Integrand size = 151, antiderivative size = 31 \[ \int \frac {125 x^3+50 x^4+5 x^5+e^{\frac {-20-12 x-125 x^2-25 x^3+e \left (25 x^2+5 x^3\right )}{25 x^2+5 x^3}} \left (200+120 x+24 x^2\right )}{125 x^4+50 x^5+5 x^6+e^{\frac {-20-12 x-125 x^2-25 x^3+e \left (25 x^2+5 x^3\right )}{25 x^2+5 x^3}} \left (125 x^3+50 x^4+5 x^5\right )} \, dx=\log \left (e^{-5+e+\frac {4 \left (-5+\frac {-5+2 x}{x}\right )}{5 x (5+x)}}+x\right ) \]
Time = 0.37 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {125 x^3+50 x^4+5 x^5+e^{\frac {-20-12 x-125 x^2-25 x^3+e \left (25 x^2+5 x^3\right )}{25 x^2+5 x^3}} \left (200+120 x+24 x^2\right )}{125 x^4+50 x^5+5 x^6+e^{\frac {-20-12 x-125 x^2-25 x^3+e \left (25 x^2+5 x^3\right )}{25 x^2+5 x^3}} \left (125 x^3+50 x^4+5 x^5\right )} \, dx=\frac {1}{5} \left (-\frac {4 (5+3 x)}{x^2 (5+x)}+5 \log \left (e^e+e^{5+\frac {4}{5 x^2}+\frac {8}{25 x}-\frac {8}{25 (5+x)}} x\right )\right ) \]
Integrate[(125*x^3 + 50*x^4 + 5*x^5 + E^((-20 - 12*x - 125*x^2 - 25*x^3 + E*(25*x^2 + 5*x^3))/(25*x^2 + 5*x^3))*(200 + 120*x + 24*x^2))/(125*x^4 + 5 0*x^5 + 5*x^6 + E^((-20 - 12*x - 125*x^2 - 25*x^3 + E*(25*x^2 + 5*x^3))/(2 5*x^2 + 5*x^3))*(125*x^3 + 50*x^4 + 5*x^5)),x]
((-4*(5 + 3*x))/(x^2*(5 + x)) + 5*Log[E^E + E^(5 + 4/(5*x^2) + 8/(25*x) - 8/(25*(5 + x)))*x])/5
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.
\(\displaystyle \int \frac {\left (24 x^2+120 x+200\right ) \exp \left (\frac {-25 x^3-125 x^2+e \left (5 x^3+25 x^2\right )-12 x-20}{5 x^3+25 x^2}\right )+5 x^5+50 x^4+125 x^3}{\left (5 x^5+50 x^4+125 x^3\right ) \exp \left (\frac {-25 x^3-125 x^2+e \left (5 x^3+25 x^2\right )-12 x-20}{5 x^3+25 x^2}\right )+5 x^6+50 x^5+125 x^4} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{x}-\frac {e^e \left (5 x^4+50 x^3+101 x^2-120 x-200\right )}{5 x^3 (x+5)^2 \left (x \exp \left (\frac {4}{(x+5) x^2}+\frac {5 x}{x+5}+\frac {25}{x+5}+\frac {12}{5 (x+5) x}\right )+e^e\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {8}{25} e^e \int \frac {1}{x^2 \left (\exp \left (\frac {5 x}{x+5}+\frac {25}{x+5}+\frac {12}{5 (x+5) x}+\frac {4}{(x+5) x^2}\right ) x+e^e\right )}dx-e^e \int \frac {1}{x \left (\exp \left (\frac {5 x}{x+5}+\frac {25}{x+5}+\frac {12}{5 (x+5) x}+\frac {4}{(x+5) x^2}\right ) x+e^e\right )}dx-\frac {8}{25} e^e \int \frac {1}{(x+5)^2 \left (\exp \left (\frac {5 x}{x+5}+\frac {25}{x+5}+\frac {12}{5 (x+5) x}+\frac {4}{(x+5) x^2}\right ) x+e^e\right )}dx+\frac {8}{5} e^e \int \frac {1}{x^3 \left (\exp \left (\frac {5 x}{x+5}+\frac {25}{x+5}+\frac {12}{5 (x+5) x}+\frac {4}{(x+5) x^2}\right ) x+e^e\right )}dx+\log (x)\) |
Int[(125*x^3 + 50*x^4 + 5*x^5 + E^((-20 - 12*x - 125*x^2 - 25*x^3 + E*(25* x^2 + 5*x^3))/(25*x^2 + 5*x^3))*(200 + 120*x + 24*x^2))/(125*x^4 + 50*x^5 + 5*x^6 + E^((-20 - 12*x - 125*x^2 - 25*x^3 + E*(25*x^2 + 5*x^3))/(25*x^2 + 5*x^3))*(125*x^3 + 50*x^4 + 5*x^5)),x]
3.14.41.3.1 Defintions of rubi rules used
Time = 0.36 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42
method | result | size |
parallelrisch | \(\ln \left (x +{\mathrm e}^{\frac {\left (5 x^{3}+25 x^{2}\right ) {\mathrm e}-25 x^{3}-125 x^{2}-12 x -20}{5 x^{2} \left (5+x \right )}}\right )\) | \(44\) |
norman | \(\ln \left (x +{\mathrm e}^{\frac {\left (5 x^{3}+25 x^{2}\right ) {\mathrm e}-25 x^{3}-125 x^{2}-12 x -20}{5 x^{3}+25 x^{2}}}\right )\) | \(48\) |
risch | \(\frac {-\frac {12 x}{5}-4}{x^{2} \left (5+x \right )}-\frac {\left (5 x^{3}+25 x^{2}\right ) {\mathrm e}-25 x^{3}-125 x^{2}-12 x -20}{5 x^{3}+25 x^{2}}+\ln \left ({\mathrm e}^{\frac {5 x^{3} {\mathrm e}+25 x^{2} {\mathrm e}-25 x^{3}-125 x^{2}-12 x -20}{5 x^{2} \left (5+x \right )}}+x \right )\) | \(103\) |
int(((24*x^2+120*x+200)*exp(((5*x^3+25*x^2)*exp(1)-25*x^3-125*x^2-12*x-20) /(5*x^3+25*x^2))+5*x^5+50*x^4+125*x^3)/((5*x^5+50*x^4+125*x^3)*exp(((5*x^3 +25*x^2)*exp(1)-25*x^3-125*x^2-12*x-20)/(5*x^3+25*x^2))+5*x^6+50*x^5+125*x ^4),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.45 \[ \int \frac {125 x^3+50 x^4+5 x^5+e^{\frac {-20-12 x-125 x^2-25 x^3+e \left (25 x^2+5 x^3\right )}{25 x^2+5 x^3}} \left (200+120 x+24 x^2\right )}{125 x^4+50 x^5+5 x^6+e^{\frac {-20-12 x-125 x^2-25 x^3+e \left (25 x^2+5 x^3\right )}{25 x^2+5 x^3}} \left (125 x^3+50 x^4+5 x^5\right )} \, dx=\log \left (x + e^{\left (-\frac {25 \, x^{3} + 125 \, x^{2} - 5 \, {\left (x^{3} + 5 \, x^{2}\right )} e + 12 \, x + 20}{5 \, {\left (x^{3} + 5 \, x^{2}\right )}}\right )}\right ) \]
integrate(((24*x^2+120*x+200)*exp(((5*x^3+25*x^2)*exp(1)-25*x^3-125*x^2-12 *x-20)/(5*x^3+25*x^2))+5*x^5+50*x^4+125*x^3)/((5*x^5+50*x^4+125*x^3)*exp(( (5*x^3+25*x^2)*exp(1)-25*x^3-125*x^2-12*x-20)/(5*x^3+25*x^2))+5*x^6+50*x^5 +125*x^4),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {125 x^3+50 x^4+5 x^5+e^{\frac {-20-12 x-125 x^2-25 x^3+e \left (25 x^2+5 x^3\right )}{25 x^2+5 x^3}} \left (200+120 x+24 x^2\right )}{125 x^4+50 x^5+5 x^6+e^{\frac {-20-12 x-125 x^2-25 x^3+e \left (25 x^2+5 x^3\right )}{25 x^2+5 x^3}} \left (125 x^3+50 x^4+5 x^5\right )} \, dx=\log {\left (x + e^{\frac {- 25 x^{3} - 125 x^{2} - 12 x + e \left (5 x^{3} + 25 x^{2}\right ) - 20}{5 x^{3} + 25 x^{2}}} \right )} \]
integrate(((24*x**2+120*x+200)*exp(((5*x**3+25*x**2)*exp(1)-25*x**3-125*x* *2-12*x-20)/(5*x**3+25*x**2))+5*x**5+50*x**4+125*x**3)/((5*x**5+50*x**4+12 5*x**3)*exp(((5*x**3+25*x**2)*exp(1)-25*x**3-125*x**2-12*x-20)/(5*x**3+25* x**2))+5*x**6+50*x**5+125*x**4),x)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (29) = 58\).
Time = 0.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10 \[ \int \frac {125 x^3+50 x^4+5 x^5+e^{\frac {-20-12 x-125 x^2-25 x^3+e \left (25 x^2+5 x^3\right )}{25 x^2+5 x^3}} \left (200+120 x+24 x^2\right )}{125 x^4+50 x^5+5 x^6+e^{\frac {-20-12 x-125 x^2-25 x^3+e \left (25 x^2+5 x^3\right )}{25 x^2+5 x^3}} \left (125 x^3+50 x^4+5 x^5\right )} \, dx=-\frac {633 \, x^{2} + 60 \, x + 100}{25 \, {\left (x^{3} + 5 \, x^{2}\right )}} + \log \left ({\left (x e^{\left (\frac {25}{x + 5} + \frac {8}{25 \, x} + \frac {4}{5 \, x^{2}} + 5\right )} + e^{\left (\frac {633}{25 \, {\left (x + 5\right )}} + e\right )}\right )} e^{\left (-e\right )}\right ) \]
integrate(((24*x^2+120*x+200)*exp(((5*x^3+25*x^2)*exp(1)-25*x^3-125*x^2-12 *x-20)/(5*x^3+25*x^2))+5*x^5+50*x^4+125*x^3)/((5*x^5+50*x^4+125*x^3)*exp(( (5*x^3+25*x^2)*exp(1)-25*x^3-125*x^2-12*x-20)/(5*x^3+25*x^2))+5*x^6+50*x^5 +125*x^4),x, algorithm=\
-1/25*(633*x^2 + 60*x + 100)/(x^3 + 5*x^2) + log((x*e^(25/(x + 5) + 8/25/x + 4/5/x^2 + 5) + e^(633/25/(x + 5) + e))*e^(-e))
Time = 0.52 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {125 x^3+50 x^4+5 x^5+e^{\frac {-20-12 x-125 x^2-25 x^3+e \left (25 x^2+5 x^3\right )}{25 x^2+5 x^3}} \left (200+120 x+24 x^2\right )}{125 x^4+50 x^5+5 x^6+e^{\frac {-20-12 x-125 x^2-25 x^3+e \left (25 x^2+5 x^3\right )}{25 x^2+5 x^3}} \left (125 x^3+50 x^4+5 x^5\right )} \, dx=\log \left (x + e^{\left (\frac {5 \, x^{3} e - 25 \, x^{3} + 25 \, x^{2} e - 125 \, x^{2} - 12 \, x - 20}{5 \, {\left (x^{3} + 5 \, x^{2}\right )}}\right )}\right ) \]
integrate(((24*x^2+120*x+200)*exp(((5*x^3+25*x^2)*exp(1)-25*x^3-125*x^2-12 *x-20)/(5*x^3+25*x^2))+5*x^5+50*x^4+125*x^3)/((5*x^5+50*x^4+125*x^3)*exp(( (5*x^3+25*x^2)*exp(1)-25*x^3-125*x^2-12*x-20)/(5*x^3+25*x^2))+5*x^6+50*x^5 +125*x^4),x, algorithm=\
Time = 14.84 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int \frac {125 x^3+50 x^4+5 x^5+e^{\frac {-20-12 x-125 x^2-25 x^3+e \left (25 x^2+5 x^3\right )}{25 x^2+5 x^3}} \left (200+120 x+24 x^2\right )}{125 x^4+50 x^5+5 x^6+e^{\frac {-20-12 x-125 x^2-25 x^3+e \left (25 x^2+5 x^3\right )}{25 x^2+5 x^3}} \left (125 x^3+50 x^4+5 x^5\right )} \, dx=\ln \left (x+{\mathrm {e}}^{-\frac {12\,x-25\,x^2\,\mathrm {e}-5\,x^3\,\mathrm {e}+125\,x^2+25\,x^3+20}{5\,x^2\,\left (x+5\right )}}\right ) \]
int((125*x^3 + 50*x^4 + 5*x^5 + exp(-(12*x - exp(1)*(25*x^2 + 5*x^3) + 125 *x^2 + 25*x^3 + 20)/(25*x^2 + 5*x^3))*(120*x + 24*x^2 + 200))/(exp(-(12*x - exp(1)*(25*x^2 + 5*x^3) + 125*x^2 + 25*x^3 + 20)/(25*x^2 + 5*x^3))*(125* x^3 + 50*x^4 + 5*x^5) + 125*x^4 + 50*x^5 + 5*x^6),x)