3.16.59 \(\int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} (2500+500 x+20 x^3-4 x^4)+e^6 (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5))}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 (4 x^3+2 x^4))} \, dx\) [1559]

3.16.59.1 Optimal result

Integrand size = 200, antiderivative size = 32 \[ \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx=\log \left (\frac {5}{\left (-e^{\left (x-\frac {5 (5+x)}{x}\right )^2}+\frac {x}{2+e^6+x}\right )^2}\right ) \]

output

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

3.16.59.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(32)=64\).

Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.56 \[ \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx=2 \log \left (2+e^6+x\right )-2 \log \left (2 e^{\frac {625}{x^2}+\frac {250}{x}-10 x+x^2}+e^{6+\frac {625}{x^2}+\frac {250}{x}-10 x+x^2}-e^{25} x+e^{\frac {625}{x^2}+\frac {250}{x}-10 x+x^2} x\right ) \]

input

Integrate[(4*x^3 + 2*E^6*x^3 + E^((625 + 250*x - 25*x^2 - 10*x^3 + x^4)/x^ 
2)*(10000 + 12000*x + 4500*x^2 + 580*x^3 + 64*x^4 + 4*x^5 - 4*x^6 + E^12*( 
2500 + 500*x + 20*x^3 - 4*x^4) + E^6*(10000 + 7000*x + 1000*x^2 + 80*x^3 + 
 24*x^4 - 8*x^5)))/(-2*x^4 - E^6*x^4 - x^5 + E^((625 + 250*x - 25*x^2 - 10 
*x^3 + x^4)/x^2)*(4*x^3 + E^12*x^3 + 4*x^4 + x^5 + E^6*(4*x^3 + 2*x^4))),x 
]
 

output

2*Log[2 + E^6 + x] - 2*Log[2*E^(625/x^2 + 250/x - 10*x + x^2) + E^(6 + 625 
/x^2 + 250/x - 10*x + x^2) - E^25*x + E^(625/x^2 + 250/x - 10*x + x^2)*x]
 

3.16.59.3 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.

\(\displaystyle \int \frac {2 e^6 x^3+4 x^3+e^{\frac {x^4-10 x^3-25 x^2+250 x+625}{x^2}} \left (-4 x^6+4 x^5+64 x^4+580 x^3+4500 x^2+e^{12} \left (-4 x^4+20 x^3+500 x+2500\right )+e^6 \left (-8 x^5+24 x^4+80 x^3+1000 x^2+7000 x+10000\right )+12000 x+10000\right )}{-x^5-e^6 x^4-2 x^4+e^{\frac {x^4-10 x^3-25 x^2+250 x+625}{x^2}} \left (x^5+4 x^4+e^{12} x^3+4 x^3+e^6 \left (2 x^4+4 x^3\right )\right )} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {2 e^6 x^3+4 x^3+e^{\frac {x^4-10 x^3-25 x^2+250 x+625}{x^2}} \left (-4 x^6+4 x^5+64 x^4+580 x^3+4500 x^2+e^{12} \left (-4 x^4+20 x^3+500 x+2500\right )+e^6 \left (-8 x^5+24 x^4+80 x^3+1000 x^2+7000 x+10000\right )+12000 x+10000\right )}{-x^5+\left (-2-e^6\right ) x^4+e^{\frac {x^4-10 x^3-25 x^2+250 x+625}{x^2}} \left (x^5+4 x^4+e^{12} x^3+4 x^3+e^6 \left (2 x^4+4 x^3\right )\right )}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {\left (4+2 e^6\right ) x^3+e^{\frac {x^4-10 x^3-25 x^2+250 x+625}{x^2}} \left (-4 x^6+4 x^5+64 x^4+580 x^3+4500 x^2+e^{12} \left (-4 x^4+20 x^3+500 x+2500\right )+e^6 \left (-8 x^5+24 x^4+80 x^3+1000 x^2+7000 x+10000\right )+12000 x+10000\right )}{-x^5+\left (-2-e^6\right ) x^4+e^{\frac {x^4-10 x^3-25 x^2+250 x+625}{x^2}} \left (x^5+4 x^4+e^{12} x^3+4 x^3+e^6 \left (2 x^4+4 x^3\right )\right )}dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {4 e^{\frac {\left (x^2-5 x-25\right )^2}{x^2}} \left (x+e^6+2\right )^2 \left (x^4-5 x^3-125 x-625\right )-2 \left (2+e^6\right ) x^3}{x^3 \left (x^2-e^{\frac {\left (x^2-5 x-25\right )^2}{x^2}} \left (x+e^6+2\right )^2+\left (2+e^6\right ) x\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 \left (-2 x^5+6 \left (1-\frac {e^6}{3}\right ) x^4+20 \left (1+\frac {e^6}{2}\right ) x^3+252 \left (1+\frac {e^6}{252}\right ) x^2+1750 \left (1+\frac {e^6}{7}\right ) x+2500 \left (1+\frac {e^6}{2}\right )\right )}{x^2 \left (x+e^6+2\right ) \left (e^{\frac {\left (x^2-5 x-25\right )^2}{x^2}} x+2 \left (1+\frac {e^6}{2}\right ) e^{\frac {\left (x^2-5 x-25\right )^2}{x^2}}-x\right )}-\frac {4 \left (x^2+25\right ) \left (x^2-5 x-25\right )}{x^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 4 \int \frac {x^2}{-e^{\frac {\left (x^2-5 x-25\right )^2}{x^2}} x+x-2 e^{\frac {\left (x^2-5 x-25\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )}dx+2500 \int \frac {1}{x^2 \left (e^{\frac {\left (x^2-5 x-25\right )^2}{x^2}} x-x+2 e^{\frac {\left (x^2-5 x-25\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )\right )}dx+500 \int \frac {1}{x \left (e^{\frac {\left (x^2-5 x-25\right )^2}{x^2}} x-x+2 e^{\frac {\left (x^2-5 x-25\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )\right )}dx+20 \int \frac {x}{e^{\frac {\left (x^2-5 x-25\right )^2}{x^2}} x-x+2 e^{\frac {\left (x^2-5 x-25\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )}dx+2 \left (2+e^6\right ) \int \frac {1}{\left (x+e^6+2\right ) \left (e^{\frac {\left (x^2-5 x-25\right )^2}{x^2}} x-x+2 e^{\frac {\left (x^2-5 x-25\right )^2}{x^2}} \left (1+\frac {e^6}{2}\right )\right )}dx-\frac {2 \left (-x^2+5 x+25\right )^2}{x^2}\)

input

Int[(4*x^3 + 2*E^6*x^3 + E^((625 + 250*x - 25*x^2 - 10*x^3 + x^4)/x^2)*(10 
000 + 12000*x + 4500*x^2 + 580*x^3 + 64*x^4 + 4*x^5 - 4*x^6 + E^12*(2500 + 
 500*x + 20*x^3 - 4*x^4) + E^6*(10000 + 7000*x + 1000*x^2 + 80*x^3 + 24*x^ 
4 - 8*x^5)))/(-2*x^4 - E^6*x^4 - x^5 + E^((625 + 250*x - 25*x^2 - 10*x^3 + 
 x^4)/x^2)*(4*x^3 + E^12*x^3 + 4*x^4 + x^5 + E^6*(4*x^3 + 2*x^4))),x]
 

output

$Aborted
 

3.16.59.3.1 Defintions of rubi rules used

rule 6

Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v 
+ (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] &&  !FreeQ[Fx, x]
 

rule 2009

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

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7293

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

3.16.59.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(70\) vs. \(2(32)=64\).

Time = 1.01 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.22

method	result	size
risch	\(-2 x^{2}+20 x +\frac {-500 x -1250}{x^{2}}+\frac {2 x^{4}-20 x^{3}-50 x^{2}+500 x +1250}{x^{2}}-2 \ln \left ({\mathrm e}^{\frac {\left (x^{2}-5 x -25\right )^{2}}{x^{2}}}-\frac {x}{{\mathrm e}^{6}+2+x}\right )\)	\(71\)
norman	\(2 \ln \left ({\mathrm e}^{6}+2+x \right )-2 \ln \left ({\mathrm e}^{\frac {x^{4}-10 x^{3}-25 x^{2}+250 x +625}{x^{2}}} {\mathrm e}^{6}+{\mathrm e}^{\frac {x^{4}-10 x^{3}-25 x^{2}+250 x +625}{x^{2}}} x -x +2 \,{\mathrm e}^{\frac {x^{4}-10 x^{3}-25 x^{2}+250 x +625}{x^{2}}}\right )\)	\(97\)
parallelrisch	\(2 \ln \left ({\mathrm e}^{6}+2+x \right )-2 \ln \left ({\mathrm e}^{\frac {x^{4}-10 x^{3}-25 x^{2}+250 x +625}{x^{2}}} {\mathrm e}^{6}+{\mathrm e}^{\frac {x^{4}-10 x^{3}-25 x^{2}+250 x +625}{x^{2}}} x -x +2 \,{\mathrm e}^{\frac {x^{4}-10 x^{3}-25 x^{2}+250 x +625}{x^{2}}}\right )\)	\(97\)

input

int((((-4*x^4+20*x^3+500*x+2500)*exp(3)^4+(-8*x^5+24*x^4+80*x^3+1000*x^2+7 
000*x+10000)*exp(3)^2-4*x^6+4*x^5+64*x^4+580*x^3+4500*x^2+12000*x+10000)*e 
xp((x^4-10*x^3-25*x^2+250*x+625)/x^2)+2*x^3*exp(3)^2+4*x^3)/((x^3*exp(3)^4 
+(2*x^4+4*x^3)*exp(3)^2+x^5+4*x^4+4*x^3)*exp((x^4-10*x^3-25*x^2+250*x+625) 
/x^2)-x^4*exp(3)^2-x^5-2*x^4),x,method=_RETURNVERBOSE)
 

output

-2*x^2+20*x+(-500*x-1250)/x^2+2*(x^4-10*x^3-25*x^2+250*x+625)/x^2-2*ln(exp 
((x^2-5*x-25)^2/x^2)-x/(exp(6)+2+x))
 

3.16.59.5 Fricas [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx=-2 \, \log \left (\frac {{\left (x + e^{6} + 2\right )} e^{\left (\frac {x^{4} - 10 \, x^{3} - 25 \, x^{2} + 250 \, x + 625}{x^{2}}\right )} - x}{x + e^{6} + 2}\right ) \]

input

integrate((((-4*x^4+20*x^3+500*x+2500)*exp(3)^4+(-8*x^5+24*x^4+80*x^3+1000 
*x^2+7000*x+10000)*exp(3)^2-4*x^6+4*x^5+64*x^4+580*x^3+4500*x^2+12000*x+10 
000)*exp((x^4-10*x^3-25*x^2+250*x+625)/x^2)+2*x^3*exp(3)^2+4*x^3)/((x^3*ex 
p(3)^4+(2*x^4+4*x^3)*exp(3)^2+x^5+4*x^4+4*x^3)*exp((x^4-10*x^3-25*x^2+250* 
x+625)/x^2)-x^4*exp(3)^2-x^5-2*x^4),x, algorithm=\
 

output

-2*log(((x + e^6 + 2)*e^((x^4 - 10*x^3 - 25*x^2 + 250*x + 625)/x^2) - x)/( 
x + e^6 + 2))
 

3.16.59.6 Sympy [A] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.12 \[ \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx=- 2 \log {\left (- \frac {x}{x + 2 + e^{6}} + e^{\frac {x^{4} - 10 x^{3} - 25 x^{2} + 250 x + 625}{x^{2}}} \right )} \]

input

integrate((((-4*x**4+20*x**3+500*x+2500)*exp(3)**4+(-8*x**5+24*x**4+80*x** 
3+1000*x**2+7000*x+10000)*exp(3)**2-4*x**6+4*x**5+64*x**4+580*x**3+4500*x* 
*2+12000*x+10000)*exp((x**4-10*x**3-25*x**2+250*x+625)/x**2)+2*x**3*exp(3) 
**2+4*x**3)/((x**3*exp(3)**4+(2*x**4+4*x**3)*exp(3)**2+x**5+4*x**4+4*x**3) 
*exp((x**4-10*x**3-25*x**2+250*x+625)/x**2)-x**4*exp(3)**2-x**5-2*x**4),x)
 

output

-2*log(-x/(x + 2 + exp(6)) + exp((x**4 - 10*x**3 - 25*x**2 + 250*x + 625)/ 
x**2))
 

3.16.59.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (30) = 60\).

Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.19 \[ \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx=-\frac {2 \, {\left (x^{3} - 10 \, x^{2} + 250\right )}}{x} - 2 \, \log \left (\frac {{\left ({\left (x + e^{6} + 2\right )} e^{\left (x^{2} + \frac {250}{x} + \frac {625}{x^{2}}\right )} - x e^{\left (10 \, x + 25\right )}\right )} e^{\left (-x^{2} - \frac {250}{x}\right )}}{x + e^{6} + 2}\right ) \]

input

integrate((((-4*x^4+20*x^3+500*x+2500)*exp(3)^4+(-8*x^5+24*x^4+80*x^3+1000 
*x^2+7000*x+10000)*exp(3)^2-4*x^6+4*x^5+64*x^4+580*x^3+4500*x^2+12000*x+10 
000)*exp((x^4-10*x^3-25*x^2+250*x+625)/x^2)+2*x^3*exp(3)^2+4*x^3)/((x^3*ex 
p(3)^4+(2*x^4+4*x^3)*exp(3)^2+x^5+4*x^4+4*x^3)*exp((x^4-10*x^3-25*x^2+250* 
x+625)/x^2)-x^4*exp(3)^2-x^5-2*x^4),x, algorithm=\
 

output

-2*(x^3 - 10*x^2 + 250)/x - 2*log(((x + e^6 + 2)*e^(x^2 + 250/x + 625/x^2) 
 - x*e^(10*x + 25))*e^(-x^2 - 250/x)/(x + e^6 + 2))
 

3.16.59.8 Giac [F(-1)]

Timed out. \[ \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx=\text {Timed out} \]

input

integrate((((-4*x^4+20*x^3+500*x+2500)*exp(3)^4+(-8*x^5+24*x^4+80*x^3+1000 
*x^2+7000*x+10000)*exp(3)^2-4*x^6+4*x^5+64*x^4+580*x^3+4500*x^2+12000*x+10 
000)*exp((x^4-10*x^3-25*x^2+250*x+625)/x^2)+2*x^3*exp(3)^2+4*x^3)/((x^3*ex 
p(3)^4+(2*x^4+4*x^3)*exp(3)^2+x^5+4*x^4+4*x^3)*exp((x^4-10*x^3-25*x^2+250* 
x+625)/x^2)-x^4*exp(3)^2-x^5-2*x^4),x, algorithm=\
 

output

Timed out
 

3.16.59.9 Mupad [B] (verification not implemented)

Time = 9.65 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.97 \[ \int \frac {4 x^3+2 e^6 x^3+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (10000+12000 x+4500 x^2+580 x^3+64 x^4+4 x^5-4 x^6+e^{12} \left (2500+500 x+20 x^3-4 x^4\right )+e^6 \left (10000+7000 x+1000 x^2+80 x^3+24 x^4-8 x^5\right )\right )}{-2 x^4-e^6 x^4-x^5+e^{\frac {625+250 x-25 x^2-10 x^3+x^4}{x^2}} \left (4 x^3+e^{12} x^3+4 x^4+x^5+e^6 \left (4 x^3+2 x^4\right )\right )} \, dx=-2\,\ln \left (\frac {2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{250/x}\,{\mathrm {e}}^{\frac {625}{x^2}}-x\,{\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{25}+{\mathrm {e}}^{x^2}\,{\mathrm {e}}^6\,{\mathrm {e}}^{250/x}\,{\mathrm {e}}^{\frac {625}{x^2}}+x\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{250/x}\,{\mathrm {e}}^{\frac {625}{x^2}}}{2\,{\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{25}+{\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{31}+x\,{\mathrm {e}}^{10\,x}\,{\mathrm {e}}^{25}}\right ) \]

input

int(-(exp((250*x - 25*x^2 - 10*x^3 + x^4 + 625)/x^2)*(12000*x + exp(12)*(5 
00*x + 20*x^3 - 4*x^4 + 2500) + exp(6)*(7000*x + 1000*x^2 + 80*x^3 + 24*x^ 
4 - 8*x^5 + 10000) + 4500*x^2 + 580*x^3 + 64*x^4 + 4*x^5 - 4*x^6 + 10000) 
+ 2*x^3*exp(6) + 4*x^3)/(x^4*exp(6) - exp((250*x - 25*x^2 - 10*x^3 + x^4 + 
 625)/x^2)*(exp(6)*(4*x^3 + 2*x^4) + x^3*exp(12) + 4*x^3 + 4*x^4 + x^5) + 
2*x^4 + x^5),x)
 

output

-2*log((2*exp(x^2)*exp(250/x)*exp(625/x^2) - x*exp(10*x)*exp(25) + exp(x^2 
)*exp(6)*exp(250/x)*exp(625/x^2) + x*exp(x^2)*exp(250/x)*exp(625/x^2))/(2* 
exp(10*x)*exp(25) + exp(10*x)*exp(31) + x*exp(10*x)*exp(25)))