Integrand size = 134, antiderivative size = 26 \[ \int \frac {-15 e^2+e^{\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} \left (-60 e^2-20 x+30 e^2 \log (5)\right )}{3 e^{2+\frac {4 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}}+e^{2+\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} (-6+6 x)+e^2 \left (3-6 x+3 x^2\right )} \, dx=\frac {5}{-1+e^{2 x \left (2+\frac {x}{3 e^2}-\log (5)\right )}+x} \] Output:
5/(x+exp((1/3*x/exp(2)-ln(5)+2)*x)^2-1)
Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-15 e^2+e^{\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} \left (-60 e^2-20 x+30 e^2 \log (5)\right )}{3 e^{2+\frac {4 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}}+e^{2+\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} (-6+6 x)+e^2 \left (3-6 x+3 x^2\right )} \, dx=\frac {5^{1+2 x}}{e^{\frac {2}{3} x \left (6+\frac {x}{e^2}\right )}+25^x (-1+x)} \] Input:
Integrate[(-15*E^2 + E^((2*(6*E^2*x + x^2 - 3*E^2*x*Log[5]))/(3*E^2))*(-60 *E^2 - 20*x + 30*E^2*Log[5]))/(3*E^(2 + (4*(6*E^2*x + x^2 - 3*E^2*x*Log[5] ))/(3*E^2)) + E^(2 + (2*(6*E^2*x + x^2 - 3*E^2*x*Log[5]))/(3*E^2))*(-6 + 6 *x) + E^2*(3 - 6*x + 3*x^2)),x]
Output:
5^(1 + 2*x)/(E^((2*x*(6 + x/E^2))/3) + 25^x*(-1 + 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.
| \(\displaystyle \int \frac {e^{\frac {2 \left (x^2+6 e^2 x-3 e^2 x \log (5)\right )}{3 e^2}} \left (-20 x-60 e^2+30 e^2 \log (5)\right )-15 e^2}{(6 x-6) \exp \left (\frac {2 \left (x^2+6 e^2 x-3 e^2 x \log (5)\right )}{3 e^2}+2\right )+3 \exp \left (\frac {4 \left (x^2+6 e^2 x-3 e^2 x \log (5)\right )}{3 e^2}+2\right )+e^2 \left (3 x^2-6 x+3\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {25^{2 x} \left (e^{\frac {2 \left (x^2+6 e^2 x-3 e^2 x \log (5)\right )}{3 e^2}} \left (-20 x-60 e^2+30 e^2 \log (5)\right )-15 e^2\right )}{3 e^2 \left (-25^x x+25^x-e^{\frac {2}{3} x \left (\frac {x}{e^2}+6\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {25^{2 x} \left (2\ 5^{1-2 x} e^{\frac {2 \left (x^2+6 e^2 x\right )}{3 e^2}} \left (2 x+3 e^2 (2-\log (5))\right )+15 e^2\right )}{\left (-25^x x+25^x-e^{\frac {2 x \left (x+6 e^2\right )}{3 e^2}}\right )^2}dx}{3 e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {25^{2 x} \left (2\ 5^{1-2 x} e^{\frac {2 \left (x^2+6 e^2 x\right )}{3 e^2}} \left (2 x+3 e^2 (2-\log (5))\right )+15 e^2\right )}{\left (-25^x x+25^x-e^{\frac {2 x \left (x+6 e^2\right )}{3 e^2}}\right )^2}dx}{3 e^2}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\int \left (\frac {4\ 5^{2 x+1} e^{\frac {2}{3} x \left (\frac {x}{e^2}+6\right )} x}{\left (5^{2 x} x-5^{2 x}+e^{\frac {2}{3} x \left (\frac {x}{e^2}+6\right )}\right )^2}+\frac {3\ 5^{4 x+1} e^2}{\left (25^x x-25^x+e^{\frac {2}{3} x \left (\frac {x}{e^2}+6\right )}\right )^2}-\frac {6\ 5^{2 x+1} e^{\frac {2 x^2}{3 e^2}+4 x+2} (-2+\log (5))}{\left (5^{2 x} x-5^{2 x}+e^{\frac {2}{3} x \left (\frac {x}{e^2}+6\right )}\right )^2}\right )dx}{3 e^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {30 (2-\log (5)) \int \frac {e^{\frac {2 x^2}{3 e^2}+2 (2+\log (5)) x+2}}{\left (5^{2 x} x-5^{2 x}+e^{\frac {2}{3} x \left (\frac {x}{e^2}+6\right )}\right )^2}dx+20 \int \frac {e^{\frac {2 x^2}{3 e^2}+2 (2+\log (5)) x} x}{\left (5^{2 x} x-5^{2 x}+e^{\frac {2}{3} x \left (\frac {x}{e^2}+6\right )}\right )^2}dx+3 e^2 \int \frac {5^{4 x+1}}{\left (25^x x-25^x+e^{\frac {2}{3} x \left (\frac {x}{e^2}+6\right )}\right )^2}dx}{3 e^2}\) |
Input:
Int[(-15*E^2 + E^((2*(6*E^2*x + x^2 - 3*E^2*x*Log[5]))/(3*E^2))*(-60*E^2 - 20*x + 30*E^2*Log[5]))/(3*E^(2 + (4*(6*E^2*x + x^2 - 3*E^2*x*Log[5]))/(3* E^2)) + E^(2 + (2*(6*E^2*x + x^2 - 3*E^2*x*Log[5]))/(3*E^2))*(-6 + 6*x) + E^2*(3 - 6*x + 3*x^2)),x]
Output:
$Aborted
Time = 0.94 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \(\frac {5}{5^{-2 x} {\mathrm e}^{\frac {2 x \left (x \,{\mathrm e}^{-2}+6\right )}{3}}+x -1}\) | \(24\) |
| parallelrisch | \(\frac {5}{{\mathrm e}^{-\frac {2 x \left (3 \,{\mathrm e}^{2} \ln \left (5\right )-6 \,{\mathrm e}^{2}-x \right ) {\mathrm e}^{-2}}{3}}+x -1}\) | \(32\) |
| norman | \(\frac {5}{{\mathrm e}^{\frac {2 \left (-3 x \,{\mathrm e}^{2} \ln \left (5\right )+6 \,{\mathrm e}^{2} x +x^{2}\right ) {\mathrm e}^{-2}}{3}}+x -1}\) | \(33\) |
Input:
int(((30*exp(2)*ln(5)-60*exp(2)-20*x)*exp(1/3*(-3*x*exp(2)*ln(5)+6*exp(2)* x+x^2)/exp(2))^2-15*exp(2))/(3*exp(2)*exp(1/3*(-3*x*exp(2)*ln(5)+6*exp(2)* x+x^2)/exp(2))^4+(6*x-6)*exp(2)*exp(1/3*(-3*x*exp(2)*ln(5)+6*exp(2)*x+x^2) /exp(2))^2+(3*x^2-6*x+3)*exp(2)),x,method=_RETURNVERBOSE)
Output:
5/(((1/5)^x)^2*exp(2/3*x*(x*exp(-2)+6))+x-1)
Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.54 \[ \int \frac {-15 e^2+e^{\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} \left (-60 e^2-20 x+30 e^2 \log (5)\right )}{3 e^{2+\frac {4 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}}+e^{2+\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} (-6+6 x)+e^2 \left (3-6 x+3 x^2\right )} \, dx=\frac {5 \, e^{2}}{{\left (x - 1\right )} e^{2} + e^{\left (-\frac {2}{3} \, {\left (3 \, x e^{2} \log \left (5\right ) - x^{2} - 3 \, {\left (2 \, x + 1\right )} e^{2}\right )} e^{\left (-2\right )}\right )}} \] Input:
integrate(((30*exp(2)*log(5)-60*exp(2)-20*x)*exp(1/3*(-3*x*exp(2)*log(5)+6 *exp(2)*x+x^2)/exp(2))^2-15*exp(2))/(3*exp(2)*exp(1/3*(-3*x*exp(2)*log(5)+ 6*exp(2)*x+x^2)/exp(2))^4+(6*x-6)*exp(2)*exp(1/3*(-3*x*exp(2)*log(5)+6*exp (2)*x+x^2)/exp(2))^2+(3*x^2-6*x+3)*exp(2)),x, algorithm="fricas")
Output:
5*e^2/((x - 1)*e^2 + e^(-2/3*(3*x*e^2*log(5) - x^2 - 3*(2*x + 1)*e^2)*e^(- 2)))
Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-15 e^2+e^{\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} \left (-60 e^2-20 x+30 e^2 \log (5)\right )}{3 e^{2+\frac {4 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}}+e^{2+\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} (-6+6 x)+e^2 \left (3-6 x+3 x^2\right )} \, dx=\frac {5}{x + e^{\frac {2 \left (\frac {x^{2}}{3} - x e^{2} \log {\left (5 \right )} + 2 x e^{2}\right )}{e^{2}}} - 1} \] Input:
integrate(((30*exp(2)*ln(5)-60*exp(2)-20*x)*exp(1/3*(-3*x*exp(2)*ln(5)+6*e xp(2)*x+x**2)/exp(2))**2-15*exp(2))/(3*exp(2)*exp(1/3*(-3*x*exp(2)*ln(5)+6 *exp(2)*x+x**2)/exp(2))**4+(6*x-6)*exp(2)*exp(1/3*(-3*x*exp(2)*ln(5)+6*exp (2)*x+x**2)/exp(2))**2+(3*x**2-6*x+3)*exp(2)),x)
Output:
5/(x + exp(2*(x**2/3 - x*exp(2)*log(5) + 2*x*exp(2))*exp(-2)) - 1)
Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {-15 e^2+e^{\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} \left (-60 e^2-20 x+30 e^2 \log (5)\right )}{3 e^{2+\frac {4 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}}+e^{2+\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} (-6+6 x)+e^2 \left (3-6 x+3 x^2\right )} \, dx=\frac {5 \cdot 5^{2 \, x}}{5^{2 \, x} {\left (x - 1\right )} + e^{\left (\frac {2}{3} \, x^{2} e^{\left (-2\right )} + 4 \, x\right )}} \] Input:
integrate(((30*exp(2)*log(5)-60*exp(2)-20*x)*exp(1/3*(-3*x*exp(2)*log(5)+6 *exp(2)*x+x^2)/exp(2))^2-15*exp(2))/(3*exp(2)*exp(1/3*(-3*x*exp(2)*log(5)+ 6*exp(2)*x+x^2)/exp(2))^4+(6*x-6)*exp(2)*exp(1/3*(-3*x*exp(2)*log(5)+6*exp (2)*x+x^2)/exp(2))^2+(3*x^2-6*x+3)*exp(2)),x, algorithm="maxima")
Output:
5*5^(2*x)/(5^(2*x)*(x - 1) + e^(2/3*x^2*e^(-2) + 4*x))
Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {-15 e^2+e^{\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} \left (-60 e^2-20 x+30 e^2 \log (5)\right )}{3 e^{2+\frac {4 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}}+e^{2+\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} (-6+6 x)+e^2 \left (3-6 x+3 x^2\right )} \, dx=\frac {5}{x + e^{\left (-\frac {2}{3} \, {\left (3 \, x e^{2} \log \left (5\right ) - x^{2} - 6 \, x e^{2}\right )} e^{\left (-2\right )}\right )} - 1} \] Input:
integrate(((30*exp(2)*log(5)-60*exp(2)-20*x)*exp(1/3*(-3*x*exp(2)*log(5)+6 *exp(2)*x+x^2)/exp(2))^2-15*exp(2))/(3*exp(2)*exp(1/3*(-3*x*exp(2)*log(5)+ 6*exp(2)*x+x^2)/exp(2))^4+(6*x-6)*exp(2)*exp(1/3*(-3*x*exp(2)*log(5)+6*exp (2)*x+x^2)/exp(2))^2+(3*x^2-6*x+3)*exp(2)),x, algorithm="giac")
Output:
5/(x + e^(-2/3*(3*x*e^2*log(5) - x^2 - 6*x*e^2)*e^(-2)) - 1)
Time = 1.65 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {-15 e^2+e^{\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} \left (-60 e^2-20 x+30 e^2 \log (5)\right )}{3 e^{2+\frac {4 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}}+e^{2+\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} (-6+6 x)+e^2 \left (3-6 x+3 x^2\right )} \, dx=\frac {15\,{\mathrm {e}}^2}{3\,x\,{\mathrm {e}}^2-3\,{\mathrm {e}}^2+\frac {3\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{-2}\,x^2}{3}+4\,x+2}}{5^{2\,x}}} \] Input:
int(-(15*exp(2) + exp(2*exp(-2)*(2*x*exp(2) + x^2/3 - x*exp(2)*log(5)))*(2 0*x + 60*exp(2) - 30*exp(2)*log(5)))/(exp(2)*(3*x^2 - 6*x + 3) + 3*exp(2)* exp(4*exp(-2)*(2*x*exp(2) + x^2/3 - x*exp(2)*log(5))) + exp(2)*exp(2*exp(- 2)*(2*x*exp(2) + x^2/3 - x*exp(2)*log(5)))*(6*x - 6)),x)
Output:
(15*exp(2))/(3*x*exp(2) - 3*exp(2) + (3*exp(4*x + (2*x^2*exp(-2))/3 + 2))/ 5^(2*x))
Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {-15 e^2+e^{\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} \left (-60 e^2-20 x+30 e^2 \log (5)\right )}{3 e^{2+\frac {4 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}}+e^{2+\frac {2 \left (6 e^2 x+x^2-3 e^2 x \log (5)\right )}{3 e^2}} (-6+6 x)+e^2 \left (3-6 x+3 x^2\right )} \, dx=\frac {5 \,5^{2 x}}{e^{\frac {12 e^{2} x +2 x^{2}}{3 e^{2}}}+5^{2 x} x -5^{2 x}} \] Input:
int(((30*exp(2)*log(5)-60*exp(2)-20*x)*exp(1/3*(-3*x*exp(2)*log(5)+6*exp(2 )*x+x^2)/exp(2))^2-15*exp(2))/(3*exp(2)*exp(1/3*(-3*x*exp(2)*log(5)+6*exp( 2)*x+x^2)/exp(2))^4+(6*x-6)*exp(2)*exp(1/3*(-3*x*exp(2)*log(5)+6*exp(2)*x+ x^2)/exp(2))^2+(3*x^2-6*x+3)*exp(2)),x)
Output:
(5*5**(2*x))/(e**((12*e**2*x + 2*x**2)/(3*e**2)) + 5**(2*x)*x - 5**(2*x))