Integrand size = 198, antiderivative size = 30 \[ \int \frac {5^{\frac {5 x^2}{(-80+25 x) \log (5)+e^x \left (5 x^3+\left (3 x^2+5 x^3\right ) \log (5)\right )}} \left (\left (-800 x+125 x^2\right ) \log ^2(5)+e^x \left (\left (-25 x^4-25 x^5\right ) \log (5)+\left (-40 x^4-25 x^5\right ) \log ^2(5)\right )\right )}{\left (6400-4000 x+625 x^2\right ) \log ^2(5)+e^x \left (\left (-800 x^3+250 x^4\right ) \log (5)+\left (-480 x^2-650 x^3+250 x^4\right ) \log ^2(5)\right )+e^{2 x} \left (25 x^6+\left (30 x^5+50 x^6\right ) \log (5)+\left (9 x^4+30 x^5+25 x^6\right ) \log ^2(5)\right )} \, dx=e^{\frac {x^2}{-16+x \left (5+e^x x \left (\frac {3}{5}+x+\frac {x}{\log (5)}\right )\right )}} \] Output:
exp(x^2/((x*(x+x/ln(5)+3/5)*exp(x)+5)*x-16))
\[ \int \frac {5^{\frac {5 x^2}{(-80+25 x) \log (5)+e^x \left (5 x^3+\left (3 x^2+5 x^3\right ) \log (5)\right )}} \left (\left (-800 x+125 x^2\right ) \log ^2(5)+e^x \left (\left (-25 x^4-25 x^5\right ) \log (5)+\left (-40 x^4-25 x^5\right ) \log ^2(5)\right )\right )}{\left (6400-4000 x+625 x^2\right ) \log ^2(5)+e^x \left (\left (-800 x^3+250 x^4\right ) \log (5)+\left (-480 x^2-650 x^3+250 x^4\right ) \log ^2(5)\right )+e^{2 x} \left (25 x^6+\left (30 x^5+50 x^6\right ) \log (5)+\left (9 x^4+30 x^5+25 x^6\right ) \log ^2(5)\right )} \, dx=\int \frac {5^{\frac {5 x^2}{(-80+25 x) \log (5)+e^x \left (5 x^3+\left (3 x^2+5 x^3\right ) \log (5)\right )}} \left (\left (-800 x+125 x^2\right ) \log ^2(5)+e^x \left (\left (-25 x^4-25 x^5\right ) \log (5)+\left (-40 x^4-25 x^5\right ) \log ^2(5)\right )\right )}{\left (6400-4000 x+625 x^2\right ) \log ^2(5)+e^x \left (\left (-800 x^3+250 x^4\right ) \log (5)+\left (-480 x^2-650 x^3+250 x^4\right ) \log ^2(5)\right )+e^{2 x} \left (25 x^6+\left (30 x^5+50 x^6\right ) \log (5)+\left (9 x^4+30 x^5+25 x^6\right ) \log ^2(5)\right )} \, dx \] Input:
Integrate[(5^((5*x^2)/((-80 + 25*x)*Log[5] + E^x*(5*x^3 + (3*x^2 + 5*x^3)* Log[5])))*((-800*x + 125*x^2)*Log[5]^2 + E^x*((-25*x^4 - 25*x^5)*Log[5] + (-40*x^4 - 25*x^5)*Log[5]^2)))/((6400 - 4000*x + 625*x^2)*Log[5]^2 + E^x*( (-800*x^3 + 250*x^4)*Log[5] + (-480*x^2 - 650*x^3 + 250*x^4)*Log[5]^2) + E ^(2*x)*(25*x^6 + (30*x^5 + 50*x^6)*Log[5] + (9*x^4 + 30*x^5 + 25*x^6)*Log[ 5]^2)),x]
Output:
Integrate[(5^((5*x^2)/((-80 + 25*x)*Log[5] + E^x*(5*x^3 + (3*x^2 + 5*x^3)* Log[5])))*((-800*x + 125*x^2)*Log[5]^2 + E^x*((-25*x^4 - 25*x^5)*Log[5] + (-40*x^4 - 25*x^5)*Log[5]^2)))/((6400 - 4000*x + 625*x^2)*Log[5]^2 + E^x*( (-800*x^3 + 250*x^4)*Log[5] + (-480*x^2 - 650*x^3 + 250*x^4)*Log[5]^2) + E ^(2*x)*(25*x^6 + (30*x^5 + 50*x^6)*Log[5] + (9*x^4 + 30*x^5 + 25*x^6)*Log[ 5]^2)), x]
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.
| \(\displaystyle \int \frac {5^{\frac {5 x^2}{e^x \left (5 x^3+\left (5 x^3+3 x^2\right ) \log (5)\right )+(25 x-80) \log (5)}} \left (\left (125 x^2-800 x\right ) \log ^2(5)+e^x \left (\left (-25 x^5-40 x^4\right ) \log ^2(5)+\left (-25 x^5-25 x^4\right ) \log (5)\right )\right )}{\left (625 x^2-4000 x+6400\right ) \log ^2(5)+e^{2 x} \left (25 x^6+\left (50 x^6+30 x^5\right ) \log (5)+\left (25 x^6+30 x^5+9 x^4\right ) \log ^2(5)\right )+e^x \left (\left (250 x^4-800 x^3\right ) \log (5)+\left (250 x^4-650 x^3-480 x^2\right ) \log ^2(5)\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \log (5) 5^{\frac {5 x^2}{e^x x^2 (5 x (1+\log (5))+\log (125))+5 (5 x-16) \log (5)}+1} \left (-e^x x^3 (5 x (1+\log (5))+5+8 \log (5))-5 (32-5 x) \log (5)\right )}{\left (e^x x^2 (5 x (1+\log (5))+\log (125))+5 (5 x-16) \log (5)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \log (5) \int -\frac {5^{1-\frac {5 x^2}{5 (16-5 x) \log (5)-e^x x^2 (5 (1+\log (5)) x+\log (125))}} x \left (e^x (5 (1+\log (5)) x+8 \log (5)+5) x^3+5 (32-5 x) \log (5)\right )}{\left (5 (16-5 x) \log (5)-e^x x^2 (5 (1+\log (5)) x+\log (125))\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\log (5) \int \frac {5^{1-\frac {5 x^2}{5 (16-5 x) \log (5)-e^x x^2 (5 (1+\log (5)) x+\log (125))}} x \left (e^x (5 (1+\log (5)) x+8 \log (5)+5) x^3+5 (32-5 x) \log (5)\right )}{\left (5 (16-5 x) \log (5)-e^x x^2 (5 (1+\log (5)) x+\log (125))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {5^{1-\frac {5 x^2}{5 (16-5 x) \log (5)-e^x x^2 (5 (1+\log (5)) x+\log (125))}} (-5 (1+\log (5)) x-8 \log (5)-5) x^2}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )}+\frac {5^{2-\frac {5 x^2}{5 (16-5 x) \log (5)-e^x x^2 (5 (1+\log (5)) x+\log (125))}} \log (5) \left (-25 (1+\log (5)) x^3+15 (2+\log (5)) x^2+(288 \log (5)+5 (48-\log (125))) x+32 \log (125)\right ) x}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} x \left (e^x (5 (1+\log (5)) x+8 \log (5)+5) x^3+5 (32-5 x) \log (5)\right )}{\left (e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} (-5 (1+\log (5)) x-8 \log (5)-5) x^2}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )}+\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+2} \log (5) \left (-25 (1+\log (5)) x^3+15 (2+\log (5)) x^2+(288 \log (5)+5 (48-\log (125))) x+32 \log (125)\right ) x}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} x \left (e^x (5 (1+\log (5)) x+8 \log (5)+5) x^3+5 (32-5 x) \log (5)\right )}{\left (e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} (-5 (1+\log (5)) x-8 \log (5)-5) x^2}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )}+\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+2} \log (5) \left (-25 (1+\log (5)) x^3+15 (2+\log (5)) x^2+(288 \log (5)+5 (48-\log (125))) x+32 \log (125)\right ) x}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} x \left (e^x (5 (1+\log (5)) x+8 \log (5)+5) x^3+5 (32-5 x) \log (5)\right )}{\left (e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} (-5 (1+\log (5)) x-8 \log (5)-5) x^2}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )}+\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+2} \log (5) \left (-25 (1+\log (5)) x^3+15 (2+\log (5)) x^2+(288 \log (5)+5 (48-\log (125))) x+32 \log (125)\right ) x}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} x \left (e^x (5 (1+\log (5)) x+8 \log (5)+5) x^3+5 (32-5 x) \log (5)\right )}{\left (e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} (-5 (1+\log (5)) x-8 \log (5)-5) x^2}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )}+\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+2} \log (5) \left (-25 (1+\log (5)) x^3+15 (2+\log (5)) x^2+(288 \log (5)+5 (48-\log (125))) x+32 \log (125)\right ) x}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} x \left (e^x (5 (1+\log (5)) x+8 \log (5)+5) x^3+5 (32-5 x) \log (5)\right )}{\left (e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} (-5 (1+\log (5)) x-8 \log (5)-5) x^2}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )}+\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+2} \log (5) \left (-25 (1+\log (5)) x^3+15 (2+\log (5)) x^2+(288 \log (5)+5 (48-\log (125))) x+32 \log (125)\right ) x}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} x \left (e^x (5 (1+\log (5)) x+8 \log (5)+5) x^3+5 (32-5 x) \log (5)\right )}{\left (e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} (-5 (1+\log (5)) x-8 \log (5)-5) x^2}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )}+\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+2} \log (5) \left (-25 (1+\log (5)) x^3+15 (2+\log (5)) x^2+(288 \log (5)+5 (48-\log (125))) x+32 \log (125)\right ) x}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} x \left (e^x (5 (1+\log (5)) x+8 \log (5)+5) x^3+5 (32-5 x) \log (5)\right )}{\left (e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} (-5 (1+\log (5)) x-8 \log (5)-5) x^2}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )}+\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+2} \log (5) \left (-25 (1+\log (5)) x^3+15 (2+\log (5)) x^2+(288 \log (5)+5 (48-\log (125))) x+32 \log (125)\right ) x}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} x \left (e^x (5 (1+\log (5)) x+8 \log (5)+5) x^3+5 (32-5 x) \log (5)\right )}{\left (e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} (-5 (1+\log (5)) x-8 \log (5)-5) x^2}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )}+\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+2} \log (5) \left (-25 (1+\log (5)) x^3+15 (2+\log (5)) x^2+(288 \log (5)+5 (48-\log (125))) x+32 \log (125)\right ) x}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} x \left (e^x (5 (1+\log (5)) x+8 \log (5)+5) x^3+5 (32-5 x) \log (5)\right )}{\left (e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} (-5 (1+\log (5)) x-8 \log (5)-5) x^2}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )}+\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+2} \log (5) \left (-25 (1+\log (5)) x^3+15 (2+\log (5)) x^2+(288 \log (5)+5 (48-\log (125))) x+32 \log (125)\right ) x}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} x \left (e^x (5 (1+\log (5)) x+8 \log (5)+5) x^3+5 (32-5 x) \log (5)\right )}{\left (e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} (-5 (1+\log (5)) x-8 \log (5)-5) x^2}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )}+\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+2} \log (5) \left (-25 (1+\log (5)) x^3+15 (2+\log (5)) x^2+(288 \log (5)+5 (48-\log (125))) x+32 \log (125)\right ) x}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} x \left (e^x (5 (1+\log (5)) x+8 \log (5)+5) x^3+5 (32-5 x) \log (5)\right )}{\left (e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} (-5 (1+\log (5)) x-8 \log (5)-5) x^2}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )}+\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+2} \log (5) \left (-25 (1+\log (5)) x^3+15 (2+\log (5)) x^2+(288 \log (5)+5 (48-\log (125))) x+32 \log (125)\right ) x}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} x \left (e^x (5 (1+\log (5)) x+8 \log (5)+5) x^3+5 (32-5 x) \log (5)\right )}{\left (e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} (-5 (1+\log (5)) x-8 \log (5)-5) x^2}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )}+\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+2} \log (5) \left (-25 (1+\log (5)) x^3+15 (2+\log (5)) x^2+(288 \log (5)+5 (48-\log (125))) x+32 \log (125)\right ) x}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\log (5) \int \frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} x \left (e^x (5 (1+\log (5)) x+8 \log (5)+5) x^3+5 (32-5 x) \log (5)\right )}{\left (e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\log (5) \int \left (\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+1} (-5 (1+\log (5)) x-8 \log (5)-5) x^2}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )}+\frac {5^{\frac {5 x^2}{e^x (5 (1+\log (5)) x+\log (125)) x^2+5 (5 x-16) \log (5)}+2} \log (5) \left (-25 (1+\log (5)) x^3+15 (2+\log (5)) x^2+(288 \log (5)+5 (48-\log (125))) x+32 \log (125)\right ) x}{(5 (1+\log (5)) x+\log (125)) \left (-5 e^x (1+\log (5)) x^3-e^x \log (125) x^2-25 \log (5) x+80 \log (5)\right )^2}\right )dx\) |
Input:
Int[(5^((5*x^2)/((-80 + 25*x)*Log[5] + E^x*(5*x^3 + (3*x^2 + 5*x^3)*Log[5] )))*((-800*x + 125*x^2)*Log[5]^2 + E^x*((-25*x^4 - 25*x^5)*Log[5] + (-40*x ^4 - 25*x^5)*Log[5]^2)))/((6400 - 4000*x + 625*x^2)*Log[5]^2 + E^x*((-800* x^3 + 250*x^4)*Log[5] + (-480*x^2 - 650*x^3 + 250*x^4)*Log[5]^2) + E^(2*x) *(25*x^6 + (30*x^5 + 50*x^6)*Log[5] + (9*x^4 + 30*x^5 + 25*x^6)*Log[5]^2)) ,x]
Output:
$Aborted
Time = 43.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47
| method | result | size |
| risch | \(3125^{\frac {x^{2}}{5 \,{\mathrm e}^{x} \ln \left (5\right ) x^{3}+3 \,{\mathrm e}^{x} \ln \left (5\right ) x^{2}+5 \,{\mathrm e}^{x} x^{3}+25 x \ln \left (5\right )-80 \ln \left (5\right )}}\) | \(44\) |
| parallelrisch | \({\mathrm e}^{\frac {5 x^{2} \ln \left (5\right )}{5 \,{\mathrm e}^{x} \ln \left (5\right ) x^{3}+3 \,{\mathrm e}^{x} \ln \left (5\right ) x^{2}+5 \,{\mathrm e}^{x} x^{3}+25 x \ln \left (5\right )-80 \ln \left (5\right )}}\) | \(46\) |
Input:
int((((-25*x^5-40*x^4)*ln(5)^2+(-25*x^5-25*x^4)*ln(5))*exp(x)+(125*x^2-800 *x)*ln(5)^2)*exp(5*x^2*ln(5)/(((5*x^3+3*x^2)*ln(5)+5*x^3)*exp(x)+(25*x-80) *ln(5)))/(((25*x^6+30*x^5+9*x^4)*ln(5)^2+(50*x^6+30*x^5)*ln(5)+25*x^6)*exp (x)^2+((250*x^4-650*x^3-480*x^2)*ln(5)^2+(250*x^4-800*x^3)*ln(5))*exp(x)+( 625*x^2-4000*x+6400)*ln(5)^2),x,method=_RETURNVERBOSE)
Output:
3125^(x^2/(5*exp(x)*ln(5)*x^3+3*exp(x)*ln(5)*x^2+5*exp(x)*x^3+25*x*ln(5)-8 0*ln(5)))
Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {5^{\frac {5 x^2}{(-80+25 x) \log (5)+e^x \left (5 x^3+\left (3 x^2+5 x^3\right ) \log (5)\right )}} \left (\left (-800 x+125 x^2\right ) \log ^2(5)+e^x \left (\left (-25 x^4-25 x^5\right ) \log (5)+\left (-40 x^4-25 x^5\right ) \log ^2(5)\right )\right )}{\left (6400-4000 x+625 x^2\right ) \log ^2(5)+e^x \left (\left (-800 x^3+250 x^4\right ) \log (5)+\left (-480 x^2-650 x^3+250 x^4\right ) \log ^2(5)\right )+e^{2 x} \left (25 x^6+\left (30 x^5+50 x^6\right ) \log (5)+\left (9 x^4+30 x^5+25 x^6\right ) \log ^2(5)\right )} \, dx=5^{\frac {5 \, x^{2}}{{\left (5 \, x^{3} + {\left (5 \, x^{3} + 3 \, x^{2}\right )} \log \left (5\right )\right )} e^{x} + 5 \, {\left (5 \, x - 16\right )} \log \left (5\right )}} \] Input:
integrate((((-25*x^5-40*x^4)*log(5)^2+(-25*x^5-25*x^4)*log(5))*exp(x)+(125 *x^2-800*x)*log(5)^2)*exp(5*x^2*log(5)/(((5*x^3+3*x^2)*log(5)+5*x^3)*exp(x )+(25*x-80)*log(5)))/(((25*x^6+30*x^5+9*x^4)*log(5)^2+(50*x^6+30*x^5)*log( 5)+25*x^6)*exp(x)^2+((250*x^4-650*x^3-480*x^2)*log(5)^2+(250*x^4-800*x^3)* log(5))*exp(x)+(625*x^2-4000*x+6400)*log(5)^2),x, algorithm="fricas")
Output:
5^(5*x^2/((5*x^3 + (5*x^3 + 3*x^2)*log(5))*e^x + 5*(5*x - 16)*log(5)))
Time = 0.62 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {5^{\frac {5 x^2}{(-80+25 x) \log (5)+e^x \left (5 x^3+\left (3 x^2+5 x^3\right ) \log (5)\right )}} \left (\left (-800 x+125 x^2\right ) \log ^2(5)+e^x \left (\left (-25 x^4-25 x^5\right ) \log (5)+\left (-40 x^4-25 x^5\right ) \log ^2(5)\right )\right )}{\left (6400-4000 x+625 x^2\right ) \log ^2(5)+e^x \left (\left (-800 x^3+250 x^4\right ) \log (5)+\left (-480 x^2-650 x^3+250 x^4\right ) \log ^2(5)\right )+e^{2 x} \left (25 x^6+\left (30 x^5+50 x^6\right ) \log (5)+\left (9 x^4+30 x^5+25 x^6\right ) \log ^2(5)\right )} \, dx=e^{\frac {5 x^{2} \log {\left (5 \right )}}{\left (25 x - 80\right ) \log {\left (5 \right )} + \left (5 x^{3} + \left (5 x^{3} + 3 x^{2}\right ) \log {\left (5 \right )}\right ) e^{x}}} \] Input:
integrate((((-25*x**5-40*x**4)*ln(5)**2+(-25*x**5-25*x**4)*ln(5))*exp(x)+( 125*x**2-800*x)*ln(5)**2)*exp(5*x**2*ln(5)/(((5*x**3+3*x**2)*ln(5)+5*x**3) *exp(x)+(25*x-80)*ln(5)))/(((25*x**6+30*x**5+9*x**4)*ln(5)**2+(50*x**6+30* x**5)*ln(5)+25*x**6)*exp(x)**2+((250*x**4-650*x**3-480*x**2)*ln(5)**2+(250 *x**4-800*x**3)*ln(5))*exp(x)+(625*x**2-4000*x+6400)*ln(5)**2),x)
Output:
exp(5*x**2*log(5)/((25*x - 80)*log(5) + (5*x**3 + (5*x**3 + 3*x**2)*log(5) )*exp(x)))
Time = 0.37 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {5^{\frac {5 x^2}{(-80+25 x) \log (5)+e^x \left (5 x^3+\left (3 x^2+5 x^3\right ) \log (5)\right )}} \left (\left (-800 x+125 x^2\right ) \log ^2(5)+e^x \left (\left (-25 x^4-25 x^5\right ) \log (5)+\left (-40 x^4-25 x^5\right ) \log ^2(5)\right )\right )}{\left (6400-4000 x+625 x^2\right ) \log ^2(5)+e^x \left (\left (-800 x^3+250 x^4\right ) \log (5)+\left (-480 x^2-650 x^3+250 x^4\right ) \log ^2(5)\right )+e^{2 x} \left (25 x^6+\left (30 x^5+50 x^6\right ) \log (5)+\left (9 x^4+30 x^5+25 x^6\right ) \log ^2(5)\right )} \, dx=5^{\frac {5 \, x^{2}}{{\left (5 \, x^{3} {\left (\log \left (5\right ) + 1\right )} + 3 \, x^{2} \log \left (5\right )\right )} e^{x} + 25 \, x \log \left (5\right ) - 80 \, \log \left (5\right )}} \] Input:
integrate((((-25*x^5-40*x^4)*log(5)^2+(-25*x^5-25*x^4)*log(5))*exp(x)+(125 *x^2-800*x)*log(5)^2)*exp(5*x^2*log(5)/(((5*x^3+3*x^2)*log(5)+5*x^3)*exp(x )+(25*x-80)*log(5)))/(((25*x^6+30*x^5+9*x^4)*log(5)^2+(50*x^6+30*x^5)*log( 5)+25*x^6)*exp(x)^2+((250*x^4-650*x^3-480*x^2)*log(5)^2+(250*x^4-800*x^3)* log(5))*exp(x)+(625*x^2-4000*x+6400)*log(5)^2),x, algorithm="maxima")
Output:
5^(5*x^2/((5*x^3*(log(5) + 1) + 3*x^2*log(5))*e^x + 25*x*log(5) - 80*log(5 )))
Time = 0.15 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {5^{\frac {5 x^2}{(-80+25 x) \log (5)+e^x \left (5 x^3+\left (3 x^2+5 x^3\right ) \log (5)\right )}} \left (\left (-800 x+125 x^2\right ) \log ^2(5)+e^x \left (\left (-25 x^4-25 x^5\right ) \log (5)+\left (-40 x^4-25 x^5\right ) \log ^2(5)\right )\right )}{\left (6400-4000 x+625 x^2\right ) \log ^2(5)+e^x \left (\left (-800 x^3+250 x^4\right ) \log (5)+\left (-480 x^2-650 x^3+250 x^4\right ) \log ^2(5)\right )+e^{2 x} \left (25 x^6+\left (30 x^5+50 x^6\right ) \log (5)+\left (9 x^4+30 x^5+25 x^6\right ) \log ^2(5)\right )} \, dx=5^{\frac {5 \, x^{2}}{5 \, x^{3} e^{x} \log \left (5\right ) + 5 \, x^{3} e^{x} + 3 \, x^{2} e^{x} \log \left (5\right ) + 25 \, x \log \left (5\right ) - 80 \, \log \left (5\right )}} \] Input:
integrate((((-25*x^5-40*x^4)*log(5)^2+(-25*x^5-25*x^4)*log(5))*exp(x)+(125 *x^2-800*x)*log(5)^2)*exp(5*x^2*log(5)/(((5*x^3+3*x^2)*log(5)+5*x^3)*exp(x )+(25*x-80)*log(5)))/(((25*x^6+30*x^5+9*x^4)*log(5)^2+(50*x^6+30*x^5)*log( 5)+25*x^6)*exp(x)^2+((250*x^4-650*x^3-480*x^2)*log(5)^2+(250*x^4-800*x^3)* log(5))*exp(x)+(625*x^2-4000*x+6400)*log(5)^2),x, algorithm="giac")
Output:
5^(5*x^2/(5*x^3*e^x*log(5) + 5*x^3*e^x + 3*x^2*e^x*log(5) + 25*x*log(5) - 80*log(5)))
Time = 8.49 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.50 \[ \int \frac {5^{\frac {5 x^2}{(-80+25 x) \log (5)+e^x \left (5 x^3+\left (3 x^2+5 x^3\right ) \log (5)\right )}} \left (\left (-800 x+125 x^2\right ) \log ^2(5)+e^x \left (\left (-25 x^4-25 x^5\right ) \log (5)+\left (-40 x^4-25 x^5\right ) \log ^2(5)\right )\right )}{\left (6400-4000 x+625 x^2\right ) \log ^2(5)+e^x \left (\left (-800 x^3+250 x^4\right ) \log (5)+\left (-480 x^2-650 x^3+250 x^4\right ) \log ^2(5)\right )+e^{2 x} \left (25 x^6+\left (30 x^5+50 x^6\right ) \log (5)+\left (9 x^4+30 x^5+25 x^6\right ) \log ^2(5)\right )} \, dx={\mathrm {e}}^{\frac {5\,x^2\,\ln \left (5\right )}{5\,x^3\,{\mathrm {e}}^x-80\,\ln \left (5\right )+25\,x\,\ln \left (5\right )+3\,x^2\,{\mathrm {e}}^x\,\ln \left (5\right )+5\,x^3\,{\mathrm {e}}^x\,\ln \left (5\right )}} \] Input:
int(-(exp((5*x^2*log(5))/(log(5)*(25*x - 80) + exp(x)*(log(5)*(3*x^2 + 5*x ^3) + 5*x^3)))*(exp(x)*(log(5)*(25*x^4 + 25*x^5) + log(5)^2*(40*x^4 + 25*x ^5)) + log(5)^2*(800*x - 125*x^2)))/(log(5)^2*(625*x^2 - 4000*x + 6400) - exp(x)*(log(5)^2*(480*x^2 + 650*x^3 - 250*x^4) + log(5)*(800*x^3 - 250*x^4 )) + exp(2*x)*(log(5)^2*(9*x^4 + 30*x^5 + 25*x^6) + log(5)*(30*x^5 + 50*x^ 6) + 25*x^6)),x)
Output:
exp((5*x^2*log(5))/(5*x^3*exp(x) - 80*log(5) + 25*x*log(5) + 3*x^2*exp(x)* log(5) + 5*x^3*exp(x)*log(5)))
Time = 0.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {5^{\frac {5 x^2}{(-80+25 x) \log (5)+e^x \left (5 x^3+\left (3 x^2+5 x^3\right ) \log (5)\right )}} \left (\left (-800 x+125 x^2\right ) \log ^2(5)+e^x \left (\left (-25 x^4-25 x^5\right ) \log (5)+\left (-40 x^4-25 x^5\right ) \log ^2(5)\right )\right )}{\left (6400-4000 x+625 x^2\right ) \log ^2(5)+e^x \left (\left (-800 x^3+250 x^4\right ) \log (5)+\left (-480 x^2-650 x^3+250 x^4\right ) \log ^2(5)\right )+e^{2 x} \left (25 x^6+\left (30 x^5+50 x^6\right ) \log (5)+\left (9 x^4+30 x^5+25 x^6\right ) \log ^2(5)\right )} \, dx=e^{\frac {5 \,\mathrm {log}\left (5\right ) x^{2}}{5 e^{x} \mathrm {log}\left (5\right ) x^{3}+3 e^{x} \mathrm {log}\left (5\right ) x^{2}+5 e^{x} x^{3}+25 \,\mathrm {log}\left (5\right ) x -80 \,\mathrm {log}\left (5\right )}} \] Input:
int((((-25*x^5-40*x^4)*log(5)^2+(-25*x^5-25*x^4)*log(5))*exp(x)+(125*x^2-8 00*x)*log(5)^2)*exp(5*x^2*log(5)/(((5*x^3+3*x^2)*log(5)+5*x^3)*exp(x)+(25* x-80)*log(5)))/(((25*x^6+30*x^5+9*x^4)*log(5)^2+(50*x^6+30*x^5)*log(5)+25* x^6)*exp(x)^2+((250*x^4-650*x^3-480*x^2)*log(5)^2+(250*x^4-800*x^3)*log(5) )*exp(x)+(625*x^2-4000*x+6400)*log(5)^2),x)
Output:
e**((5*log(5)*x**2)/(5*e**x*log(5)*x**3 + 3*e**x*log(5)*x**2 + 5*e**x*x**3 + 25*log(5)*x - 80*log(5)))