Integrand size = 166, antiderivative size = 29 \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=\left (5-\left (21-\frac {5}{\log \left (2+e^{5-\frac {2 x}{5+x}}\right )}\right )^2\right )^2 \] Output:
(5-(21-5/ln(exp(5-2*x/(5+x))+2))^2)^2
Leaf count is larger than twice the leaf count of optimal. \(91\) vs. \(2(29)=58\).
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.14 \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=-200 \left (-\frac {25}{8 \log ^4\left (2+e^{\frac {25+3 x}{5+x}}\right )}+\frac {105}{2 \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}-\frac {659}{2 \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )}+\frac {4578}{5 \log \left (2+e^{\frac {25+3 x}{5+x}}\right )}\right ) \] Input:
Integrate[(25000*E^((25 + 3*x)/(5 + x)) - 315000*E^((25 + 3*x)/(5 + x))*Lo g[2 + E^((25 + 3*x)/(5 + x))] + 1318000*E^((25 + 3*x)/(5 + x))*Log[2 + E^( (25 + 3*x)/(5 + x))]^2 - 1831200*E^((25 + 3*x)/(5 + x))*Log[2 + E^((25 + 3 *x)/(5 + x))]^3)/((50 + 20*x + 2*x^2 + E^((25 + 3*x)/(5 + x))*(25 + 10*x + x^2))*Log[2 + E^((25 + 3*x)/(5 + x))]^5),x]
Output:
-200*(-25/(8*Log[2 + E^((25 + 3*x)/(5 + x))]^4) + 105/(2*Log[2 + E^((25 + 3*x)/(5 + x))]^3) - 659/(2*Log[2 + E^((25 + 3*x)/(5 + x))]^2) + 4578/(5*Lo g[2 + E^((25 + 3*x)/(5 + x))]))
Leaf count is larger than twice the leaf count of optimal. \(91\) vs. \(2(29)=58\).
Time = 3.64 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.14, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {7292, 27, 7293, 2009}
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.
\(\displaystyle \int \frac {25000 e^{\frac {3 x+25}{x+5}}-1831200 e^{\frac {3 x+25}{x+5}} \log ^3\left (e^{\frac {3 x+25}{x+5}}+2\right )+1318000 e^{\frac {3 x+25}{x+5}} \log ^2\left (e^{\frac {3 x+25}{x+5}}+2\right )-315000 e^{\frac {3 x+25}{x+5}} \log \left (e^{\frac {3 x+25}{x+5}}+2\right )}{\left (2 x^2+e^{\frac {3 x+25}{x+5}} \left (x^2+10 x+25\right )+20 x+50\right ) \log ^5\left (e^{\frac {3 x+25}{x+5}}+2\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {200 e^{\frac {3 x+25}{x+5}} \left (-9156 \log ^3\left (e^{\frac {3 x+25}{x+5}}+2\right )+6590 \log ^2\left (e^{\frac {3 x+25}{x+5}}+2\right )-1575 \log \left (e^{\frac {3 x+25}{x+5}}+2\right )+125\right )}{\left (e^{\frac {3 x}{x+5}+\frac {25}{x+5}}+2\right ) (x+5)^2 \log ^5\left (e^{\frac {3 x+25}{x+5}}+2\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 200 \int \frac {e^{\frac {3 x+25}{x+5}} \left (-9156 \log ^3\left (2+e^{\frac {3 x+25}{x+5}}\right )+6590 \log ^2\left (2+e^{\frac {3 x+25}{x+5}}\right )-1575 \log \left (2+e^{\frac {3 x+25}{x+5}}\right )+125\right )}{\left (2+e^{\frac {3 x}{x+5}+\frac {25}{x+5}}\right ) (x+5)^2 \log ^5\left (2+e^{\frac {3 x+25}{x+5}}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 200 \int \left (-\frac {9156 e^{\frac {3 x+25}{x+5}}}{(x+5)^2 \log ^2\left (2+e^{\frac {3 x+25}{x+5}}\right ) \left (2+e^{\frac {3 x}{x+5}+\frac {25}{x+5}}\right )}+\frac {6590 e^{\frac {3 x+25}{x+5}}}{(x+5)^2 \log ^3\left (2+e^{\frac {3 x+25}{x+5}}\right ) \left (2+e^{\frac {3 x}{x+5}+\frac {25}{x+5}}\right )}-\frac {1575 e^{\frac {3 x+25}{x+5}}}{(x+5)^2 \log ^4\left (2+e^{\frac {3 x+25}{x+5}}\right ) \left (2+e^{\frac {3 x}{x+5}+\frac {25}{x+5}}\right )}+\frac {125 e^{\frac {3 x+25}{x+5}}}{(x+5)^2 \log ^5\left (2+e^{\frac {3 x+25}{x+5}}\right ) \left (2+e^{\frac {3 x}{x+5}+\frac {25}{x+5}}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 200 \left (\frac {25}{8 \log ^4\left (e^{\frac {3 x+25}{x+5}}+2\right )}-\frac {105}{2 \log ^3\left (e^{\frac {3 x+25}{x+5}}+2\right )}+\frac {659}{2 \log ^2\left (e^{\frac {3 x+25}{x+5}}+2\right )}-\frac {4578}{5 \log \left (e^{\frac {3 x+25}{x+5}}+2\right )}\right )\) |
Input:
Int[(25000*E^((25 + 3*x)/(5 + x)) - 315000*E^((25 + 3*x)/(5 + x))*Log[2 + E^((25 + 3*x)/(5 + x))] + 1318000*E^((25 + 3*x)/(5 + x))*Log[2 + E^((25 + 3*x)/(5 + x))]^2 - 1831200*E^((25 + 3*x)/(5 + x))*Log[2 + E^((25 + 3*x)/(5 + x))]^3)/((50 + 20*x + 2*x^2 + E^((25 + 3*x)/(5 + x))*(25 + 10*x + x^2)) *Log[2 + E^((25 + 3*x)/(5 + x))]^5),x]
Output:
200*(25/(8*Log[2 + E^((25 + 3*x)/(5 + x))]^4) - 105/(2*Log[2 + E^((25 + 3* x)/(5 + x))]^3) + 659/(2*Log[2 + E^((25 + 3*x)/(5 + x))]^2) - 4578/(5*Log[ 2 + E^((25 + 3*x)/(5 + x))]))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(76\) vs. \(2(28)=56\).
Time = 0.81 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.66
method | result | size |
risch | \(-\frac {5 \left (36624 \ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )^{3}-13180 \ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )^{2}+2100 \ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )-125\right )}{\ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )^{4}}\) | \(77\) |
parallelrisch | \(-\frac {-50000+14649600 \ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )^{3}-5272000 \ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )^{2}+840000 \ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )}{80 \ln \left ({\mathrm e}^{\frac {25+3 x}{5+x}}+2\right )^{4}}\) | \(77\) |
Input:
int((-1831200*exp((25+3*x)/(5+x))*ln(exp((25+3*x)/(5+x))+2)^3+1318000*exp( (25+3*x)/(5+x))*ln(exp((25+3*x)/(5+x))+2)^2-315000*exp((25+3*x)/(5+x))*ln( exp((25+3*x)/(5+x))+2)+25000*exp((25+3*x)/(5+x)))/((x^2+10*x+25)*exp((25+3 *x)/(5+x))+2*x^2+20*x+50)/ln(exp((25+3*x)/(5+x))+2)^5,x,method=_RETURNVERB OSE)
Output:
-5*(36624*ln(exp((25+3*x)/(5+x))+2)^3-13180*ln(exp((25+3*x)/(5+x))+2)^2+21 00*ln(exp((25+3*x)/(5+x))+2)-125)/ln(exp((25+3*x)/(5+x))+2)^4
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (26) = 52\).
Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.62 \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=-\frac {5 \, {\left (36624 \, \log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right )^{3} - 13180 \, \log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right )^{2} + 2100 \, \log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right ) - 125\right )}}{\log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right )^{4}} \] Input:
integrate((-1831200*exp((25+3*x)/(5+x))*log(exp((25+3*x)/(5+x))+2)^3+13180 00*exp((25+3*x)/(5+x))*log(exp((25+3*x)/(5+x))+2)^2-315000*exp((25+3*x)/(5 +x))*log(exp((25+3*x)/(5+x))+2)+25000*exp((25+3*x)/(5+x)))/((x^2+10*x+25)* exp((25+3*x)/(5+x))+2*x^2+20*x+50)/log(exp((25+3*x)/(5+x))+2)^5,x, algorit hm="fricas")
Output:
-5*(36624*log(e^((3*x + 25)/(x + 5)) + 2)^3 - 13180*log(e^((3*x + 25)/(x + 5)) + 2)^2 + 2100*log(e^((3*x + 25)/(x + 5)) + 2) - 125)/log(e^((3*x + 25 )/(x + 5)) + 2)^4
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (20) = 40\).
Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.24 \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=\frac {- 183120 \log {\left (e^{\frac {3 x + 25}{x + 5}} + 2 \right )}^{3} + 65900 \log {\left (e^{\frac {3 x + 25}{x + 5}} + 2 \right )}^{2} - 10500 \log {\left (e^{\frac {3 x + 25}{x + 5}} + 2 \right )} + 625}{\log {\left (e^{\frac {3 x + 25}{x + 5}} + 2 \right )}^{4}} \] Input:
integrate((-1831200*exp((25+3*x)/(5+x))*ln(exp((25+3*x)/(5+x))+2)**3+13180 00*exp((25+3*x)/(5+x))*ln(exp((25+3*x)/(5+x))+2)**2-315000*exp((25+3*x)/(5 +x))*ln(exp((25+3*x)/(5+x))+2)+25000*exp((25+3*x)/(5+x)))/((x**2+10*x+25)* exp((25+3*x)/(5+x))+2*x**2+20*x+50)/ln(exp((25+3*x)/(5+x))+2)**5,x)
Output:
(-183120*log(exp((3*x + 25)/(x + 5)) + 2)**3 + 65900*log(exp((3*x + 25)/(x + 5)) + 2)**2 - 10500*log(exp((3*x + 25)/(x + 5)) + 2) + 625)/log(exp((3* x + 25)/(x + 5)) + 2)**4
Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (26) = 52\).
Time = 0.14 (sec) , antiderivative size = 297, normalized size of antiderivative = 10.24 \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=-\frac {45780 \, \log \left (e^{\left (\frac {3 \, x}{x + 5} + \frac {25}{x + 5}\right )} + 2\right )^{3}}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{4}} - \frac {45780 \, \log \left (e^{\left (\frac {3 \, x}{x + 5} + \frac {25}{x + 5}\right )} + 2\right )^{2}}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{3}} - \frac {45780 \, \log \left (e^{\left (\frac {3 \, x}{x + 5} + \frac {25}{x + 5}\right )} + 2\right )}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{2}} - \frac {45780}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )} + \frac {32950 \, \log \left (e^{\left (\frac {3 \, x}{x + 5} + \frac {25}{x + 5}\right )} + 2\right )^{2}}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{4}} + \frac {65900 \, \log \left (e^{\left (\frac {3 \, x}{x + 5} + \frac {25}{x + 5}\right )} + 2\right )}{3 \, \log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{3}} + \frac {32950}{3 \, \log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{2}} - \frac {7875 \, \log \left (e^{\left (\frac {3 \, x}{x + 5} + \frac {25}{x + 5}\right )} + 2\right )}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{4}} - \frac {2625}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{3}} + \frac {625}{\log \left (e^{\left (\frac {10}{x + 5} + 3\right )} + 2\right )^{4}} \] Input:
integrate((-1831200*exp((25+3*x)/(5+x))*log(exp((25+3*x)/(5+x))+2)^3+13180 00*exp((25+3*x)/(5+x))*log(exp((25+3*x)/(5+x))+2)^2-315000*exp((25+3*x)/(5 +x))*log(exp((25+3*x)/(5+x))+2)+25000*exp((25+3*x)/(5+x)))/((x^2+10*x+25)* exp((25+3*x)/(5+x))+2*x^2+20*x+50)/log(exp((25+3*x)/(5+x))+2)^5,x, algorit hm="maxima")
Output:
-45780*log(e^(3*x/(x + 5) + 25/(x + 5)) + 2)^3/log(e^(10/(x + 5) + 3) + 2) ^4 - 45780*log(e^(3*x/(x + 5) + 25/(x + 5)) + 2)^2/log(e^(10/(x + 5) + 3) + 2)^3 - 45780*log(e^(3*x/(x + 5) + 25/(x + 5)) + 2)/log(e^(10/(x + 5) + 3 ) + 2)^2 - 45780/log(e^(10/(x + 5) + 3) + 2) + 32950*log(e^(3*x/(x + 5) + 25/(x + 5)) + 2)^2/log(e^(10/(x + 5) + 3) + 2)^4 + 65900/3*log(e^(3*x/(x + 5) + 25/(x + 5)) + 2)/log(e^(10/(x + 5) + 3) + 2)^3 + 32950/3/log(e^(10/( x + 5) + 3) + 2)^2 - 7875*log(e^(3*x/(x + 5) + 25/(x + 5)) + 2)/log(e^(10/ (x + 5) + 3) + 2)^4 - 2625/log(e^(10/(x + 5) + 3) + 2)^3 + 625/log(e^(10/( x + 5) + 3) + 2)^4
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (26) = 52\).
Time = 0.14 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.62 \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=-\frac {5 \, {\left (36624 \, \log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right )^{3} - 13180 \, \log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right )^{2} + 2100 \, \log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right ) - 125\right )}}{\log \left (e^{\left (\frac {3 \, x + 25}{x + 5}\right )} + 2\right )^{4}} \] Input:
integrate((-1831200*exp((25+3*x)/(5+x))*log(exp((25+3*x)/(5+x))+2)^3+13180 00*exp((25+3*x)/(5+x))*log(exp((25+3*x)/(5+x))+2)^2-315000*exp((25+3*x)/(5 +x))*log(exp((25+3*x)/(5+x))+2)+25000*exp((25+3*x)/(5+x)))/((x^2+10*x+25)* exp((25+3*x)/(5+x))+2*x^2+20*x+50)/log(exp((25+3*x)/(5+x))+2)^5,x, algorit hm="giac")
Output:
-5*(36624*log(e^((3*x + 25)/(x + 5)) + 2)^3 - 13180*log(e^((3*x + 25)/(x + 5)) + 2)^2 + 2100*log(e^((3*x + 25)/(x + 5)) + 2) - 125)/log(e^((3*x + 25 )/(x + 5)) + 2)^4
Time = 1.82 (sec) , antiderivative size = 101, normalized size of antiderivative = 3.48 \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=\frac {65900}{{\ln \left ({\mathrm {e}}^{\frac {3\,x}{x+5}}\,{\mathrm {e}}^{\frac {25}{x+5}}+2\right )}^2}-\frac {183120}{\ln \left ({\mathrm {e}}^{\frac {3\,x}{x+5}}\,{\mathrm {e}}^{\frac {25}{x+5}}+2\right )}-\frac {10500}{{\ln \left ({\mathrm {e}}^{\frac {3\,x}{x+5}}\,{\mathrm {e}}^{\frac {25}{x+5}}+2\right )}^3}+\frac {625}{{\ln \left ({\mathrm {e}}^{\frac {3\,x}{x+5}}\,{\mathrm {e}}^{\frac {25}{x+5}}+2\right )}^4} \] Input:
int((25000*exp((3*x + 25)/(x + 5)) - 315000*exp((3*x + 25)/(x + 5))*log(ex p((3*x + 25)/(x + 5)) + 2) + 1318000*exp((3*x + 25)/(x + 5))*log(exp((3*x + 25)/(x + 5)) + 2)^2 - 1831200*exp((3*x + 25)/(x + 5))*log(exp((3*x + 25) /(x + 5)) + 2)^3)/(log(exp((3*x + 25)/(x + 5)) + 2)^5*(20*x + exp((3*x + 2 5)/(x + 5))*(10*x + x^2 + 25) + 2*x^2 + 50)),x)
Output:
65900/log(exp((3*x)/(x + 5))*exp(25/(x + 5)) + 2)^2 - 183120/log(exp((3*x) /(x + 5))*exp(25/(x + 5)) + 2) - 10500/log(exp((3*x)/(x + 5))*exp(25/(x + 5)) + 2)^3 + 625/log(exp((3*x)/(x + 5))*exp(25/(x + 5)) + 2)^4
Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.76 \[ \int \frac {25000 e^{\frac {25+3 x}{5+x}}-315000 e^{\frac {25+3 x}{5+x}} \log \left (2+e^{\frac {25+3 x}{5+x}}\right )+1318000 e^{\frac {25+3 x}{5+x}} \log ^2\left (2+e^{\frac {25+3 x}{5+x}}\right )-1831200 e^{\frac {25+3 x}{5+x}} \log ^3\left (2+e^{\frac {25+3 x}{5+x}}\right )}{\left (50+20 x+2 x^2+e^{\frac {25+3 x}{5+x}} \left (25+10 x+x^2\right )\right ) \log ^5\left (2+e^{\frac {25+3 x}{5+x}}\right )} \, dx=\frac {-183120 \mathrm {log}\left (e^{\frac {10}{x +5}} e^{3}+2\right )^{3}+65900 \mathrm {log}\left (e^{\frac {10}{x +5}} e^{3}+2\right )^{2}-10500 \,\mathrm {log}\left (e^{\frac {10}{x +5}} e^{3}+2\right )+625}{\mathrm {log}\left (e^{\frac {10}{x +5}} e^{3}+2\right )^{4}} \] Input:
int((-1831200*exp((25+3*x)/(5+x))*log(exp((25+3*x)/(5+x))+2)^3+1318000*exp ((25+3*x)/(5+x))*log(exp((25+3*x)/(5+x))+2)^2-315000*exp((25+3*x)/(5+x))*l og(exp((25+3*x)/(5+x))+2)+25000*exp((25+3*x)/(5+x)))/((x^2+10*x+25)*exp((2 5+3*x)/(5+x))+2*x^2+20*x+50)/log(exp((25+3*x)/(5+x))+2)^5,x)
Output:
(5*( - 36624*log(e**(10/(x + 5))*e**3 + 2)**3 + 13180*log(e**(10/(x + 5))* e**3 + 2)**2 - 2100*log(e**(10/(x + 5))*e**3 + 2) + 125))/log(e**(10/(x + 5))*e**3 + 2)**4