\(\int \frac {x^{\frac {x}{x^2+(25 x^2+10 x^3+11 x^4+2 x^5+x^6)^x \log (x)}} (5 x^2+x^3+x^4+(-5 x^2-x^3-x^4) \log (x)+(25 x^2+10 x^3+11 x^4+2 x^5+x^6)^x ((5-9 x-3 x^2-6 x^3) \log ^2(x)+(-5 x-x^2-x^3) \log ^2(x) \log (25 x^2+10 x^3+11 x^4+2 x^5+x^6)))}{5 x^4+x^5+x^6+(10 x^2+2 x^3+2 x^4) (25 x^2+10 x^3+11 x^4+2 x^5+x^6)^x \log (x)+(5+x+x^2) (25 x^2+10 x^3+11 x^4+2 x^5+x^6)^{2 x} \log ^2(x)} \, dx\) [698]

Optimal result

Integrand size = 261, antiderivative size = 29 \[ \int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx=e^{\frac {x}{\left (\left (x+x \left (4+x+x^2\right )\right )^2\right )^x+\frac {x^2}{\log (x)}}} \]

[Out]

exp(x/(exp(ln((x+x*(x^2+x+4))^2)*x)+x^2/ln(x)))

Rubi [F]

\[ \int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx=\int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx \]

[In]

Int[(x^(x/(x^2 + (25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)^x*Log[x]))*(5*x^2 + x^3 + x^4 + (-5*x^2 - x^3 - x^4)
*Log[x] + (25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)^x*((5 - 9*x - 3*x^2 - 6*x^3)*Log[x]^2 + (-5*x - x^2 - x^3)*
Log[x]^2*Log[25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6])))/(5*x^4 + x^5 + x^6 + (10*x^2 + 2*x^3 + 2*x^4)*(25*x^2
+ 10*x^3 + 11*x^4 + 2*x^5 + x^6)^x*Log[x] + (5 + x + x^2)*(25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)^(2*x)*Log[x
]^2),x]

[Out]

((10*I)*Defer[Int][x^(2 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))/((-1 + I*Sqrt[19] - 2*x)*(x^2 + (x^2*(5 +
x + x^2)^2)^x*Log[x])^2), x])/Sqrt[19] + ((2*I)*Defer[Int][x^(3 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))/((
-1 + I*Sqrt[19] - 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])^2), x])/Sqrt[19] + ((2*I)*Defer[Int][x^(4 + x/(x
^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))/((-1 + I*Sqrt[19] - 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])^2), x])/
Sqrt[19] + ((10*I)*Defer[Int][x^(2 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))/((1 + I*Sqrt[19] + 2*x)*(x^2 +
(x^2*(5 + x + x^2)^2)^x*Log[x])^2), x])/Sqrt[19] + ((2*I)*Defer[Int][x^(3 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*L
og[x]))/((1 + I*Sqrt[19] + 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])^2), x])/Sqrt[19] + ((2*I)*Defer[Int][x^
(4 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))/((1 + I*Sqrt[19] + 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])^
2), x])/Sqrt[19] - ((20*I)*Defer[Int][(x^(2 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x])/((-1 + I*Sqrt[
19] - 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])^2), x])/Sqrt[19] + ((16*I)*Defer[Int][(x^(3 + x/(x^2 + (x^2*
(5 + x + x^2)^2)^x*Log[x]))*Log[x])/((-1 + I*Sqrt[19] - 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])^2), x])/Sq
rt[19] + ((4*I)*Defer[Int][(x^(4 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x])/((-1 + I*Sqrt[19] - 2*x)*
(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])^2), x])/Sqrt[19] + ((12*I)*Defer[Int][(x^(5 + x/(x^2 + (x^2*(5 + x + x^
2)^2)^x*Log[x]))*Log[x])/((-1 + I*Sqrt[19] - 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])^2), x])/Sqrt[19] - ((
20*I)*Defer[Int][(x^(2 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x])/((1 + I*Sqrt[19] + 2*x)*(x^2 + (x^2
*(5 + x + x^2)^2)^x*Log[x])^2), x])/Sqrt[19] + ((16*I)*Defer[Int][(x^(3 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log
[x]))*Log[x])/((1 + I*Sqrt[19] + 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])^2), x])/Sqrt[19] + ((4*I)*Defer[I
nt][(x^(4 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x])/((1 + I*Sqrt[19] + 2*x)*(x^2 + (x^2*(5 + x + x^2
)^2)^x*Log[x])^2), x])/Sqrt[19] + ((12*I)*Defer[Int][(x^(5 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x])
/((1 + I*Sqrt[19] + 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])^2), x])/Sqrt[19] + ((10*I)*Defer[Int][(x^(x/(x
^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x])/((-1 + I*Sqrt[19] - 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))
, x])/Sqrt[19] - ((18*I)*Defer[Int][(x^(1 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x])/((-1 + I*Sqrt[19
] - 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])), x])/Sqrt[19] - ((6*I)*Defer[Int][(x^(2 + x/(x^2 + (x^2*(5 +
x + x^2)^2)^x*Log[x]))*Log[x])/((-1 + I*Sqrt[19] - 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])), x])/Sqrt[19]
- ((12*I)*Defer[Int][(x^(3 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x])/((-1 + I*Sqrt[19] - 2*x)*(x^2 +
(x^2*(5 + x + x^2)^2)^x*Log[x])), x])/Sqrt[19] + ((10*I)*Defer[Int][(x^(x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[
x]))*Log[x])/((1 + I*Sqrt[19] + 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])), x])/Sqrt[19] - ((18*I)*Defer[Int
][(x^(1 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x])/((1 + I*Sqrt[19] + 2*x)*(x^2 + (x^2*(5 + x + x^2)^
2)^x*Log[x])), x])/Sqrt[19] - ((6*I)*Defer[Int][(x^(2 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x])/((1
+ I*Sqrt[19] + 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])), x])/Sqrt[19] - ((12*I)*Defer[Int][(x^(3 + x/(x^2
+ (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x])/((1 + I*Sqrt[19] + 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])), x]
)/Sqrt[19] + ((10*I)*Defer[Int][(x^(3 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x]*Log[x^2*(5 + x + x^2)
^2])/((-1 + I*Sqrt[19] - 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])^2), x])/Sqrt[19] + ((2*I)*Defer[Int][(x^(
4 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x]*Log[x^2*(5 + x + x^2)^2])/((-1 + I*Sqrt[19] - 2*x)*(x^2 +
(x^2*(5 + x + x^2)^2)^x*Log[x])^2), x])/Sqrt[19] + ((2*I)*Defer[Int][(x^(5 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x
*Log[x]))*Log[x]*Log[x^2*(5 + x + x^2)^2])/((-1 + I*Sqrt[19] - 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])^2),
x])/Sqrt[19] + ((10*I)*Defer[Int][(x^(3 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x]*Log[x^2*(5 + x + x
^2)^2])/((1 + I*Sqrt[19] + 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])^2), x])/Sqrt[19] + ((2*I)*Defer[Int][(x
^(4 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x]*Log[x^2*(5 + x + x^2)^2])/((1 + I*Sqrt[19] + 2*x)*(x^2
+ (x^2*(5 + x + x^2)^2)^x*Log[x])^2), x])/Sqrt[19] + ((2*I)*Defer[Int][(x^(5 + x/(x^2 + (x^2*(5 + x + x^2)^2)^
x*Log[x]))*Log[x]*Log[x^2*(5 + x + x^2)^2])/((1 + I*Sqrt[19] + 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])^2),
x])/Sqrt[19] - ((10*I)*Defer[Int][(x^(1 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x]*Log[x^2*(5 + x + x
^2)^2])/((-1 + I*Sqrt[19] - 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])), x])/Sqrt[19] - ((2*I)*Defer[Int][(x^
(2 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x]*Log[x^2*(5 + x + x^2)^2])/((-1 + I*Sqrt[19] - 2*x)*(x^2
+ (x^2*(5 + x + x^2)^2)^x*Log[x])), x])/Sqrt[19] - ((2*I)*Defer[Int][(x^(3 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*
Log[x]))*Log[x]*Log[x^2*(5 + x + x^2)^2])/((-1 + I*Sqrt[19] - 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])), x]
)/Sqrt[19] - ((10*I)*Defer[Int][(x^(1 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x]*Log[x^2*(5 + x + x^2)
^2])/((1 + I*Sqrt[19] + 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])), x])/Sqrt[19] - ((2*I)*Defer[Int][(x^(2 +
x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))*Log[x]*Log[x^2*(5 + x + x^2)^2])/((1 + I*Sqrt[19] + 2*x)*(x^2 + (x^
2*(5 + x + x^2)^2)^x*Log[x])), x])/Sqrt[19] - ((2*I)*Defer[Int][(x^(3 + x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x
]))*Log[x]*Log[x^2*(5 + x + x^2)^2])/((1 + I*Sqrt[19] + 2*x)*(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x])), x])/Sqrt
[19]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^{\frac {x}{x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)}} \left (x^2 \left (5+x+x^2\right )-x^2 \left (5+x+x^2\right ) \log (x)-\left (x^2 \left (5+x+x^2\right )^2\right )^x \log ^2(x) \left (-5+9 x+3 x^2+6 x^3+x \left (5+x+x^2\right ) \log \left (x^2 \left (5+x+x^2\right )^2\right )\right )\right )}{\left (5+x+x^2\right ) \left (x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)\right )^2} \, dx \\ & = \int \left (-\frac {x^{\frac {x}{x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)}} \log (x) \left (-5+9 x+3 x^2+6 x^3+5 x \log \left (x^2 \left (5+x+x^2\right )^2\right )+x^2 \log \left (x^2 \left (5+x+x^2\right )^2\right )+x^3 \log \left (x^2 \left (5+x+x^2\right )^2\right )\right )}{\left (5+x+x^2\right ) \left (x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)\right )}+\frac {x^{2+\frac {x}{x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)}} \left (5+x+x^2-10 \log (x)+8 x \log (x)+2 x^2 \log (x)+6 x^3 \log (x)+5 x \log (x) \log \left (x^2 \left (5+x+x^2\right )^2\right )+x^2 \log (x) \log \left (x^2 \left (5+x+x^2\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (5+x+x^2\right )^2\right )\right )}{\left (5+x+x^2\right ) \left (x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)\right )^2}\right ) \, dx \\ & = -\int \frac {x^{\frac {x}{x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)}} \log (x) \left (-5+9 x+3 x^2+6 x^3+5 x \log \left (x^2 \left (5+x+x^2\right )^2\right )+x^2 \log \left (x^2 \left (5+x+x^2\right )^2\right )+x^3 \log \left (x^2 \left (5+x+x^2\right )^2\right )\right )}{\left (5+x+x^2\right ) \left (x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)\right )} \, dx+\int \frac {x^{2+\frac {x}{x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)}} \left (5+x+x^2-10 \log (x)+8 x \log (x)+2 x^2 \log (x)+6 x^3 \log (x)+5 x \log (x) \log \left (x^2 \left (5+x+x^2\right )^2\right )+x^2 \log (x) \log \left (x^2 \left (5+x+x^2\right )^2\right )+x^3 \log (x) \log \left (x^2 \left (5+x+x^2\right )^2\right )\right )}{\left (5+x+x^2\right ) \left (x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)\right )^2} \, dx \\ & = -\int \frac {x^{\frac {x}{x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)}} \log (x) \left (-5+9 x+3 x^2+6 x^3+x \left (5+x+x^2\right ) \log \left (x^2 \left (5+x+x^2\right )^2\right )\right )}{\left (5+x+x^2\right ) \left (x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)\right )} \, dx+\int \frac {x^{2+\frac {x}{x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)}} \left (5+x+x^2+\log (x) \left (2 \left (-5+4 x+x^2+3 x^3\right )+x \left (5+x+x^2\right ) \log \left (x^2 \left (5+x+x^2\right )^2\right )\right )\right )}{\left (5+x+x^2\right ) \left (x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)\right )^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx=x^{\frac {x}{x^2+\left (x^2 \left (5+x+x^2\right )^2\right )^x \log (x)}} \]

[In]

Integrate[(x^(x/(x^2 + (25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)^x*Log[x]))*(5*x^2 + x^3 + x^4 + (-5*x^2 - x^3
- x^4)*Log[x] + (25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)^x*((5 - 9*x - 3*x^2 - 6*x^3)*Log[x]^2 + (-5*x - x^2 -
x^3)*Log[x]^2*Log[25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6])))/(5*x^4 + x^5 + x^6 + (10*x^2 + 2*x^3 + 2*x^4)*(2
5*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)^x*Log[x] + (5 + x + x^2)*(25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)^(2*x)
*Log[x]^2),x]

[Out]

x^(x/(x^2 + (x^2*(5 + x + x^2)^2)^x*Log[x]))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.10 (sec) , antiderivative size = 124, normalized size of antiderivative = 4.28

\[x^{\frac {x}{\ln \left (x \right ) x^{2 x} \left (x^{2}+x +5\right )^{2 x} {\mathrm e}^{-\frac {i \pi x \left (\operatorname {csgn}\left (i \left (x^{2}+x +5\right )^{2}\right )-2 \,\operatorname {csgn}\left (i \left (x^{2}+x +5\right )\right )+\operatorname {csgn}\left (i \left (x^{2}+x +5\right )^{2}\right ) \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{2} \left (x^{2}+x +5\right )^{2}\right )+\operatorname {csgn}\left (i x^{2}\right )-2 \,\operatorname {csgn}\left (i x \right )+\operatorname {csgn}\left (i x^{2} \left (x^{2}+x +5\right )^{2}\right )\right )}{2}}+x^{2}}}\]

[In]

int((((-x^3-x^2-5*x)*ln(x)^2*ln(x^6+2*x^5+11*x^4+10*x^3+25*x^2)+(-6*x^3-3*x^2-9*x+5)*ln(x)^2)*exp(x*ln(x^6+2*x
^5+11*x^4+10*x^3+25*x^2))+(-x^4-x^3-5*x^2)*ln(x)+x^4+x^3+5*x^2)*exp(x*ln(x)/(ln(x)*exp(x*ln(x^6+2*x^5+11*x^4+1
0*x^3+25*x^2))+x^2))/((x^2+x+5)*ln(x)^2*exp(x*ln(x^6+2*x^5+11*x^4+10*x^3+25*x^2))^2+(2*x^4+2*x^3+10*x^2)*ln(x)
*exp(x*ln(x^6+2*x^5+11*x^4+10*x^3+25*x^2))+x^6+x^5+5*x^4),x)

[Out]

x^(x/(ln(x)*x^(2*x)*(x^2+x+5)^(2*x)*exp(-1/2*I*Pi*x*(csgn(I*(x^2+x+5)^2)-2*csgn(I*(x^2+x+5))+csgn(I*(x^2+x+5)^
2)*csgn(I*x^2)*csgn(I*x^2*(x^2+x+5)^2)+csgn(I*x^2)-2*csgn(I*x)+csgn(I*x^2*(x^2+x+5)^2)))+x^2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx=x^{\frac {x}{x^{2} + {\left (x^{6} + 2 \, x^{5} + 11 \, x^{4} + 10 \, x^{3} + 25 \, x^{2}\right )}^{x} \log \left (x\right )}} \]

[In]

integrate((((-x^3-x^2-5*x)*log(x)^2*log(x^6+2*x^5+11*x^4+10*x^3+25*x^2)+(-6*x^3-3*x^2-9*x+5)*log(x)^2)*exp(x*l
og(x^6+2*x^5+11*x^4+10*x^3+25*x^2))+(-x^4-x^3-5*x^2)*log(x)+x^4+x^3+5*x^2)*exp(x*log(x)/(log(x)*exp(x*log(x^6+
2*x^5+11*x^4+10*x^3+25*x^2))+x^2))/((x^2+x+5)*log(x)^2*exp(x*log(x^6+2*x^5+11*x^4+10*x^3+25*x^2))^2+(2*x^4+2*x
^3+10*x^2)*log(x)*exp(x*log(x^6+2*x^5+11*x^4+10*x^3+25*x^2))+x^6+x^5+5*x^4),x, algorithm="fricas")

[Out]

x^(x/(x^2 + (x^6 + 2*x^5 + 11*x^4 + 10*x^3 + 25*x^2)^x*log(x)))

Sympy [F(-1)]

Timed out. \[ \int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx=\text {Timed out} \]

[In]

integrate((((-x**3-x**2-5*x)*ln(x)**2*ln(x**6+2*x**5+11*x**4+10*x**3+25*x**2)+(-6*x**3-3*x**2-9*x+5)*ln(x)**2)
*exp(x*ln(x**6+2*x**5+11*x**4+10*x**3+25*x**2))+(-x**4-x**3-5*x**2)*ln(x)+x**4+x**3+5*x**2)*exp(x*ln(x)/(ln(x)
*exp(x*ln(x**6+2*x**5+11*x**4+10*x**3+25*x**2))+x**2))/((x**2+x+5)*ln(x)**2*exp(x*ln(x**6+2*x**5+11*x**4+10*x*
*3+25*x**2))**2+(2*x**4+2*x**3+10*x**2)*ln(x)*exp(x*ln(x**6+2*x**5+11*x**4+10*x**3+25*x**2))+x**6+x**5+5*x**4)
,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx=x^{\frac {x}{x^{2} + e^{\left (2 \, x \log \left (x^{2} + x + 5\right ) + 2 \, x \log \left (x\right )\right )} \log \left (x\right )}} \]

[In]

integrate((((-x^3-x^2-5*x)*log(x)^2*log(x^6+2*x^5+11*x^4+10*x^3+25*x^2)+(-6*x^3-3*x^2-9*x+5)*log(x)^2)*exp(x*l
og(x^6+2*x^5+11*x^4+10*x^3+25*x^2))+(-x^4-x^3-5*x^2)*log(x)+x^4+x^3+5*x^2)*exp(x*log(x)/(log(x)*exp(x*log(x^6+
2*x^5+11*x^4+10*x^3+25*x^2))+x^2))/((x^2+x+5)*log(x)^2*exp(x*log(x^6+2*x^5+11*x^4+10*x^3+25*x^2))^2+(2*x^4+2*x
^3+10*x^2)*log(x)*exp(x*log(x^6+2*x^5+11*x^4+10*x^3+25*x^2))+x^6+x^5+5*x^4),x, algorithm="maxima")

[Out]

x^(x/(x^2 + e^(2*x*log(x^2 + x + 5) + 2*x*log(x))*log(x)))

Giac [F]

\[ \int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx=\int { \frac {{\left (x^{4} + x^{3} - {\left ({\left (x^{3} + x^{2} + 5 \, x\right )} \log \left (x^{6} + 2 \, x^{5} + 11 \, x^{4} + 10 \, x^{3} + 25 \, x^{2}\right ) \log \left (x\right )^{2} + {\left (6 \, x^{3} + 3 \, x^{2} + 9 \, x - 5\right )} \log \left (x\right )^{2}\right )} {\left (x^{6} + 2 \, x^{5} + 11 \, x^{4} + 10 \, x^{3} + 25 \, x^{2}\right )}^{x} + 5 \, x^{2} - {\left (x^{4} + x^{3} + 5 \, x^{2}\right )} \log \left (x\right )\right )} x^{\frac {x}{x^{2} + {\left (x^{6} + 2 \, x^{5} + 11 \, x^{4} + 10 \, x^{3} + 25 \, x^{2}\right )}^{x} \log \left (x\right )}}}{x^{6} + x^{5} + 5 \, x^{4} + {\left (x^{2} + x + 5\right )} {\left (x^{6} + 2 \, x^{5} + 11 \, x^{4} + 10 \, x^{3} + 25 \, x^{2}\right )}^{2 \, x} \log \left (x\right )^{2} + 2 \, {\left (x^{4} + x^{3} + 5 \, x^{2}\right )} {\left (x^{6} + 2 \, x^{5} + 11 \, x^{4} + 10 \, x^{3} + 25 \, x^{2}\right )}^{x} \log \left (x\right )} \,d x } \]

[In]

integrate((((-x^3-x^2-5*x)*log(x)^2*log(x^6+2*x^5+11*x^4+10*x^3+25*x^2)+(-6*x^3-3*x^2-9*x+5)*log(x)^2)*exp(x*l
og(x^6+2*x^5+11*x^4+10*x^3+25*x^2))+(-x^4-x^3-5*x^2)*log(x)+x^4+x^3+5*x^2)*exp(x*log(x)/(log(x)*exp(x*log(x^6+
2*x^5+11*x^4+10*x^3+25*x^2))+x^2))/((x^2+x+5)*log(x)^2*exp(x*log(x^6+2*x^5+11*x^4+10*x^3+25*x^2))^2+(2*x^4+2*x
^3+10*x^2)*log(x)*exp(x*log(x^6+2*x^5+11*x^4+10*x^3+25*x^2))+x^6+x^5+5*x^4),x, algorithm="giac")

[Out]

integrate((x^4 + x^3 - ((x^3 + x^2 + 5*x)*log(x^6 + 2*x^5 + 11*x^4 + 10*x^3 + 25*x^2)*log(x)^2 + (6*x^3 + 3*x^
2 + 9*x - 5)*log(x)^2)*(x^6 + 2*x^5 + 11*x^4 + 10*x^3 + 25*x^2)^x + 5*x^2 - (x^4 + x^3 + 5*x^2)*log(x))*x^(x/(
x^2 + (x^6 + 2*x^5 + 11*x^4 + 10*x^3 + 25*x^2)^x*log(x)))/(x^6 + x^5 + 5*x^4 + (x^2 + x + 5)*(x^6 + 2*x^5 + 11
*x^4 + 10*x^3 + 25*x^2)^(2*x)*log(x)^2 + 2*(x^4 + x^3 + 5*x^2)*(x^6 + 2*x^5 + 11*x^4 + 10*x^3 + 25*x^2)^x*log(
x)), x)

Mupad [B] (verification not implemented)

Time = 8.73 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {x^{\frac {x}{x^2+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)}} \left (5 x^2+x^3+x^4+\left (-5 x^2-x^3-x^4\right ) \log (x)+\left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \left (\left (5-9 x-3 x^2-6 x^3\right ) \log ^2(x)+\left (-5 x-x^2-x^3\right ) \log ^2(x) \log \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )\right )\right )}{5 x^4+x^5+x^6+\left (10 x^2+2 x^3+2 x^4\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^x \log (x)+\left (5+x+x^2\right ) \left (25 x^2+10 x^3+11 x^4+2 x^5+x^6\right )^{2 x} \log ^2(x)} \, dx=x^{\frac {x}{\ln \left (x\right )\,{\left (x^6+2\,x^5+11\,x^4+10\,x^3+25\,x^2\right )}^x+x^2}} \]

[In]

int((exp((x*log(x))/(x^2 + exp(x*log(25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6))*log(x)))*(5*x^2 - exp(x*log(25*x
^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6))*(log(x)^2*(9*x + 3*x^2 + 6*x^3 - 5) + log(25*x^2 + 10*x^3 + 11*x^4 + 2*x^
5 + x^6)*log(x)^2*(5*x + x^2 + x^3)) - log(x)*(5*x^2 + x^3 + x^4) + x^3 + x^4))/(5*x^4 + x^5 + x^6 + exp(x*log
(25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6))*log(x)*(10*x^2 + 2*x^3 + 2*x^4) + exp(2*x*log(25*x^2 + 10*x^3 + 11*x
^4 + 2*x^5 + x^6))*log(x)^2*(x + x^2 + 5)),x)

[Out]

x^(x/(log(x)*(25*x^2 + 10*x^3 + 11*x^4 + 2*x^5 + x^6)^x + x^2))