\(\int \frac {300 e^{10} x+216 x^3+510 x^4+300 x^5+e^5 (-510 x^2-600 x^3)+(-100 e^{10} x-72 x^3-160 x^4-100 x^5+e^5 (180 x^2+200 x^3)) \log (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2})+(100 e^{10} x+72 x^3+170 x^4+100 x^5+e^5 (-170 x^2-200 x^3)) \log ^2(\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2})}{450 e^{10}+324 x^2+765 x^3+450 x^4+e^5 (-765 x-900 x^2)+(300 e^{10}+216 x^2+510 x^3+300 x^4+e^5 (-510 x-600 x^2)) \log ^2(\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2})+(50 e^{10}+36 x^2+85 x^3+50 x^4+e^5 (-85 x-100 x^2)) \log ^4(\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2})} \, dx\) [915]

Optimal result

Integrand size = 366, antiderivative size = 34 \[ \int \frac {300 e^{10} x+216 x^3+510 x^4+300 x^5+e^5 \left (-510 x^2-600 x^3\right )+\left (-100 e^{10} x-72 x^3-160 x^4-100 x^5+e^5 \left (180 x^2+200 x^3\right )\right ) \log \left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (100 e^{10} x+72 x^3+170 x^4+100 x^5+e^5 \left (-170 x^2-200 x^3\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )}{450 e^{10}+324 x^2+765 x^3+450 x^4+e^5 \left (-765 x-900 x^2\right )+\left (300 e^{10}+216 x^2+510 x^3+300 x^4+e^5 \left (-510 x-600 x^2\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (50 e^{10}+36 x^2+85 x^3+50 x^4+e^5 \left (-85 x-100 x^2\right )\right ) \log ^4\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )} \, dx=\frac {x^2}{3+\log ^2\left (2 x+\frac {x}{4-5 \left (\frac {e^5}{x}-x\right )}\right )} \] Output:

x^2/(3+ln(2*x+x/(4-5*exp(5)/x+5*x))^2)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.21 \[ \int \frac {300 e^{10} x+216 x^3+510 x^4+300 x^5+e^5 \left (-510 x^2-600 x^3\right )+\left (-100 e^{10} x-72 x^3-160 x^4-100 x^5+e^5 \left (180 x^2+200 x^3\right )\right ) \log \left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (100 e^{10} x+72 x^3+170 x^4+100 x^5+e^5 \left (-170 x^2-200 x^3\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )}{450 e^{10}+324 x^2+765 x^3+450 x^4+e^5 \left (-765 x-900 x^2\right )+\left (300 e^{10}+216 x^2+510 x^3+300 x^4+e^5 \left (-510 x-600 x^2\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (50 e^{10}+36 x^2+85 x^3+50 x^4+e^5 \left (-85 x-100 x^2\right )\right ) \log ^4\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )} \, dx=\frac {x^2}{3+\log ^2\left (\frac {x \left (-10 e^5+x (9+10 x)\right )}{-5 e^5+x (4+5 x)}\right )} \] Input:

Integrate[(300*E^10*x + 216*x^3 + 510*x^4 + 300*x^5 + E^5*(-510*x^2 - 600* 
x^3) + (-100*E^10*x - 72*x^3 - 160*x^4 - 100*x^5 + E^5*(180*x^2 + 200*x^3) 
)*Log[(10*E^5*x - 9*x^2 - 10*x^3)/(5*E^5 - 4*x - 5*x^2)] + (100*E^10*x + 7 
2*x^3 + 170*x^4 + 100*x^5 + E^5*(-170*x^2 - 200*x^3))*Log[(10*E^5*x - 9*x^ 
2 - 10*x^3)/(5*E^5 - 4*x - 5*x^2)]^2)/(450*E^10 + 324*x^2 + 765*x^3 + 450* 
x^4 + E^5*(-765*x - 900*x^2) + (300*E^10 + 216*x^2 + 510*x^3 + 300*x^4 + E 
^5*(-510*x - 600*x^2))*Log[(10*E^5*x - 9*x^2 - 10*x^3)/(5*E^5 - 4*x - 5*x^ 
2)]^2 + (50*E^10 + 36*x^2 + 85*x^3 + 50*x^4 + E^5*(-85*x - 100*x^2))*Log[( 
10*E^5*x - 9*x^2 - 10*x^3)/(5*E^5 - 4*x - 5*x^2)]^4),x]
 

Output:

x^2/(3 + Log[(x*(-10*E^5 + x*(9 + 10*x)))/(-5*E^5 + x*(4 + 5*x))]^2)
 

Rubi [F]

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.

\(\displaystyle \int \frac {300 x^5+510 x^4+216 x^3+e^5 \left (-600 x^3-510 x^2\right )+\left (100 x^5+170 x^4+72 x^3+e^5 \left (-200 x^3-170 x^2\right )+100 e^{10} x\right ) \log ^2\left (\frac {-10 x^3-9 x^2+10 e^5 x}{-5 x^2-4 x+5 e^5}\right )+\left (-100 x^5-160 x^4-72 x^3+e^5 \left (200 x^3+180 x^2\right )-100 e^{10} x\right ) \log \left (\frac {-10 x^3-9 x^2+10 e^5 x}{-5 x^2-4 x+5 e^5}\right )+300 e^{10} x}{450 x^4+765 x^3+324 x^2+e^5 \left (-900 x^2-765 x\right )+\left (50 x^4+85 x^3+36 x^2+e^5 \left (-100 x^2-85 x\right )+50 e^{10}\right ) \log ^4\left (\frac {-10 x^3-9 x^2+10 e^5 x}{-5 x^2-4 x+5 e^5}\right )+\left (300 x^4+510 x^3+216 x^2+e^5 \left (-600 x^2-510 x\right )+300 e^{10}\right ) \log ^2\left (\frac {-10 x^3-9 x^2+10 e^5 x}{-5 x^2-4 x+5 e^5}\right )+450 e^{10}} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {300 x^5+510 x^4+216 x^3+e^5 \left (-600 x^3-510 x^2\right )+\left (100 x^5+170 x^4+72 x^3+e^5 \left (-200 x^3-170 x^2\right )+100 e^{10} x\right ) \log ^2\left (\frac {-10 x^3-9 x^2+10 e^5 x}{-5 x^2-4 x+5 e^5}\right )+\left (-100 x^5-160 x^4-72 x^3+e^5 \left (200 x^3+180 x^2\right )-100 e^{10} x\right ) \log \left (\frac {-10 x^3-9 x^2+10 e^5 x}{-5 x^2-4 x+5 e^5}\right )+300 e^{10} x}{\left (50 x^4+85 x^3+4 \left (9-25 e^5\right ) x^2-85 e^5 x+50 e^{10}\right ) \left (\log ^2\left (\frac {x \left (x (10 x+9)-10 e^5\right )}{x (5 x+4)-5 e^5}\right )+3\right )^2}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {(-5 x-4) \left (300 x^5+510 x^4+216 x^3+e^5 \left (-600 x^3-510 x^2\right )+\left (100 x^5+170 x^4+72 x^3+e^5 \left (-200 x^3-170 x^2\right )+100 e^{10} x\right ) \log ^2\left (\frac {-10 x^3-9 x^2+10 e^5 x}{-5 x^2-4 x+5 e^5}\right )+\left (-100 x^5-160 x^4-72 x^3+e^5 \left (200 x^3+180 x^2\right )-100 e^{10} x\right ) \log \left (\frac {-10 x^3-9 x^2+10 e^5 x}{-5 x^2-4 x+5 e^5}\right )+300 e^{10} x\right )}{5 e^5 \left (-5 x^2-4 x+5 e^5\right ) \left (\log ^2\left (\frac {x \left (x (10 x+9)-10 e^5\right )}{x (5 x+4)-5 e^5}\right )+3\right )^2}-\frac {(-10 x-9) \left (300 x^5+510 x^4+216 x^3+e^5 \left (-600 x^3-510 x^2\right )+\left (100 x^5+170 x^4+72 x^3+e^5 \left (-200 x^3-170 x^2\right )+100 e^{10} x\right ) \log ^2\left (\frac {-10 x^3-9 x^2+10 e^5 x}{-5 x^2-4 x+5 e^5}\right )+\left (-100 x^5-160 x^4-72 x^3+e^5 \left (200 x^3+180 x^2\right )-100 e^{10} x\right ) \log \left (\frac {-10 x^3-9 x^2+10 e^5 x}{-5 x^2-4 x+5 e^5}\right )+300 e^{10} x\right )}{5 e^5 \left (-10 x^2-9 x+10 e^5\right ) \left (\log ^2\left (\frac {x \left (x (10 x+9)-10 e^5\right )}{x (5 x+4)-5 e^5}\right )+3\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 x \left (3 \left (\left (50 x^2+85 x+36\right ) x^2-5 e^5 (20 x+17) x+50 e^{10}\right )+\left (\left (50 x^2+85 x+36\right ) x^2-5 e^5 (20 x+17) x+50 e^{10}\right ) \log ^2\left (\frac {x \left (x (10 x+9)-10 e^5\right )}{x (5 x+4)-5 e^5}\right )-2 \left (\left (25 x^2+40 x+18\right ) x^2-5 e^5 (10 x+9) x+25 e^{10}\right ) \log \left (\frac {x \left (x (10 x+9)-10 e^5\right )}{x (5 x+4)-5 e^5}\right )\right )}{\left (-10 x^2-9 x+10 e^5\right ) \left (-5 x^2-4 x+5 e^5\right ) \left (\log ^2\left (\frac {x \left (x (10 x+9)-10 e^5\right )}{x (5 x+4)-5 e^5}\right )+3\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {x \left (\left (\left (50 x^2+85 x+36\right ) x^2-5 e^5 (20 x+17) x+50 e^{10}\right ) \log ^2\left (\frac {x \left (10 e^5-x (10 x+9)\right )}{5 e^5-x (5 x+4)}\right )-2 \left (\left (25 x^2+40 x+18\right ) x^2-5 e^5 (10 x+9) x+25 e^{10}\right ) \log \left (\frac {x \left (10 e^5-x (10 x+9)\right )}{5 e^5-x (5 x+4)}\right )+3 \left (\left (50 x^2+85 x+36\right ) x^2-5 e^5 (20 x+17) x+50 e^{10}\right )\right )}{\left (-10 x^2-9 x+10 e^5\right ) \left (-5 x^2-4 x+5 e^5\right ) \left (\log ^2\left (\frac {x \left (10 e^5-x (10 x+9)\right )}{5 e^5-x (5 x+4)}\right )+3\right )^2}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle 2 \int \left (\frac {x}{\log ^2\left (\frac {x \left (x (10 x+9)-10 e^5\right )}{x (5 x+4)-5 e^5}\right )+3}+\frac {2 \left (-25 x^4-40 x^3-2 \left (9-25 e^5\right ) x^2+45 e^5 x-25 e^{10}\right ) \log \left (\frac {x \left (10 x^2+9 x-10 e^5\right )}{5 x^2+4 x-5 e^5}\right ) x}{\left (-10 x^2-9 x+10 e^5\right ) \left (-5 x^2-4 x+5 e^5\right ) \left (\log ^2\left (\frac {x \left (x (10 x+9)-10 e^5\right )}{x (5 x+4)-5 e^5}\right )+3\right )^2}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle 2 \int \left (\frac {x}{\log ^2\left (\frac {x \left (x (10 x+9)-10 e^5\right )}{x (5 x+4)-5 e^5}\right )+3}+\frac {2 \left (-25 x^4-40 x^3-2 \left (9-25 e^5\right ) x^2+45 e^5 x-25 e^{10}\right ) \log \left (\frac {x \left (10 x^2+9 x-10 e^5\right )}{5 x^2+4 x-5 e^5}\right ) x}{\left (-10 x^2-9 x+10 e^5\right ) \left (-5 x^2-4 x+5 e^5\right ) \left (\log ^2\left (\frac {x \left (x (10 x+9)-10 e^5\right )}{x (5 x+4)-5 e^5}\right )+3\right )^2}\right )dx\)

Input:

Int[(300*E^10*x + 216*x^3 + 510*x^4 + 300*x^5 + E^5*(-510*x^2 - 600*x^3) + 
 (-100*E^10*x - 72*x^3 - 160*x^4 - 100*x^5 + E^5*(180*x^2 + 200*x^3))*Log[ 
(10*E^5*x - 9*x^2 - 10*x^3)/(5*E^5 - 4*x - 5*x^2)] + (100*E^10*x + 72*x^3 
+ 170*x^4 + 100*x^5 + E^5*(-170*x^2 - 200*x^3))*Log[(10*E^5*x - 9*x^2 - 10 
*x^3)/(5*E^5 - 4*x - 5*x^2)]^2)/(450*E^10 + 324*x^2 + 765*x^3 + 450*x^4 + 
E^5*(-765*x - 900*x^2) + (300*E^10 + 216*x^2 + 510*x^3 + 300*x^4 + E^5*(-5 
10*x - 600*x^2))*Log[(10*E^5*x - 9*x^2 - 10*x^3)/(5*E^5 - 4*x - 5*x^2)]^2 
+ (50*E^10 + 36*x^2 + 85*x^3 + 50*x^4 + E^5*(-85*x - 100*x^2))*Log[(10*E^5 
*x - 9*x^2 - 10*x^3)/(5*E^5 - 4*x - 5*x^2)]^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29

\[\frac {x^{2}}{\ln \left (\frac {10 x \,{\mathrm e}^{5}-10 x^{3}-9 x^{2}}{5 \,{\mathrm e}^{5}-5 x^{2}-4 x}\right )^{2}+3}\]

Input:

int(((100*x*exp(5)^2+(-200*x^3-170*x^2)*exp(5)+100*x^5+170*x^4+72*x^3)*ln( 
(10*x*exp(5)-10*x^3-9*x^2)/(5*exp(5)-5*x^2-4*x))^2+(-100*x*exp(5)^2+(200*x 
^3+180*x^2)*exp(5)-100*x^5-160*x^4-72*x^3)*ln((10*x*exp(5)-10*x^3-9*x^2)/( 
5*exp(5)-5*x^2-4*x))+300*x*exp(5)^2+(-600*x^3-510*x^2)*exp(5)+300*x^5+510* 
x^4+216*x^3)/((50*exp(5)^2+(-100*x^2-85*x)*exp(5)+50*x^4+85*x^3+36*x^2)*ln 
((10*x*exp(5)-10*x^3-9*x^2)/(5*exp(5)-5*x^2-4*x))^4+(300*exp(5)^2+(-600*x^ 
2-510*x)*exp(5)+300*x^4+510*x^3+216*x^2)*ln((10*x*exp(5)-10*x^3-9*x^2)/(5* 
exp(5)-5*x^2-4*x))^2+450*exp(5)^2+(-900*x^2-765*x)*exp(5)+450*x^4+765*x^3+ 
324*x^2),x)
 

Output:

x^2/(ln((10*x*exp(5)-10*x^3-9*x^2)/(5*exp(5)-5*x^2-4*x))^2+3)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26 \[ \int \frac {300 e^{10} x+216 x^3+510 x^4+300 x^5+e^5 \left (-510 x^2-600 x^3\right )+\left (-100 e^{10} x-72 x^3-160 x^4-100 x^5+e^5 \left (180 x^2+200 x^3\right )\right ) \log \left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (100 e^{10} x+72 x^3+170 x^4+100 x^5+e^5 \left (-170 x^2-200 x^3\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )}{450 e^{10}+324 x^2+765 x^3+450 x^4+e^5 \left (-765 x-900 x^2\right )+\left (300 e^{10}+216 x^2+510 x^3+300 x^4+e^5 \left (-510 x-600 x^2\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (50 e^{10}+36 x^2+85 x^3+50 x^4+e^5 \left (-85 x-100 x^2\right )\right ) \log ^4\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )} \, dx=\frac {x^{2}}{\log \left (\frac {10 \, x^{3} + 9 \, x^{2} - 10 \, x e^{5}}{5 \, x^{2} + 4 \, x - 5 \, e^{5}}\right )^{2} + 3} \] Input:

integrate(((100*x*exp(5)^2+(-200*x^3-170*x^2)*exp(5)+100*x^5+170*x^4+72*x^ 
3)*log((10*x*exp(5)-10*x^3-9*x^2)/(5*exp(5)-5*x^2-4*x))^2+(-100*x*exp(5)^2 
+(200*x^3+180*x^2)*exp(5)-100*x^5-160*x^4-72*x^3)*log((10*x*exp(5)-10*x^3- 
9*x^2)/(5*exp(5)-5*x^2-4*x))+300*x*exp(5)^2+(-600*x^3-510*x^2)*exp(5)+300* 
x^5+510*x^4+216*x^3)/((50*exp(5)^2+(-100*x^2-85*x)*exp(5)+50*x^4+85*x^3+36 
*x^2)*log((10*x*exp(5)-10*x^3-9*x^2)/(5*exp(5)-5*x^2-4*x))^4+(300*exp(5)^2 
+(-600*x^2-510*x)*exp(5)+300*x^4+510*x^3+216*x^2)*log((10*x*exp(5)-10*x^3- 
9*x^2)/(5*exp(5)-5*x^2-4*x))^2+450*exp(5)^2+(-900*x^2-765*x)*exp(5)+450*x^ 
4+765*x^3+324*x^2),x, algorithm="fricas")
 

Output:

x^2/(log((10*x^3 + 9*x^2 - 10*x*e^5)/(5*x^2 + 4*x - 5*e^5))^2 + 3)
 

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \frac {300 e^{10} x+216 x^3+510 x^4+300 x^5+e^5 \left (-510 x^2-600 x^3\right )+\left (-100 e^{10} x-72 x^3-160 x^4-100 x^5+e^5 \left (180 x^2+200 x^3\right )\right ) \log \left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (100 e^{10} x+72 x^3+170 x^4+100 x^5+e^5 \left (-170 x^2-200 x^3\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )}{450 e^{10}+324 x^2+765 x^3+450 x^4+e^5 \left (-765 x-900 x^2\right )+\left (300 e^{10}+216 x^2+510 x^3+300 x^4+e^5 \left (-510 x-600 x^2\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (50 e^{10}+36 x^2+85 x^3+50 x^4+e^5 \left (-85 x-100 x^2\right )\right ) \log ^4\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )} \, dx=\frac {x^{2}}{\log {\left (\frac {- 10 x^{3} - 9 x^{2} + 10 x e^{5}}{- 5 x^{2} - 4 x + 5 e^{5}} \right )}^{2} + 3} \] Input:

integrate(((100*x*exp(5)**2+(-200*x**3-170*x**2)*exp(5)+100*x**5+170*x**4+ 
72*x**3)*ln((10*x*exp(5)-10*x**3-9*x**2)/(5*exp(5)-5*x**2-4*x))**2+(-100*x 
*exp(5)**2+(200*x**3+180*x**2)*exp(5)-100*x**5-160*x**4-72*x**3)*ln((10*x* 
exp(5)-10*x**3-9*x**2)/(5*exp(5)-5*x**2-4*x))+300*x*exp(5)**2+(-600*x**3-5 
10*x**2)*exp(5)+300*x**5+510*x**4+216*x**3)/((50*exp(5)**2+(-100*x**2-85*x 
)*exp(5)+50*x**4+85*x**3+36*x**2)*ln((10*x*exp(5)-10*x**3-9*x**2)/(5*exp(5 
)-5*x**2-4*x))**4+(300*exp(5)**2+(-600*x**2-510*x)*exp(5)+300*x**4+510*x** 
3+216*x**2)*ln((10*x*exp(5)-10*x**3-9*x**2)/(5*exp(5)-5*x**2-4*x))**2+450* 
exp(5)**2+(-900*x**2-765*x)*exp(5)+450*x**4+765*x**3+324*x**2),x)
 

Output:

x**2/(log((-10*x**3 - 9*x**2 + 10*x*exp(5))/(-5*x**2 - 4*x + 5*exp(5)))**2 
 + 3)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (31) = 62\).

Time = 0.18 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.06 \[ \int \frac {300 e^{10} x+216 x^3+510 x^4+300 x^5+e^5 \left (-510 x^2-600 x^3\right )+\left (-100 e^{10} x-72 x^3-160 x^4-100 x^5+e^5 \left (180 x^2+200 x^3\right )\right ) \log \left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (100 e^{10} x+72 x^3+170 x^4+100 x^5+e^5 \left (-170 x^2-200 x^3\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )}{450 e^{10}+324 x^2+765 x^3+450 x^4+e^5 \left (-765 x-900 x^2\right )+\left (300 e^{10}+216 x^2+510 x^3+300 x^4+e^5 \left (-510 x-600 x^2\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (50 e^{10}+36 x^2+85 x^3+50 x^4+e^5 \left (-85 x-100 x^2\right )\right ) \log ^4\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )} \, dx=-\frac {x^{2}}{2 \, {\left (\log \left (5 \, x^{2} + 4 \, x - 5 \, e^{5}\right ) - \log \left (x\right )\right )} \log \left (10 \, x^{2} + 9 \, x - 10 \, e^{5}\right ) - \log \left (10 \, x^{2} + 9 \, x - 10 \, e^{5}\right )^{2} - \log \left (5 \, x^{2} + 4 \, x - 5 \, e^{5}\right )^{2} + 2 \, \log \left (5 \, x^{2} + 4 \, x - 5 \, e^{5}\right ) \log \left (x\right ) - \log \left (x\right )^{2} - 3} \] Input:

integrate(((100*x*exp(5)^2+(-200*x^3-170*x^2)*exp(5)+100*x^5+170*x^4+72*x^ 
3)*log((10*x*exp(5)-10*x^3-9*x^2)/(5*exp(5)-5*x^2-4*x))^2+(-100*x*exp(5)^2 
+(200*x^3+180*x^2)*exp(5)-100*x^5-160*x^4-72*x^3)*log((10*x*exp(5)-10*x^3- 
9*x^2)/(5*exp(5)-5*x^2-4*x))+300*x*exp(5)^2+(-600*x^3-510*x^2)*exp(5)+300* 
x^5+510*x^4+216*x^3)/((50*exp(5)^2+(-100*x^2-85*x)*exp(5)+50*x^4+85*x^3+36 
*x^2)*log((10*x*exp(5)-10*x^3-9*x^2)/(5*exp(5)-5*x^2-4*x))^4+(300*exp(5)^2 
+(-600*x^2-510*x)*exp(5)+300*x^4+510*x^3+216*x^2)*log((10*x*exp(5)-10*x^3- 
9*x^2)/(5*exp(5)-5*x^2-4*x))^2+450*exp(5)^2+(-900*x^2-765*x)*exp(5)+450*x^ 
4+765*x^3+324*x^2),x, algorithm="maxima")
 

Output:

-x^2/(2*(log(5*x^2 + 4*x - 5*e^5) - log(x))*log(10*x^2 + 9*x - 10*e^5) - l 
og(10*x^2 + 9*x - 10*e^5)^2 - log(5*x^2 + 4*x - 5*e^5)^2 + 2*log(5*x^2 + 4 
*x - 5*e^5)*log(x) - log(x)^2 - 3)
 

Giac [A] (verification not implemented)

Time = 2.83 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29 \[ \int \frac {300 e^{10} x+216 x^3+510 x^4+300 x^5+e^5 \left (-510 x^2-600 x^3\right )+\left (-100 e^{10} x-72 x^3-160 x^4-100 x^5+e^5 \left (180 x^2+200 x^3\right )\right ) \log \left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (100 e^{10} x+72 x^3+170 x^4+100 x^5+e^5 \left (-170 x^2-200 x^3\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )}{450 e^{10}+324 x^2+765 x^3+450 x^4+e^5 \left (-765 x-900 x^2\right )+\left (300 e^{10}+216 x^2+510 x^3+300 x^4+e^5 \left (-510 x-600 x^2\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (50 e^{10}+36 x^2+85 x^3+50 x^4+e^5 \left (-85 x-100 x^2\right )\right ) \log ^4\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )} \, dx=\frac {2 \, x^{2}}{\log \left (\frac {10 \, x^{3} + 9 \, x^{2} - 10 \, x e^{5}}{5 \, x^{2} + 4 \, x - 5 \, e^{5}}\right )^{2} + 3} \] Input:

integrate(((100*x*exp(5)^2+(-200*x^3-170*x^2)*exp(5)+100*x^5+170*x^4+72*x^ 
3)*log((10*x*exp(5)-10*x^3-9*x^2)/(5*exp(5)-5*x^2-4*x))^2+(-100*x*exp(5)^2 
+(200*x^3+180*x^2)*exp(5)-100*x^5-160*x^4-72*x^3)*log((10*x*exp(5)-10*x^3- 
9*x^2)/(5*exp(5)-5*x^2-4*x))+300*x*exp(5)^2+(-600*x^3-510*x^2)*exp(5)+300* 
x^5+510*x^4+216*x^3)/((50*exp(5)^2+(-100*x^2-85*x)*exp(5)+50*x^4+85*x^3+36 
*x^2)*log((10*x*exp(5)-10*x^3-9*x^2)/(5*exp(5)-5*x^2-4*x))^4+(300*exp(5)^2 
+(-600*x^2-510*x)*exp(5)+300*x^4+510*x^3+216*x^2)*log((10*x*exp(5)-10*x^3- 
9*x^2)/(5*exp(5)-5*x^2-4*x))^2+450*exp(5)^2+(-900*x^2-765*x)*exp(5)+450*x^ 
4+765*x^3+324*x^2),x, algorithm="giac")
 

Output:

2*x^2/(log((10*x^3 + 9*x^2 - 10*x*e^5)/(5*x^2 + 4*x - 5*e^5))^2 + 3)
 

Mupad [B] (verification not implemented)

Time = 9.92 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26 \[ \int \frac {300 e^{10} x+216 x^3+510 x^4+300 x^5+e^5 \left (-510 x^2-600 x^3\right )+\left (-100 e^{10} x-72 x^3-160 x^4-100 x^5+e^5 \left (180 x^2+200 x^3\right )\right ) \log \left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (100 e^{10} x+72 x^3+170 x^4+100 x^5+e^5 \left (-170 x^2-200 x^3\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )}{450 e^{10}+324 x^2+765 x^3+450 x^4+e^5 \left (-765 x-900 x^2\right )+\left (300 e^{10}+216 x^2+510 x^3+300 x^4+e^5 \left (-510 x-600 x^2\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (50 e^{10}+36 x^2+85 x^3+50 x^4+e^5 \left (-85 x-100 x^2\right )\right ) \log ^4\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )} \, dx=\frac {x^2}{{\ln \left (\frac {10\,x^3+9\,x^2-10\,{\mathrm {e}}^5\,x}{5\,x^2+4\,x-5\,{\mathrm {e}}^5}\right )}^2+3} \] Input:

int((300*x*exp(10) - exp(5)*(510*x^2 + 600*x^3) - log((9*x^2 - 10*x*exp(5) 
 + 10*x^3)/(4*x - 5*exp(5) + 5*x^2))*(100*x*exp(10) - exp(5)*(180*x^2 + 20 
0*x^3) + 72*x^3 + 160*x^4 + 100*x^5) + 216*x^3 + 510*x^4 + 300*x^5 + log(( 
9*x^2 - 10*x*exp(5) + 10*x^3)/(4*x - 5*exp(5) + 5*x^2))^2*(100*x*exp(10) - 
 exp(5)*(170*x^2 + 200*x^3) + 72*x^3 + 170*x^4 + 100*x^5))/(450*exp(10) + 
log((9*x^2 - 10*x*exp(5) + 10*x^3)/(4*x - 5*exp(5) + 5*x^2))^4*(50*exp(10) 
 - exp(5)*(85*x + 100*x^2) + 36*x^2 + 85*x^3 + 50*x^4) + log((9*x^2 - 10*x 
*exp(5) + 10*x^3)/(4*x - 5*exp(5) + 5*x^2))^2*(300*exp(10) - exp(5)*(510*x 
 + 600*x^2) + 216*x^2 + 510*x^3 + 300*x^4) - exp(5)*(765*x + 900*x^2) + 32 
4*x^2 + 765*x^3 + 450*x^4),x)
 

Output:

x^2/(log((9*x^2 - 10*x*exp(5) + 10*x^3)/(4*x - 5*exp(5) + 5*x^2))^2 + 3)
 

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.32 \[ \int \frac {300 e^{10} x+216 x^3+510 x^4+300 x^5+e^5 \left (-510 x^2-600 x^3\right )+\left (-100 e^{10} x-72 x^3-160 x^4-100 x^5+e^5 \left (180 x^2+200 x^3\right )\right ) \log \left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (100 e^{10} x+72 x^3+170 x^4+100 x^5+e^5 \left (-170 x^2-200 x^3\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )}{450 e^{10}+324 x^2+765 x^3+450 x^4+e^5 \left (-765 x-900 x^2\right )+\left (300 e^{10}+216 x^2+510 x^3+300 x^4+e^5 \left (-510 x-600 x^2\right )\right ) \log ^2\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )+\left (50 e^{10}+36 x^2+85 x^3+50 x^4+e^5 \left (-85 x-100 x^2\right )\right ) \log ^4\left (\frac {10 e^5 x-9 x^2-10 x^3}{5 e^5-4 x-5 x^2}\right )} \, dx=\frac {x^{2}}{\mathrm {log}\left (\frac {10 e^{5} x -10 x^{3}-9 x^{2}}{5 e^{5}-5 x^{2}-4 x}\right )^{2}+3} \] Input:

int(((100*x*exp(5)^2+(-200*x^3-170*x^2)*exp(5)+100*x^5+170*x^4+72*x^3)*log 
((10*x*exp(5)-10*x^3-9*x^2)/(5*exp(5)-5*x^2-4*x))^2+(-100*x*exp(5)^2+(200* 
x^3+180*x^2)*exp(5)-100*x^5-160*x^4-72*x^3)*log((10*x*exp(5)-10*x^3-9*x^2) 
/(5*exp(5)-5*x^2-4*x))+300*x*exp(5)^2+(-600*x^3-510*x^2)*exp(5)+300*x^5+51 
0*x^4+216*x^3)/((50*exp(5)^2+(-100*x^2-85*x)*exp(5)+50*x^4+85*x^3+36*x^2)* 
log((10*x*exp(5)-10*x^3-9*x^2)/(5*exp(5)-5*x^2-4*x))^4+(300*exp(5)^2+(-600 
*x^2-510*x)*exp(5)+300*x^4+510*x^3+216*x^2)*log((10*x*exp(5)-10*x^3-9*x^2) 
/(5*exp(5)-5*x^2-4*x))^2+450*exp(5)^2+(-900*x^2-765*x)*exp(5)+450*x^4+765* 
x^3+324*x^2),x)
 

Output:

x**2/(log((10*e**5*x - 10*x**3 - 9*x**2)/(5*e**5 - 5*x**2 - 4*x))**2 + 3)