x^3\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x \left (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6\right )+\left (e^x \left (121500-194400 x+29160 x^2+38880 x^3+4860 x^4\right ) \log (4)+\left (24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (e^x \left (-24300+19440 x+4860 x^2\right ) \log ^2(4)+\left (-4860 x+3888 x^2+972 x^3\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (1620 e^x \log ^3(4)+324 x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (5 e^x+x\right )\right )} \, dx=\frac {x (5+x)^2 \left (8 x+4 x^2+\log (16)+5 e^x (8+4 x+\log (16))\right )}{648 \left (4 x+2 x^2+\log (4)+5 e^x (4+2 x+\log (4))\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2} \]
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Time = 5.22 (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 | \(\frac {25 x^{3}+250 x^{2}+625 x}{32400 \ln \left (2\right )^{2} \ln \left (\frac {x}{5}+{\mathrm e}^{x}\right )^{2}+32400 \ln \left (2\right ) \ln \left (\frac {x}{5}+{\mathrm e}^{x}\right ) x^{2}+8100 x^{4}+129600 \ln \left (2\right ) \ln \left (\frac {x}{5}+{\mathrm e}^{x}\right ) x +64800 x^{3}-162000 \ln \left (2\right ) \ln \left (\frac {x}{5}+{\mathrm e}^{x}\right )+48600 x^{2}-324000 x +202500}\) | \(89\) |
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Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.43 \[ \int \frac {-125 x-200 x^2-90 x^3-16 x^4-x^5+\left (-50 x-20 x^2-2 x^3\right ) \log (4)+e^x \left (-625-1000 x-450 x^2-80 x^3-5 x^4+\left (-250 x-100 x^2-10 x^3\right ) \log (4)\right )+\left (e^x \left (125+100 x+15 x^2\right ) \log (4)+\left (25 x+20 x^2+3 x^3\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x \left (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6\right )+\left (e^x \left (121500-194400 x+29160 x^2+38880 x^3+4860 x^4\right ) \log (4)+\left (24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (e^x \left (-24300+19440 x+4860 x^2\right ) \log ^2(4)+\left (-4860 x+3888 x^2+972 x^3\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (1620 e^x \log ^3(4)+324 x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (5 e^x+x\right )\right )} \, dx=\frac {x^{3} + 10 \, x^{2} + 25 \, x}{324 \, {\left (x^{4} + 4 \, \log \left (2\right )^{2} \log \left (\frac {1}{5} \, x + e^{x}\right )^{2} + 8 \, x^{3} + 4 \, {\left (x^{2} + 4 \, x - 5\right )} \log \left (2\right ) \log \left (\frac {1}{5} \, x + e^{x}\right ) + 6 \, x^{2} - 40 \, x + 25\right )}} \]
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Time = 0.32 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.71 \[ \int \frac {-125 x-200 x^2-90 x^3-16 x^4-x^5+\left (-50 x-20 x^2-2 x^3\right ) \log (4)+e^x \left (-625-1000 x-450 x^2-80 x^3-5 x^4+\left (-250 x-100 x^2-10 x^3\right ) \log (4)\right )+\left (e^x \left (125+100 x+15 x^2\right ) \log (4)+\left (25 x+20 x^2+3 x^3\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x \left (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6\right )+\left (e^x \left (121500-194400 x+29160 x^2+38880 x^3+4860 x^4\right ) \log (4)+\left (24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (e^x \left (-24300+19440 x+4860 x^2\right ) \log ^2(4)+\left (-4860 x+3888 x^2+972 x^3\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (1620 e^x \log ^3(4)+324 x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (5 e^x+x\right )\right )} \, dx=\frac {x^{3} + 10 x^{2} + 25 x}{324 x^{4} + 2592 x^{3} + 1944 x^{2} - 12960 x + \left (1296 x^{2} \log {\left (2 \right )} + 5184 x \log {\left (2 \right )} - 6480 \log {\left (2 \right )}\right ) \log {\left (\frac {x}{5} + e^{x} \right )} + 1296 \log {\left (2 \right )}^{2} \log {\left (\frac {x}{5} + e^{x} \right )}^{2} + 8100} \]
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Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (24) = 48\).
Time = 0.79 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.07 \[ \int \frac {-125 x-200 x^2-90 x^3-16 x^4-x^5+\left (-50 x-20 x^2-2 x^3\right ) \log (4)+e^x \left (-625-1000 x-450 x^2-80 x^3-5 x^4+\left (-250 x-100 x^2-10 x^3\right ) \log (4)\right )+\left (e^x \left (125+100 x+15 x^2\right ) \log (4)+\left (25 x+20 x^2+3 x^3\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x \left (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6\right )+\left (e^x \left (121500-194400 x+29160 x^2+38880 x^3+4860 x^4\right ) \log (4)+\left (24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (e^x \left (-24300+19440 x+4860 x^2\right ) \log ^2(4)+\left (-4860 x+3888 x^2+972 x^3\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (1620 e^x \log ^3(4)+324 x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (5 e^x+x\right )\right )} \, dx=\frac {x^{3} + 10 \, x^{2} + 25 \, x}{324 \, {\left (x^{4} + 4 \, \log \left (5\right )^{2} \log \left (2\right )^{2} + 4 \, \log \left (2\right )^{2} \log \left (x + 5 \, e^{x}\right )^{2} - 2 \, {\left (2 \, \log \left (5\right ) \log \left (2\right ) - 3\right )} x^{2} + 8 \, x^{3} - 8 \, {\left (2 \, \log \left (5\right ) \log \left (2\right ) + 5\right )} x + 20 \, \log \left (5\right ) \log \left (2\right ) + 4 \, {\left (x^{2} \log \left (2\right ) - 2 \, \log \left (5\right ) \log \left (2\right )^{2} + 4 \, x \log \left (2\right ) - 5 \, \log \left (2\right )\right )} \log \left (x + 5 \, e^{x}\right ) + 25\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (24) = 48\).
Time = 0.80 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.75 \[ \int \frac {-125 x-200 x^2-90 x^3-16 x^4-x^5+\left (-50 x-20 x^2-2 x^3\right ) \log (4)+e^x \left (-625-1000 x-450 x^2-80 x^3-5 x^4+\left (-250 x-100 x^2-10 x^3\right ) \log (4)\right )+\left (e^x \left (125+100 x+15 x^2\right ) \log (4)+\left (25 x+20 x^2+3 x^3\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x \left (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6\right )+\left (e^x \left (121500-194400 x+29160 x^2+38880 x^3+4860 x^4\right ) \log (4)+\left (24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (e^x \left (-24300+19440 x+4860 x^2\right ) \log ^2(4)+\left (-4860 x+3888 x^2+972 x^3\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (1620 e^x \log ^3(4)+324 x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (5 e^x+x\right )\right )} \, dx=\frac {x^{3} + 10 \, x^{2} + 25 \, x}{324 \, {\left (x^{4} - 4 \, x^{2} \log \left (5\right ) \log \left (2\right ) + 4 \, \log \left (5\right )^{2} \log \left (2\right )^{2} + 4 \, x^{2} \log \left (2\right ) \log \left (x + 5 \, e^{x}\right ) - 8 \, \log \left (5\right ) \log \left (2\right )^{2} \log \left (x + 5 \, e^{x}\right ) + 4 \, \log \left (2\right )^{2} \log \left (x + 5 \, e^{x}\right )^{2} + 8 \, x^{3} - 16 \, x \log \left (5\right ) \log \left (2\right ) + 16 \, x \log \left (2\right ) \log \left (x + 5 \, e^{x}\right ) + 6 \, x^{2} + 20 \, \log \left (5\right ) \log \left (2\right ) - 20 \, \log \left (2\right ) \log \left (x + 5 \, e^{x}\right ) - 40 \, x + 25\right )}} \]
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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 \]
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