3.28.17 \(\int \frac {5^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} (-900 x+20000 x^3+20000 x^4+(-1800 x+900 x^2-40000 x^4-20000 x^5) \log (5 x))}{81-3600 x^2-3600 x^3+40000 x^4+80000 x^5+40000 x^6} \, dx\) [2717]

3.28.17.1 Optimal result

Integrand size = 125, antiderivative size = 45 \[ \int \frac {5^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} \left (-900 x+20000 x^3+20000 x^4+\left (-1800 x+900 x^2-40000 x^4-20000 x^5\right ) \log (5 x)\right )}{81-3600 x^2-3600 x^3+40000 x^4+80000 x^5+40000 x^6} \, dx=5^{\frac {e^{-x}}{2-\frac {9}{100 x^2}+2 x}} x^{\frac {e^{-x}}{2-\frac {9}{100 x^2}+2 x}} \]

output

exp(ln(5*x)/exp(x)/(2*x+2-9/100/x^2))
 

3.28.17.2 Mathematica [F]

\[ \int \frac {5^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} \left (-900 x+20000 x^3+20000 x^4+\left (-1800 x+900 x^2-40000 x^4-20000 x^5\right ) \log (5 x)\right )}{81-3600 x^2-3600 x^3+40000 x^4+80000 x^5+40000 x^6} \, dx=\int \frac {5^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} \left (-900 x+20000 x^3+20000 x^4+\left (-1800 x+900 x^2-40000 x^4-20000 x^5\right ) \log (5 x)\right )}{81-3600 x^2-3600 x^3+40000 x^4+80000 x^5+40000 x^6} \, dx \]

input

Integrate[(5^((100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*x^((100*x^2)/(E^x* 
(-9 + 200*x^2 + 200*x^3)))*(-900*x + 20000*x^3 + 20000*x^4 + (-1800*x + 90 
0*x^2 - 40000*x^4 - 20000*x^5)*Log[5*x]))/(E^x*(81 - 3600*x^2 - 3600*x^3 + 
 40000*x^4 + 80000*x^5 + 40000*x^6)),x]
 

output

Integrate[(5^((100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*x^((100*x^2)/(E^x* 
(-9 + 200*x^2 + 200*x^3)))*(-900*x + 20000*x^3 + 20000*x^4 + (-1800*x + 90 
0*x^2 - 40000*x^4 - 20000*x^5)*Log[5*x]))/(E^x*(81 - 3600*x^2 - 3600*x^3 + 
 40000*x^4 + 80000*x^5 + 40000*x^6)), x]
 

3.28.17.3 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.

\(\displaystyle \int \frac {5^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}} e^{-x} x^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}} \left (20000 x^4+20000 x^3+\left (-20000 x^5-40000 x^4+900 x^2-1800 x\right ) \log (5 x)-900 x\right )}{40000 x^6+80000 x^5+40000 x^4-3600 x^3-3600 x^2+81} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \frac {5^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}} e^{-x} x^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}} \left (20000 x^4+20000 x^3+\left (-20000 x^5-40000 x^4+900 x^2-1800 x\right ) \log (5 x)-900 x\right )}{\left (200 x^3+200 x^2-9\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {4\ 5^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+2} e^{-x} x^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+1}}{200 x^3+200 x^2-9}-\frac {4\ 5^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+2} e^{-x} x^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+1} \left (200 x^4+400 x^3-9 x+18\right ) \log (5 x)}{\left (200 x^3+200 x^2-9\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 4 \int \frac {5^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+2} e^{-x} x^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+1}}{200 x^3+200 x^2-9}dx+108 \int \frac {\int \frac {5^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+2} e^{-x} x^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+1}}{\left (200 x^3+200 x^2-9\right )^2}dx}{x}dx-32 \int \frac {\int \frac {5^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+4} e^{-x} x^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+3}}{\left (200 x^3+200 x^2-9\right )^2}dx}{x}dx+4 \int \frac {\int \frac {5^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+2} e^{-x} x^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+1}}{200 x^3+200 x^2-9}dx}{x}dx+4 \int \frac {\int \frac {5^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+2} e^{-x} x^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+2}}{200 x^3+200 x^2-9}dx}{x}dx-108 \log (5 x) \int \frac {5^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+2} e^{-x} x^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+1}}{\left (200 x^3+200 x^2-9\right )^2}dx+32 \log (5 x) \int \frac {5^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+4} e^{-x} x^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+3}}{\left (200 x^3+200 x^2-9\right )^2}dx-4 \log (5 x) \int \frac {5^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+2} e^{-x} x^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+1}}{200 x^3+200 x^2-9}dx-4 \log (5 x) \int \frac {5^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+2} e^{-x} x^{\frac {100 e^{-x} x^2}{200 x^3+200 x^2-9}+2}}{200 x^3+200 x^2-9}dx\)

input

Int[(5^((100*x^2)/(E^x*(-9 + 200*x^2 + 200*x^3)))*x^((100*x^2)/(E^x*(-9 + 
200*x^2 + 200*x^3)))*(-900*x + 20000*x^3 + 20000*x^4 + (-1800*x + 900*x^2 
- 40000*x^4 - 20000*x^5)*Log[5*x]))/(E^x*(81 - 3600*x^2 - 3600*x^3 + 40000 
*x^4 + 80000*x^5 + 40000*x^6)),x]
 

output

$Aborted
 

3.28.17.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2463

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u, Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt 
Q[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 
0]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.28.17.4 Maple [A] (verified)

Time = 161.96 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.62

method	result	size
risch	\(\left (5 x \right )^{\frac {100 x^{2} {\mathrm e}^{-x}}{200 x^{3}+200 x^{2}-9}}\)	\(28\)
parallelrisch	\({\mathrm e}^{\frac {100 x^{2} \ln \left (5 x \right ) {\mathrm e}^{-x}}{200 x^{3}+200 x^{2}-9}}\)	\(29\)

input

int(((-20000*x^5-40000*x^4+900*x^2-1800*x)*ln(5*x)+20000*x^4+20000*x^3-900 
*x)*exp(100*x^2*ln(5*x)/(200*x^3+200*x^2-9)/exp(x))/(40000*x^6+80000*x^5+4 
0000*x^4-3600*x^3-3600*x^2+81)/exp(x),x,method=_RETURNVERBOSE)
 

output

(5*x)^(100*x^2/(200*x^3+200*x^2-9)*exp(-x))
 

3.28.17.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.60 \[ \int \frac {5^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} \left (-900 x+20000 x^3+20000 x^4+\left (-1800 x+900 x^2-40000 x^4-20000 x^5\right ) \log (5 x)\right )}{81-3600 x^2-3600 x^3+40000 x^4+80000 x^5+40000 x^6} \, dx=\left (5 \, x\right )^{\frac {100 \, x^{2} e^{\left (-x\right )}}{200 \, x^{3} + 200 \, x^{2} - 9}} \]

input

integrate(((-20000*x^5-40000*x^4+900*x^2-1800*x)*log(5*x)+20000*x^4+20000* 
x^3-900*x)*exp(100*x^2*log(5*x)/(200*x^3+200*x^2-9)/exp(x))/(40000*x^6+800 
00*x^5+40000*x^4-3600*x^3-3600*x^2+81)/exp(x),x, algorithm=\
 

output

(5*x)^(100*x^2*e^(-x)/(200*x^3 + 200*x^2 - 9))
 

3.28.17.6 Sympy [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.58 \[ \int \frac {5^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} \left (-900 x+20000 x^3+20000 x^4+\left (-1800 x+900 x^2-40000 x^4-20000 x^5\right ) \log (5 x)\right )}{81-3600 x^2-3600 x^3+40000 x^4+80000 x^5+40000 x^6} \, dx=e^{\frac {100 x^{2} e^{- x} \log {\left (5 x \right )}}{200 x^{3} + 200 x^{2} - 9}} \]

input

integrate(((-20000*x**5-40000*x**4+900*x**2-1800*x)*ln(5*x)+20000*x**4+200 
00*x**3-900*x)*exp(100*x**2*ln(5*x)/(200*x**3+200*x**2-9)/exp(x))/(40000*x 
**6+80000*x**5+40000*x**4-3600*x**3-3600*x**2+81)/exp(x),x)
 

output

exp(100*x**2*exp(-x)*log(5*x)/(200*x**3 + 200*x**2 - 9))
 

3.28.17.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (22) = 44\).

Time = 0.42 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.16 \[ \int \frac {5^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} \left (-900 x+20000 x^3+20000 x^4+\left (-1800 x+900 x^2-40000 x^4-20000 x^5\right ) \log (5 x)\right )}{81-3600 x^2-3600 x^3+40000 x^4+80000 x^5+40000 x^6} \, dx=e^{\left (\frac {100 \, x^{2} e^{\left (-x\right )} \log \left (5\right )}{200 \, x^{3} + 200 \, x^{2} - 9} + \frac {100 \, x^{2} e^{\left (-x\right )} \log \left (x\right )}{200 \, x^{3} + 200 \, x^{2} - 9}\right )} \]

input

integrate(((-20000*x^5-40000*x^4+900*x^2-1800*x)*log(5*x)+20000*x^4+20000* 
x^3-900*x)*exp(100*x^2*log(5*x)/(200*x^3+200*x^2-9)/exp(x))/(40000*x^6+800 
00*x^5+40000*x^4-3600*x^3-3600*x^2+81)/exp(x),x, algorithm=\
 

output

e^(100*x^2*e^(-x)*log(5)/(200*x^3 + 200*x^2 - 9) + 100*x^2*e^(-x)*log(x)/( 
200*x^3 + 200*x^2 - 9))
 

3.28.17.8 Giac [A] (verification not implemented)

Time = 3.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.60 \[ \int \frac {5^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} \left (-900 x+20000 x^3+20000 x^4+\left (-1800 x+900 x^2-40000 x^4-20000 x^5\right ) \log (5 x)\right )}{81-3600 x^2-3600 x^3+40000 x^4+80000 x^5+40000 x^6} \, dx=\left (5 \, x\right )^{\frac {100 \, x^{2} e^{\left (-x\right )}}{200 \, x^{3} + 200 \, x^{2} - 9}} \]

input

integrate(((-20000*x^5-40000*x^4+900*x^2-1800*x)*log(5*x)+20000*x^4+20000* 
x^3-900*x)*exp(100*x^2*log(5*x)/(200*x^3+200*x^2-9)/exp(x))/(40000*x^6+800 
00*x^5+40000*x^4-3600*x^3-3600*x^2+81)/exp(x),x, algorithm=\
 

output

(5*x)^(100*x^2*e^(-x)/(200*x^3 + 200*x^2 - 9))
 

3.28.17.9 Mupad [B] (verification not implemented)

Time = 13.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.13 \[ \int \frac {5^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} e^{-x} x^{\frac {100 e^{-x} x^2}{-9+200 x^2+200 x^3}} \left (-900 x+20000 x^3+20000 x^4+\left (-1800 x+900 x^2-40000 x^4-20000 x^5\right ) \log (5 x)\right )}{81-3600 x^2-3600 x^3+40000 x^4+80000 x^5+40000 x^6} \, dx=5^{\frac {100\,x^2\,{\mathrm {e}}^{-x}}{200\,x^3+200\,x^2-9}}\,x^{\frac {100\,x^2\,{\mathrm {e}}^{-x}}{200\,x^3+200\,x^2-9}} \]

input

int(-(exp(-x)*exp((100*x^2*log(5*x)*exp(-x))/(200*x^2 + 200*x^3 - 9))*(900 
*x + log(5*x)*(1800*x - 900*x^2 + 40000*x^4 + 20000*x^5) - 20000*x^3 - 200 
00*x^4))/(40000*x^4 - 3600*x^3 - 3600*x^2 + 80000*x^5 + 40000*x^6 + 81),x)
 

output

5^((100*x^2*exp(-x))/(200*x^2 + 200*x^3 - 9))*x^((100*x^2*exp(-x))/(200*x^ 
2 + 200*x^3 - 9))