\(\int \frac {e^{\frac {25 x^4+(x+400 x^3+25 x^4) \log (x)}{25 x+(400+25 x) \log (x)}} (x+75 x^4+(2400 x^3+150 x^4) \log (x)+(16+19200 x^2+2400 x^3+75 x^4) \log ^2(x))}{25 x^2+(800 x+50 x^2) \log (x)+(6400+800 x+25 x^2) \log ^2(x)} \, dx\) [4326]

Optimal result

Integrand size = 117, antiderivative size = 25 \[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=1+e^{x \left (x^2+\frac {1}{25 \left (16+x+\frac {x}{\log (x)}\right )}\right )} \]

[Out]

1+exp((x^2+1/25/(x+16+x/ln(x)))*x)

Rubi [F]

\[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=\int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}\right ) \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx \]

[In]

Int[(E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*x + (400 + 25*x)*Log[x]))*(x + 75*x^4 + (2400*x^3 + 150*x
^4)*Log[x] + (16 + 19200*x^2 + 2400*x^3 + 75*x^4)*Log[x]^2))/(25*x^2 + (800*x + 50*x^2)*Log[x] + (6400 + 800*x
+ 25*x^2)*Log[x]^2),x]

[Out]

3*Defer[Int][E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*(x + 16*Log[x] + x*Log[x])))*x^2, x] + (16*Defer[
Int][E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*(x + 16*Log[x] + x*Log[x])))/(16 + x)^2, x])/25 + (16*Def
er[Int][E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*(x + 16*Log[x] + x*Log[x])))/(x + 16*Log[x] + x*Log[x]
)^2, x])/25 + Defer[Int][(E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*(x + 16*Log[x] + x*Log[x])))*x)/(x +
16*Log[x] + x*Log[x])^2, x]/25 + (4096*Defer[Int][E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*(x + 16*Log
[x] + x*Log[x])))/((16 + x)^2*(x + 16*Log[x] + x*Log[x])^2), x])/25 - (512*Defer[Int][E^((25*x^4 + (x + 400*x^
3 + 25*x^4)*Log[x])/(25*(x + 16*Log[x] + x*Log[x])))/((16 + x)*(x + 16*Log[x] + x*Log[x])^2), x])/25 + (512*De
fer[Int][E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*(x + 16*Log[x] + x*Log[x])))/((16 + x)^2*(x + 16*Log[
x] + x*Log[x])), x])/25 - (32*Defer[Int][E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*(x + 16*Log[x] + x*Lo
g[x])))/((16 + x)*(x + 16*Log[x] + x*Log[x])), x])/25

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 (x+16 \log (x)+x \log (x))^2} \, dx \\ & = \frac {1}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{(x+16 \log (x)+x \log (x))^2} \, dx \\ & = \frac {1}{25} \int \left (\frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) \left (16+19200 x^2+2400 x^3+75 x^4\right )}{(16+x)^2}+\frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x \left (256+48 x+x^2\right )}{(16+x)^2 (x+16 \log (x)+x \log (x))^2}-\frac {32 \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x}{(16+x)^2 (x+16 \log (x)+x \log (x))}\right ) \, dx \\ & = \frac {1}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) \left (16+19200 x^2+2400 x^3+75 x^4\right )}{(16+x)^2} \, dx+\frac {1}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x \left (256+48 x+x^2\right )}{(16+x)^2 (x+16 \log (x)+x \log (x))^2} \, dx-\frac {32}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x}{(16+x)^2 (x+16 \log (x)+x \log (x))} \, dx \\ & = \frac {1}{25} \int \left (75 \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x^2+\frac {16 \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x)^2}\right ) \, dx+\frac {1}{25} \int \left (\frac {16 \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(x+16 \log (x)+x \log (x))^2}+\frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x}{(x+16 \log (x)+x \log (x))^2}+\frac {4096 \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x)^2 (x+16 \log (x)+x \log (x))^2}-\frac {512 \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x) (x+16 \log (x)+x \log (x))^2}\right ) \, dx-\frac {32}{25} \int \left (-\frac {16 \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x)^2 (x+16 \log (x)+x \log (x))}+\frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x) (x+16 \log (x)+x \log (x))}\right ) \, dx \\ & = \frac {1}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x}{(x+16 \log (x)+x \log (x))^2} \, dx+\frac {16}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x)^2} \, dx+\frac {16}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(x+16 \log (x)+x \log (x))^2} \, dx-\frac {32}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x) (x+16 \log (x)+x \log (x))} \, dx+3 \int \exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right ) x^2 \, dx-\frac {512}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x) (x+16 \log (x)+x \log (x))^2} \, dx+\frac {512}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x)^2 (x+16 \log (x)+x \log (x))} \, dx+\frac {4096}{25} \int \frac {\exp \left (\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 (x+16 \log (x)+x \log (x))}\right )}{(16+x)^2 (x+16 \log (x)+x \log (x))^2} \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(25)=50\).

Time = 0.18 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.76 \[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=e^{x^3+\frac {x}{400+25 x}+\frac {x^4}{x+(16+x) \log (x)}} x^{-\frac {x^2+400 x^4+25 x^5}{25 (16+x) \log (x) (x+(16+x) \log (x))}} \]

[In]

Integrate[(E^((25*x^4 + (x + 400*x^3 + 25*x^4)*Log[x])/(25*x + (400 + 25*x)*Log[x]))*(x + 75*x^4 + (2400*x^3 +
150*x^4)*Log[x] + (16 + 19200*x^2 + 2400*x^3 + 75*x^4)*Log[x]^2))/(25*x^2 + (800*x + 50*x^2)*Log[x] + (6400 +
800*x + 25*x^2)*Log[x]^2),x]

[Out]

E^(x^3 + x/(400 + 25*x) + x^4/(x + (16 + x)*Log[x]))/x^((x^2 + 400*x^4 + 25*x^5)/(25*(16 + x)*Log[x]*(x + (16
+ x)*Log[x])))

Maple [A] (verified)

Time = 1.39 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.48

method	result	size
parallelrisch	\({\mathrm e}^{\frac {\left (25 x^{4}+400 x^{3}+x \right ) \ln \left (x \right )+25 x^{4}}{25 x \ln \left (x \right )+400 \ln \left (x \right )+25 x}}\)	\(37\)
risch	\({\mathrm e}^{\frac {x \left (25 x^{3} \ln \left (x \right )+400 x^{2} \ln \left (x \right )+25 x^{3}+\ln \left (x \right )\right )}{25 x \ln \left (x \right )+400 \ln \left (x \right )+25 x}}\)	\(39\)

[In]

int(((75*x^4+2400*x^3+19200*x^2+16)*ln(x)^2+(150*x^4+2400*x^3)*ln(x)+75*x^4+x)*exp(((25*x^4+400*x^3+x)*ln(x)+2
5*x^4)/((25*x+400)*ln(x)+25*x))/((25*x^2+800*x+6400)*ln(x)^2+(50*x^2+800*x)*ln(x)+25*x^2),x,method=_RETURNVERB
OSE)

[Out]

exp(1/25/(x*ln(x)+16*ln(x)+x)*((25*x^4+400*x^3+x)*ln(x)+25*x^4))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=e^{\left (\frac {25 \, x^{4} + {\left (25 \, x^{4} + 400 \, x^{3} + x\right )} \log \left (x\right )}{25 \, {\left ({\left (x + 16\right )} \log \left (x\right ) + x\right )}}\right )} \]

[In]

integrate(((75*x^4+2400*x^3+19200*x^2+16)*log(x)^2+(150*x^4+2400*x^3)*log(x)+75*x^4+x)*exp(((25*x^4+400*x^3+x)
*log(x)+25*x^4)/((25*x+400)*log(x)+25*x))/((25*x^2+800*x+6400)*log(x)^2+(50*x^2+800*x)*log(x)+25*x^2),x, algor
ithm="fricas")

[Out]

e^(1/25*(25*x^4 + (25*x^4 + 400*x^3 + x)*log(x))/((x + 16)*log(x) + x))

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=e^{\frac {25 x^{4} + \left (25 x^{4} + 400 x^{3} + x\right ) \log {\left (x \right )}}{25 x + \left (25 x + 400\right ) \log {\left (x \right )}}} \]

[In]

integrate(((75*x**4+2400*x**3+19200*x**2+16)*ln(x)**2+(150*x**4+2400*x**3)*ln(x)+75*x**4+x)*exp(((25*x**4+400*
x**3+x)*ln(x)+25*x**4)/((25*x+400)*ln(x)+25*x))/((25*x**2+800*x+6400)*ln(x)**2+(50*x**2+800*x)*ln(x)+25*x**2),
x)

[Out]

exp((25*x**4 + (25*x**4 + 400*x**3 + x)*log(x))/(25*x + (25*x + 400)*log(x)))

Maxima [F(-1)]

Timed out. \[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=\text {Timed out} \]

[In]

integrate(((75*x^4+2400*x^3+19200*x^2+16)*log(x)^2+(150*x^4+2400*x^3)*log(x)+75*x^4+x)*exp(((25*x^4+400*x^3+x)
*log(x)+25*x^4)/((25*x+400)*log(x)+25*x))/((25*x^2+800*x+6400)*log(x)^2+(50*x^2+800*x)*log(x)+25*x^2),x, algor
ithm="maxima")

[Out]

Timed out

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (23) = 46\).

Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.88 \[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=e^{\left (\frac {x^{4} \log \left (x\right )}{x \log \left (x\right ) + x + 16 \, \log \left (x\right )} + \frac {x^{4}}{x \log \left (x\right ) + x + 16 \, \log \left (x\right )} + \frac {16 \, x^{3} \log \left (x\right )}{x \log \left (x\right ) + x + 16 \, \log \left (x\right )} + \frac {x \log \left (x\right )}{25 \, {\left (x \log \left (x\right ) + x + 16 \, \log \left (x\right )\right )}}\right )} \]

[In]

integrate(((75*x^4+2400*x^3+19200*x^2+16)*log(x)^2+(150*x^4+2400*x^3)*log(x)+75*x^4+x)*exp(((25*x^4+400*x^3+x)
*log(x)+25*x^4)/((25*x+400)*log(x)+25*x))/((25*x^2+800*x+6400)*log(x)^2+(50*x^2+800*x)*log(x)+25*x^2),x, algor
ithm="giac")

[Out]

e^(x^4*log(x)/(x*log(x) + x + 16*log(x)) + x^4/(x*log(x) + x + 16*log(x)) + 16*x^3*log(x)/(x*log(x) + x + 16*l
og(x)) + 1/25*x*log(x)/(x*log(x) + x + 16*log(x)))

Mupad [B] (verification not implemented)

Time = 11.36 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.56 \[ \int \frac {e^{\frac {25 x^4+\left (x+400 x^3+25 x^4\right ) \log (x)}{25 x+(400+25 x) \log (x)}} \left (x+75 x^4+\left (2400 x^3+150 x^4\right ) \log (x)+\left (16+19200 x^2+2400 x^3+75 x^4\right ) \log ^2(x)\right )}{25 x^2+\left (800 x+50 x^2\right ) \log (x)+\left (6400+800 x+25 x^2\right ) \log ^2(x)} \, dx=x^{\frac {x^4+16\,x^3}{x+16\,\ln \left (x\right )+x\,\ln \left (x\right )}+\frac {x}{25\,x+400\,\ln \left (x\right )+25\,x\,\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {25\,x^4}{25\,x+400\,\ln \left (x\right )+25\,x\,\ln \left (x\right )}} \]

[In]

int((exp((log(x)*(x + 400*x^3 + 25*x^4) + 25*x^4)/(25*x + log(x)*(25*x + 400)))*(x + log(x)*(2400*x^3 + 150*x^
4) + log(x)^2*(19200*x^2 + 2400*x^3 + 75*x^4 + 16) + 75*x^4))/(log(x)^2*(800*x + 25*x^2 + 6400) + log(x)*(800*
x + 50*x^2) + 25*x^2),x)

[Out]

x^((16*x^3 + x^4)/(x + 16*log(x) + x*log(x)) + x/(25*x + 400*log(x) + 25*x*log(x)))*exp((25*x^4)/(25*x + 400*l
og(x) + 25*x*log(x)))