\(\int \frac {-10+8 x+12 e^x x+12 x^2+(\frac {-5+4 x+6 e^x x+6 x^2}{x})^x (-5-x+6 x^2-6 x^3+e^x (6 x-6 x^3)+(5 x-4 x^2-6 e^x x^2-6 x^3) \log (\frac {-5+4 x+6 e^x x+6 x^2}{x}))}{-60+48 x+72 e^x x+72 x^2+(\frac {-5+4 x+6 e^x x+6 x^2}{x})^{2 x} (-15+12 x+18 e^x x+18 x^2)+(\frac {-5+4 x+6 e^x x+6 x^2}{x})^x (-60+48 x+72 e^x x+72 x^2)} \, dx\) [7881]

Optimal result

Integrand size = 210, antiderivative size = 25 \[ \int \frac {-10+8 x+12 e^x x+12 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-5-x+6 x^2-6 x^3+e^x \left (6 x-6 x^3\right )+\left (5 x-4 x^2-6 e^x x^2-6 x^3\right ) \log \left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )\right )}{-60+48 x+72 e^x x+72 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^{2 x} \left (-15+12 x+18 e^x x+18 x^2\right )+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-60+48 x+72 e^x x+72 x^2\right )} \, dx=\frac {x}{3 \left (2+\left (4-\frac {5}{x}+6 \left (e^x+x\right )\right )^x\right )} \]

[Out]

x/(3*exp(x*ln(4-5/x+6*exp(x)+6*x))+6)

Rubi [F]

\[ \int \frac {-10+8 x+12 e^x x+12 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-5-x+6 x^2-6 x^3+e^x \left (6 x-6 x^3\right )+\left (5 x-4 x^2-6 e^x x^2-6 x^3\right ) \log \left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )\right )}{-60+48 x+72 e^x x+72 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^{2 x} \left (-15+12 x+18 e^x x+18 x^2\right )+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-60+48 x+72 e^x x+72 x^2\right )} \, dx=\int \frac {-10+8 x+12 e^x x+12 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-5-x+6 x^2-6 x^3+e^x \left (6 x-6 x^3\right )+\left (5 x-4 x^2-6 e^x x^2-6 x^3\right ) \log \left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )\right )}{-60+48 x+72 e^x x+72 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^{2 x} \left (-15+12 x+18 e^x x+18 x^2\right )+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-60+48 x+72 e^x x+72 x^2\right )} \, dx \]

[In]

Int[(-10 + 8*x + 12*E^x*x + 12*x^2 + ((-5 + 4*x + 6*E^x*x + 6*x^2)/x)^x*(-5 - x + 6*x^2 - 6*x^3 + E^x*(6*x - 6
*x^3) + (5*x - 4*x^2 - 6*E^x*x^2 - 6*x^3)*Log[(-5 + 4*x + 6*E^x*x + 6*x^2)/x]))/(-60 + 48*x + 72*E^x*x + 72*x^
2 + ((-5 + 4*x + 6*E^x*x + 6*x^2)/x)^(2*x)*(-15 + 12*x + 18*E^x*x + 18*x^2) + ((-5 + 4*x + 6*E^x*x + 6*x^2)/x)
^x*(-60 + 48*x + 72*E^x*x + 72*x^2)),x]

[Out]

(-10*Defer[Int][1/((-5 + 4*x + 6*E^x*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x])/3 + (8*Defer[Int][x/((
-5 + 4*x + 6*E^x*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x])/3 + 4*Defer[Int][(E^x*x)/((-5 + 4*x + 6*E^
x*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x] + 4*Defer[Int][x^2/((-5 + 4*x + 6*E^x*x + 6*x^2)*(2 + (4 +
6*E^x - 5/x + 6*x)^x)^2), x] - (5*Defer[Int][(4 + 6*E^x - 5/x + 6*x)^x/((-5 + 4*x + 6*E^x*x + 6*x^2)*(2 + (4
+ 6*E^x - 5/x + 6*x)^x)^2), x])/3 - Defer[Int][(x*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + 4*x + 6*E^x*x + 6*x^2)*(2
+ (4 + 6*E^x - 5/x + 6*x)^x)^2), x]/3 + (5*Log[4 + 6*E^x - 5/x + 6*x]*Defer[Int][(x*(4 + 6*E^x - 5/x + 6*x)^x)
/((-5 + 4*x + 6*E^x*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x])/3 + 2*Defer[Int][(E^x*x*(4 + 6*E^x - 5/
x + 6*x)^x)/((-5 + 4*x + 6*E^x*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x] + 2*Defer[Int][(x^2*(4 + 6*E^
x - 5/x + 6*x)^x)/((-5 + 4*x + 6*E^x*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x] - (4*Log[4 + 6*E^x - 5/
x + 6*x]*Defer[Int][(x^2*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + 4*x + 6*E^x*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)
^x)^2), x])/3 - 2*Log[4 + 6*E^x - 5/x + 6*x]*Defer[Int][(E^x*x^2*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + 4*x + 6*E^x
*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x] - 2*Defer[Int][(x^3*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + 4*x +
6*E^x*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x] - 2*Log[4 + 6*E^x - 5/x + 6*x]*Defer[Int][(x^3*(4 + 6
*E^x - 5/x + 6*x)^x)/((-5 + 4*x + 6*E^x*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x] - 2*Defer[Int][(E^x*
x^3*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + 4*x + 6*E^x*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x] - (5*Defer
[Int][Defer[Int][(x*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2
), x], x])/3 - (25*Defer[Int][Defer[Int][(x*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (4 +
6*E^x - 5/x + 6*x)^x)^2), x]/(-5 + 4*x + 6*E^x*x + 6*x^2), x])/3 - (25*Defer[Int][Defer[Int][(x*(4 + 6*E^x -
5/x + 6*x)^x)/((-5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x]/(x*(-5 + 4*x + 6*E^x*x + 6*
x^2)), x])/3 - (10*Defer[Int][(x*Defer[Int][(x*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (
4 + 6*E^x - 5/x + 6*x)^x)^2), x])/(-5 + 4*x + 6*E^x*x + 6*x^2), x])/3 + 10*Defer[Int][(x^2*Defer[Int][(x*(4 +
6*E^x - 5/x + 6*x)^x)/((-5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x])/(-5 + 4*x + 6*E^x*
x + 6*x^2), x] + (4*Defer[Int][Defer[Int][(x^2*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (
4 + 6*E^x - 5/x + 6*x)^x)^2), x], x])/3 + (20*Defer[Int][Defer[Int][(x^2*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + (4
+ 6*E^x)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x]/(-5 + 4*x + 6*E^x*x + 6*x^2), x])/3 + (20*Defer[Int
][Defer[Int][(x^2*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2),
x]/(x*(-5 + 4*x + 6*E^x*x + 6*x^2)), x])/3 + (8*Defer[Int][(x*Defer[Int][(x^2*(4 + 6*E^x - 5/x + 6*x)^x)/((-5
+ (4 + 6*E^x)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x])/(-5 + 4*x + 6*E^x*x + 6*x^2), x])/3 - 8*Defe
r[Int][(x^2*Defer[Int][(x^2*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6
*x)^x)^2), x])/(-5 + 4*x + 6*E^x*x + 6*x^2), x] + 2*Defer[Int][Defer[Int][(E^x*x^2*(4 + 6*E^x - 5/x + 6*x)^x)/
((-5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x], x] + 10*Defer[Int][Defer[Int][(E^x*x^2*(
4 + 6*E^x - 5/x + 6*x)^x)/((-5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x]/(-5 + 4*x + 6*E
^x*x + 6*x^2), x] + 10*Defer[Int][Defer[Int][(E^x*x^2*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + (4 + 6*E^x)*x + 6*x^2)
*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x]/(x*(-5 + 4*x + 6*E^x*x + 6*x^2)), x] + 4*Defer[Int][(x*Defer[Int][(E^x
*x^2*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x])/(-5 + 4*
x + 6*E^x*x + 6*x^2), x] - 12*Defer[Int][(x^2*Defer[Int][(E^x*x^2*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + (4 + 6*E^x
)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x])/(-5 + 4*x + 6*E^x*x + 6*x^2), x] + 2*Defer[Int][Defer[Int
][(x^3*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x], x] + 1
0*Defer[Int][Defer[Int][(x^3*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x +
6*x)^x)^2), x]/(-5 + 4*x + 6*E^x*x + 6*x^2), x] + 10*Defer[Int][Defer[Int][(x^3*(4 + 6*E^x - 5/x + 6*x)^x)/((-
5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x]/(x*(-5 + 4*x + 6*E^x*x + 6*x^2)), x] + 4*Def
er[Int][(x*Defer[Int][(x^3*(4 + 6*E^x - 5/x + 6*x)^x)/((-5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*
x)^x)^2), x])/(-5 + 4*x + 6*E^x*x + 6*x^2), x] - 12*Defer[Int][(x^2*Defer[Int][(x^3*(4 + 6*E^x - 5/x + 6*x)^x)
/((-5 + (4 + 6*E^x)*x + 6*x^2)*(2 + (4 + 6*E^x - 5/x + 6*x)^x)^2), x])/(-5 + 4*x + 6*E^x*x + 6*x^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {10-8 x-12 e^x x-12 x^2-\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-5-x+6 x^2-6 x^3+e^x \left (6 x-6 x^3\right )+\left (5 x-4 x^2-6 e^x x^2-6 x^3\right ) \log \left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )\right )}{3 \left (5-4 x-6 e^x x-6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx \\ & = \frac {1}{3} \int \frac {10-8 x-12 e^x x-12 x^2-\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-5-x+6 x^2-6 x^3+e^x \left (6 x-6 x^3\right )+\left (5 x-4 x^2-6 e^x x^2-6 x^3\right ) \log \left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )\right )}{\left (5-4 x-6 e^x x-6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx \\ & = \frac {1}{3} \int \left (-\frac {10}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2}+\frac {8 x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2}+\frac {12 e^x x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2}+\frac {12 x^2}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2}-\frac {\left (4+6 e^x-\frac {5}{x}+6 x\right )^x \left (5+x-6 e^x x-6 x^2+6 x^3+6 e^x x^3-5 x \log \left (4+6 e^x-\frac {5}{x}+6 x\right )+4 x^2 \log \left (4+6 e^x-\frac {5}{x}+6 x\right )+6 e^x x^2 \log \left (4+6 e^x-\frac {5}{x}+6 x\right )+6 x^3 \log \left (4+6 e^x-\frac {5}{x}+6 x\right )\right )}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {\left (4+6 e^x-\frac {5}{x}+6 x\right )^x \left (5+x-6 e^x x-6 x^2+6 x^3+6 e^x x^3-5 x \log \left (4+6 e^x-\frac {5}{x}+6 x\right )+4 x^2 \log \left (4+6 e^x-\frac {5}{x}+6 x\right )+6 e^x x^2 \log \left (4+6 e^x-\frac {5}{x}+6 x\right )+6 x^3 \log \left (4+6 e^x-\frac {5}{x}+6 x\right )\right )}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx\right )+\frac {8}{3} \int \frac {x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx-\frac {10}{3} \int \frac {1}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx+4 \int \frac {e^x x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx+4 \int \frac {x^2}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx \\ & = -\left (\frac {1}{3} \int \frac {\left (4+6 e^x-\frac {5}{x}+6 x\right )^x \left (-5-x+6 e^x x+6 x^2-6 \left (1+e^x\right ) x^3-x \left (-5+\left (4+6 e^x\right ) x+6 x^2\right ) \log \left (4+6 e^x-\frac {5}{x}+6 x\right )\right )}{\left (5-\left (4+6 e^x\right ) x-6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx\right )+\frac {8}{3} \int \frac {x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx-\frac {10}{3} \int \frac {1}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx+4 \int \frac {e^x x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx+4 \int \frac {x^2}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx \\ & = -\left (\frac {1}{3} \int \left (\frac {5 \left (4+6 e^x-\frac {5}{x}+6 x\right )^x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2}+\frac {x \left (4+6 e^x-\frac {5}{x}+6 x\right )^x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2}-\frac {6 e^x x \left (4+6 e^x-\frac {5}{x}+6 x\right )^x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2}-\frac {6 x^2 \left (4+6 e^x-\frac {5}{x}+6 x\right )^x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2}+\frac {6 x^3 \left (4+6 e^x-\frac {5}{x}+6 x\right )^x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2}+\frac {6 e^x x^3 \left (4+6 e^x-\frac {5}{x}+6 x\right )^x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2}-\frac {5 x \left (4+6 e^x-\frac {5}{x}+6 x\right )^x \log \left (4+6 e^x-\frac {5}{x}+6 x\right )}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2}+\frac {4 x^2 \left (4+6 e^x-\frac {5}{x}+6 x\right )^x \log \left (4+6 e^x-\frac {5}{x}+6 x\right )}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2}+\frac {6 e^x x^2 \left (4+6 e^x-\frac {5}{x}+6 x\right )^x \log \left (4+6 e^x-\frac {5}{x}+6 x\right )}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2}+\frac {6 x^3 \left (4+6 e^x-\frac {5}{x}+6 x\right )^x \log \left (4+6 e^x-\frac {5}{x}+6 x\right )}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2}\right ) \, dx\right )+\frac {8}{3} \int \frac {x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx-\frac {10}{3} \int \frac {1}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx+4 \int \frac {e^x x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx+4 \int \frac {x^2}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx \\ & = -\left (\frac {1}{3} \int \frac {x \left (4+6 e^x-\frac {5}{x}+6 x\right )^x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx\right )-\frac {4}{3} \int \frac {x^2 \left (4+6 e^x-\frac {5}{x}+6 x\right )^x \log \left (4+6 e^x-\frac {5}{x}+6 x\right )}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx-\frac {5}{3} \int \frac {\left (4+6 e^x-\frac {5}{x}+6 x\right )^x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx+\frac {5}{3} \int \frac {x \left (4+6 e^x-\frac {5}{x}+6 x\right )^x \log \left (4+6 e^x-\frac {5}{x}+6 x\right )}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx+2 \int \frac {e^x x \left (4+6 e^x-\frac {5}{x}+6 x\right )^x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx+2 \int \frac {x^2 \left (4+6 e^x-\frac {5}{x}+6 x\right )^x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx-2 \int \frac {x^3 \left (4+6 e^x-\frac {5}{x}+6 x\right )^x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx-2 \int \frac {e^x x^3 \left (4+6 e^x-\frac {5}{x}+6 x\right )^x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx-2 \int \frac {e^x x^2 \left (4+6 e^x-\frac {5}{x}+6 x\right )^x \log \left (4+6 e^x-\frac {5}{x}+6 x\right )}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx-2 \int \frac {x^3 \left (4+6 e^x-\frac {5}{x}+6 x\right )^x \log \left (4+6 e^x-\frac {5}{x}+6 x\right )}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx+\frac {8}{3} \int \frac {x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx-\frac {10}{3} \int \frac {1}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx+4 \int \frac {e^x x}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx+4 \int \frac {x^2}{\left (-5+4 x+6 e^x x+6 x^2\right ) \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {-10+8 x+12 e^x x+12 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-5-x+6 x^2-6 x^3+e^x \left (6 x-6 x^3\right )+\left (5 x-4 x^2-6 e^x x^2-6 x^3\right ) \log \left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )\right )}{-60+48 x+72 e^x x+72 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^{2 x} \left (-15+12 x+18 e^x x+18 x^2\right )+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-60+48 x+72 e^x x+72 x^2\right )} \, dx=\frac {x}{3 \left (2+\left (4+6 e^x-\frac {5}{x}+6 x\right )^x\right )} \]

[In]

Integrate[(-10 + 8*x + 12*E^x*x + 12*x^2 + ((-5 + 4*x + 6*E^x*x + 6*x^2)/x)^x*(-5 - x + 6*x^2 - 6*x^3 + E^x*(6
*x - 6*x^3) + (5*x - 4*x^2 - 6*E^x*x^2 - 6*x^3)*Log[(-5 + 4*x + 6*E^x*x + 6*x^2)/x]))/(-60 + 48*x + 72*E^x*x +
72*x^2 + ((-5 + 4*x + 6*E^x*x + 6*x^2)/x)^(2*x)*(-15 + 12*x + 18*E^x*x + 18*x^2) + ((-5 + 4*x + 6*E^x*x + 6*x
^2)/x)^x*(-60 + 48*x + 72*E^x*x + 72*x^2)),x]

[Out]

x/(3*(2 + (4 + 6*E^x - 5/x + 6*x)^x))

Maple [A] (verified)

Time = 12.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24

method	result	size
parallelrisch	\(\frac {x}{3 \,{\mathrm e}^{x \ln \left (\frac {6 \,{\mathrm e}^{x} x +6 x^{2}+4 x -5}{x}\right )}+6}\)	\(31\)
risch	\(\frac {x}{3 x^{-x} 3^{x} 2^{x} \left (-\frac {5}{6}+x^{2}+\left (\frac {2}{3}+{\mathrm e}^{x}\right ) x \right )^{x} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (-\frac {5}{6}+x^{2}+\left (\frac {2}{3}+{\mathrm e}^{x}\right ) x \right )}{x}\right ) x \left (-\operatorname {csgn}\left (\frac {i \left (-\frac {5}{6}+x^{2}+\left (\frac {2}{3}+{\mathrm e}^{x}\right ) x \right )}{x}\right )+\operatorname {csgn}\left (i \left (-\frac {5}{6}+x^{2}+\left (\frac {2}{3}+{\mathrm e}^{x}\right ) x \right )\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (-\frac {5}{6}+x^{2}+\left (\frac {2}{3}+{\mathrm e}^{x}\right ) x \right )}{x}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right )}{2}}+6}\)	\(121\)

[In]

int((((-6*exp(x)*x^2-6*x^3-4*x^2+5*x)*ln((6*exp(x)*x+6*x^2+4*x-5)/x)+(-6*x^3+6*x)*exp(x)-6*x^3+6*x^2-x-5)*exp(
x*ln((6*exp(x)*x+6*x^2+4*x-5)/x))+12*exp(x)*x+12*x^2+8*x-10)/((18*exp(x)*x+18*x^2+12*x-15)*exp(x*ln((6*exp(x)*
x+6*x^2+4*x-5)/x))^2+(72*exp(x)*x+72*x^2+48*x-60)*exp(x*ln((6*exp(x)*x+6*x^2+4*x-5)/x))+72*exp(x)*x+72*x^2+48*
x-60),x,method=_RETURNVERBOSE)

[Out]

1/3*x/(exp(x*ln((6*exp(x)*x+6*x^2+4*x-5)/x))+2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {-10+8 x+12 e^x x+12 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-5-x+6 x^2-6 x^3+e^x \left (6 x-6 x^3\right )+\left (5 x-4 x^2-6 e^x x^2-6 x^3\right ) \log \left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )\right )}{-60+48 x+72 e^x x+72 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^{2 x} \left (-15+12 x+18 e^x x+18 x^2\right )+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-60+48 x+72 e^x x+72 x^2\right )} \, dx=\frac {x}{3 \, {\left (\left (\frac {6 \, x^{2} + 6 \, x e^{x} + 4 \, x - 5}{x}\right )^{x} + 2\right )}} \]

[In]

integrate((((-6*exp(x)*x^2-6*x^3-4*x^2+5*x)*log((6*exp(x)*x+6*x^2+4*x-5)/x)+(-6*x^3+6*x)*exp(x)-6*x^3+6*x^2-x-
5)*exp(x*log((6*exp(x)*x+6*x^2+4*x-5)/x))+12*exp(x)*x+12*x^2+8*x-10)/((18*exp(x)*x+18*x^2+12*x-15)*exp(x*log((
6*exp(x)*x+6*x^2+4*x-5)/x))^2+(72*exp(x)*x+72*x^2+48*x-60)*exp(x*log((6*exp(x)*x+6*x^2+4*x-5)/x))+72*exp(x)*x+
72*x^2+48*x-60),x, algorithm="fricas")

[Out]

1/3*x/(((6*x^2 + 6*x*e^x + 4*x - 5)/x)^x + 2)

Sympy [F(-1)]

Timed out. \[ \int \frac {-10+8 x+12 e^x x+12 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-5-x+6 x^2-6 x^3+e^x \left (6 x-6 x^3\right )+\left (5 x-4 x^2-6 e^x x^2-6 x^3\right ) \log \left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )\right )}{-60+48 x+72 e^x x+72 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^{2 x} \left (-15+12 x+18 e^x x+18 x^2\right )+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-60+48 x+72 e^x x+72 x^2\right )} \, dx=\text {Timed out} \]

[In]

integrate((((-6*exp(x)*x**2-6*x**3-4*x**2+5*x)*ln((6*exp(x)*x+6*x**2+4*x-5)/x)+(-6*x**3+6*x)*exp(x)-6*x**3+6*x
**2-x-5)*exp(x*ln((6*exp(x)*x+6*x**2+4*x-5)/x))+12*exp(x)*x+12*x**2+8*x-10)/((18*exp(x)*x+18*x**2+12*x-15)*exp
(x*ln((6*exp(x)*x+6*x**2+4*x-5)/x))**2+(72*exp(x)*x+72*x**2+48*x-60)*exp(x*ln((6*exp(x)*x+6*x**2+4*x-5)/x))+72
*exp(x)*x+72*x**2+48*x-60),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {-10+8 x+12 e^x x+12 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-5-x+6 x^2-6 x^3+e^x \left (6 x-6 x^3\right )+\left (5 x-4 x^2-6 e^x x^2-6 x^3\right ) \log \left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )\right )}{-60+48 x+72 e^x x+72 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^{2 x} \left (-15+12 x+18 e^x x+18 x^2\right )+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-60+48 x+72 e^x x+72 x^2\right )} \, dx=\frac {x x^{x}}{3 \, {\left ({\left (6 \, x^{2} + 6 \, x e^{x} + 4 \, x - 5\right )}^{x} + 2 \, x^{x}\right )}} \]

[In]

integrate((((-6*exp(x)*x^2-6*x^3-4*x^2+5*x)*log((6*exp(x)*x+6*x^2+4*x-5)/x)+(-6*x^3+6*x)*exp(x)-6*x^3+6*x^2-x-
5)*exp(x*log((6*exp(x)*x+6*x^2+4*x-5)/x))+12*exp(x)*x+12*x^2+8*x-10)/((18*exp(x)*x+18*x^2+12*x-15)*exp(x*log((
6*exp(x)*x+6*x^2+4*x-5)/x))^2+(72*exp(x)*x+72*x^2+48*x-60)*exp(x*log((6*exp(x)*x+6*x^2+4*x-5)/x))+72*exp(x)*x+
72*x^2+48*x-60),x, algorithm="maxima")

[Out]

1/3*x*x^x/((6*x^2 + 6*x*e^x + 4*x - 5)^x + 2*x^x)

Giac [F]

\[ \int \frac {-10+8 x+12 e^x x+12 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-5-x+6 x^2-6 x^3+e^x \left (6 x-6 x^3\right )+\left (5 x-4 x^2-6 e^x x^2-6 x^3\right ) \log \left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )\right )}{-60+48 x+72 e^x x+72 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^{2 x} \left (-15+12 x+18 e^x x+18 x^2\right )+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-60+48 x+72 e^x x+72 x^2\right )} \, dx=\int { \frac {12 \, x^{2} - {\left (6 \, x^{3} - 6 \, x^{2} + 6 \, {\left (x^{3} - x\right )} e^{x} + {\left (6 \, x^{3} + 6 \, x^{2} e^{x} + 4 \, x^{2} - 5 \, x\right )} \log \left (\frac {6 \, x^{2} + 6 \, x e^{x} + 4 \, x - 5}{x}\right ) + x + 5\right )} \left (\frac {6 \, x^{2} + 6 \, x e^{x} + 4 \, x - 5}{x}\right )^{x} + 12 \, x e^{x} + 8 \, x - 10}{3 \, {\left (24 \, x^{2} + {\left (6 \, x^{2} + 6 \, x e^{x} + 4 \, x - 5\right )} \left (\frac {6 \, x^{2} + 6 \, x e^{x} + 4 \, x - 5}{x}\right )^{2 \, x} + 4 \, {\left (6 \, x^{2} + 6 \, x e^{x} + 4 \, x - 5\right )} \left (\frac {6 \, x^{2} + 6 \, x e^{x} + 4 \, x - 5}{x}\right )^{x} + 24 \, x e^{x} + 16 \, x - 20\right )}} \,d x } \]

[In]

integrate((((-6*exp(x)*x^2-6*x^3-4*x^2+5*x)*log((6*exp(x)*x+6*x^2+4*x-5)/x)+(-6*x^3+6*x)*exp(x)-6*x^3+6*x^2-x-
5)*exp(x*log((6*exp(x)*x+6*x^2+4*x-5)/x))+12*exp(x)*x+12*x^2+8*x-10)/((18*exp(x)*x+18*x^2+12*x-15)*exp(x*log((
6*exp(x)*x+6*x^2+4*x-5)/x))^2+(72*exp(x)*x+72*x^2+48*x-60)*exp(x*log((6*exp(x)*x+6*x^2+4*x-5)/x))+72*exp(x)*x+
72*x^2+48*x-60),x, algorithm="giac")

[Out]

integrate(1/3*(12*x^2 - (6*x^3 - 6*x^2 + 6*(x^3 - x)*e^x + (6*x^3 + 6*x^2*e^x + 4*x^2 - 5*x)*log((6*x^2 + 6*x*
e^x + 4*x - 5)/x) + x + 5)*((6*x^2 + 6*x*e^x + 4*x - 5)/x)^x + 12*x*e^x + 8*x - 10)/(24*x^2 + (6*x^2 + 6*x*e^x
+ 4*x - 5)*((6*x^2 + 6*x*e^x + 4*x - 5)/x)^(2*x) + 4*(6*x^2 + 6*x*e^x + 4*x - 5)*((6*x^2 + 6*x*e^x + 4*x - 5)
/x)^x + 24*x*e^x + 16*x - 20), x)

Mupad [B] (verification not implemented)

Time = 13.10 (sec) , antiderivative size = 254, normalized size of antiderivative = 10.16 \[ \int \frac {-10+8 x+12 e^x x+12 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-5-x+6 x^2-6 x^3+e^x \left (6 x-6 x^3\right )+\left (5 x-4 x^2-6 e^x x^2-6 x^3\right ) \log \left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )\right )}{-60+48 x+72 e^x x+72 x^2+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^{2 x} \left (-15+12 x+18 e^x x+18 x^2\right )+\left (\frac {-5+4 x+6 e^x x+6 x^2}{x}\right )^x \left (-60+48 x+72 e^x x+72 x^2\right )} \, dx=\frac {5\,x+6\,x^3\,{\mathrm {e}}^x-5\,x\,\ln \left (\frac {4\,x+6\,x\,{\mathrm {e}}^x+6\,x^2-5}{x}\right )+6\,x^3+4\,x^2\,\ln \left (\frac {4\,x+6\,x\,{\mathrm {e}}^x+6\,x^2-5}{x}\right )+6\,x^3\,\ln \left (\frac {4\,x+6\,x\,{\mathrm {e}}^x+6\,x^2-5}{x}\right )+6\,x^2\,{\mathrm {e}}^x\,\ln \left (\frac {4\,x+6\,x\,{\mathrm {e}}^x+6\,x^2-5}{x}\right )}{3\,\left ({\left (\frac {4\,x+6\,x\,{\mathrm {e}}^x+6\,x^2-5}{x}\right )}^x+2\right )\,\left (6\,x^2\,{\mathrm {e}}^x-5\,\ln \left (\frac {4\,x+6\,x\,{\mathrm {e}}^x+6\,x^2-5}{x}\right )+4\,x\,\ln \left (\frac {4\,x+6\,x\,{\mathrm {e}}^x+6\,x^2-5}{x}\right )+6\,x^2+6\,x^2\,\ln \left (\frac {4\,x+6\,x\,{\mathrm {e}}^x+6\,x^2-5}{x}\right )+6\,x\,{\mathrm {e}}^x\,\ln \left (\frac {4\,x+6\,x\,{\mathrm {e}}^x+6\,x^2-5}{x}\right )+5\right )} \]

[In]

int((8*x - exp(x*log((4*x + 6*x*exp(x) + 6*x^2 - 5)/x))*(x - exp(x)*(6*x - 6*x^3) - 6*x^2 + 6*x^3 + log((4*x +
6*x*exp(x) + 6*x^2 - 5)/x)*(6*x^2*exp(x) - 5*x + 4*x^2 + 6*x^3) + 5) + 12*x*exp(x) + 12*x^2 - 10)/(48*x + exp
(2*x*log((4*x + 6*x*exp(x) + 6*x^2 - 5)/x))*(12*x + 18*x*exp(x) + 18*x^2 - 15) + exp(x*log((4*x + 6*x*exp(x) +
6*x^2 - 5)/x))*(48*x + 72*x*exp(x) + 72*x^2 - 60) + 72*x*exp(x) + 72*x^2 - 60),x)

[Out]

(5*x + 6*x^3*exp(x) - 5*x*log((4*x + 6*x*exp(x) + 6*x^2 - 5)/x) + 6*x^3 + 4*x^2*log((4*x + 6*x*exp(x) + 6*x^2
- 5)/x) + 6*x^3*log((4*x + 6*x*exp(x) + 6*x^2 - 5)/x) + 6*x^2*exp(x)*log((4*x + 6*x*exp(x) + 6*x^2 - 5)/x))/(3
*(((4*x + 6*x*exp(x) + 6*x^2 - 5)/x)^x + 2)*(6*x^2*exp(x) - 5*log((4*x + 6*x*exp(x) + 6*x^2 - 5)/x) + 4*x*log(
(4*x + 6*x*exp(x) + 6*x^2 - 5)/x) + 6*x^2 + 6*x^2*log((4*x + 6*x*exp(x) + 6*x^2 - 5)/x) + 6*x*exp(x)*log((4*x
+ 6*x*exp(x) + 6*x^2 - 5)/x) + 5))