\(\int \frac {(\frac {6+11 x+5 x^2+e^x (8 x+5 x^2)}{2+x})^x (16 x+20 x^2+5 x^3+e^x (16 x+36 x^2+23 x^3+5 x^4)+(12+28 x+21 x^2+5 x^3+e^x (16 x+18 x^2+5 x^3)) \log (\frac {6+11 x+5 x^2+e^x (8 x+5 x^2)}{2+x}))}{12+28 x+21 x^2+5 x^3+e^x (16 x+18 x^2+5 x^3)} \, dx\) [7697]

Optimal result

Integrand size = 168, antiderivative size = 21 \[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx=\left (3+\left (x+e^x x\right ) \left (4+\frac {x}{2+x}\right )\right )^x \]

[Out]

exp(ln((x/(2+x)+4)*(exp(x)*x+x)+3)*x)

Rubi [F]

\[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx=\int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx \]

[In]

Int[(((6 + 11*x + 5*x^2 + E^x*(8*x + 5*x^2))/(2 + x))^x*(16*x + 20*x^2 + 5*x^3 + E^x*(16*x + 36*x^2 + 23*x^3 +
5*x^4) + (12 + 28*x + 21*x^2 + 5*x^3 + E^x*(16*x + 18*x^2 + 5*x^3))*Log[(6 + 11*x + 5*x^2 + E^x*(8*x + 5*x^2)
)/(2 + x)]))/(12 + 28*x + 21*x^2 + 5*x^3 + E^x*(16*x + 18*x^2 + 5*x^3)),x]

[Out]

Log[(6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x)]*Defer[Int][((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x)
)^(-1 + x), x] + 4*Defer[Int][E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x] - 2*Log[(6 + (
11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x)]*Defer[Int][E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1
+ x), x] + 5*Defer[Int][x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x] + 5*Log[(6 + (11 + 8*E
^x)*x + 5*(1 + E^x)*x^2)/(2 + x)]*Defer[Int][x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x] +
3*Defer[Int][E^x*x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x] + 5*Log[(6 + (11 + 8*E^x)*x
+ 5*(1 + E^x)*x^2)/(2 + x)]*Defer[Int][E^x*x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x] + 5
*Defer[Int][E^x*x^2*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x] + 8*Defer[Int][((6 + (11 + 8
*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x)/(2 + x)^2, x] + 8*Defer[Int][(E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E
^x)*x^2)/(2 + x))^(-1 + x))/(2 + x)^2, x] - 4*Defer[Int][((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1
+ x)/(2 + x), x] + 4*Log[(6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x)]*Defer[Int][((6 + (11 + 8*E^x)*x + 5*(
1 + E^x)*x^2)/(2 + x))^(-1 + x)/(2 + x), x] - 12*Defer[Int][(E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 +
x))^(-1 + x))/(2 + x), x] + 4*Log[(6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x)]*Defer[Int][(E^x*((6 + (11 +
8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x))/(2 + x), x] - Defer[Int][Defer[Int][((6 + (11 + 8*E^x)*x + 5*(1
+ E^x)*x^2)/(2 + x))^(-1 + x), x], x] - Defer[Int][Defer[Int][((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x)
)^(-1 + x), x]/x, x] + Defer[Int][Defer[Int][((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x]/(2
+ x), x] - 5*Defer[Int][Defer[Int][((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x]/(8 + 5*x), x]
+ 9*Defer[Int][Defer[Int][((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x]/(6 + 11*x + 8*E^x*x +
5*x^2 + 5*E^x*x^2), x] + 6*Defer[Int][Defer[Int][((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x
]/(x*(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2)), x] + 11*Defer[Int][(x*Defer[Int][((6 + (11 + 8*E^x)*x + 5*(1 +
E^x)*x^2)/(2 + x))^(-1 + x), x])/(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2), x] + 5*Defer[Int][(x^2*Defer[Int][
((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x])/(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2), x] +
6*Defer[Int][Defer[Int][((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x]/((8 + 5*x)*(6 + 11*x + 8
*E^x*x + 5*x^2 + 5*E^x*x^2)), x] + 2*Defer[Int][Defer[Int][E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x)
)^(-1 + x), x], x] + 2*Defer[Int][Defer[Int][E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x]
/x, x] - 2*Defer[Int][Defer[Int][E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x]/(2 + x), x]
+ 10*Defer[Int][Defer[Int][E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x]/(8 + 5*x), x] -
18*Defer[Int][Defer[Int][E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x]/(6 + 11*x + 8*E^x*x
+ 5*x^2 + 5*E^x*x^2), x] - 12*Defer[Int][Defer[Int][E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1
+ x), x]/(x*(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2)), x] - 22*Defer[Int][(x*Defer[Int][E^x*((6 + (11 + 8*E^x)
*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x])/(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2), x] - 10*Defer[Int][(x^2
*Defer[Int][E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x])/(6 + 11*x + 8*E^x*x + 5*x^2 + 5
*E^x*x^2), x] - 12*Defer[Int][Defer[Int][E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x]/((8
+ 5*x)*(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2)), x] - 5*Defer[Int][Defer[Int][x*((6 + (11 + 8*E^x)*x + 5*(1
+ E^x)*x^2)/(2 + x))^(-1 + x), x], x] - 5*Defer[Int][Defer[Int][x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 +
x))^(-1 + x), x]/x, x] + 5*Defer[Int][Defer[Int][x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x),
x]/(2 + x), x] - 25*Defer[Int][Defer[Int][x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x]/(8
+ 5*x), x] + 45*Defer[Int][Defer[Int][x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x]/(6 + 11*
x + 8*E^x*x + 5*x^2 + 5*E^x*x^2), x] + 30*Defer[Int][Defer[Int][x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 +
x))^(-1 + x), x]/(x*(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2)), x] + 55*Defer[Int][(x*Defer[Int][x*((6 + (11 +
8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x])/(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2), x] + 25*Defer[In
t][(x^2*Defer[Int][x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x])/(6 + 11*x + 8*E^x*x + 5*x^
2 + 5*E^x*x^2), x] + 30*Defer[Int][Defer[Int][x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x]/
((8 + 5*x)*(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2)), x] - 5*Defer[Int][Defer[Int][E^x*x*((6 + (11 + 8*E^x)*x
+ 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x], x] - 5*Defer[Int][Defer[Int][E^x*x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x
)*x^2)/(2 + x))^(-1 + x), x]/x, x] + 5*Defer[Int][Defer[Int][E^x*x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2
+ x))^(-1 + x), x]/(2 + x), x] - 25*Defer[Int][Defer[Int][E^x*x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x
))^(-1 + x), x]/(8 + 5*x), x] + 45*Defer[Int][Defer[Int][E^x*x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x)
)^(-1 + x), x]/(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2), x] + 30*Defer[Int][Defer[Int][E^x*x*((6 + (11 + 8*E^x
)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x]/(x*(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2)), x] + 55*Defer[Int][
(x*Defer[Int][E^x*x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x])/(6 + 11*x + 8*E^x*x + 5*x^2
+ 5*E^x*x^2), x] + 25*Defer[Int][(x^2*Defer[Int][E^x*x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 +
x), x])/(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2), x] + 30*Defer[Int][Defer[Int][E^x*x*((6 + (11 + 8*E^x)*x +
5*(1 + E^x)*x^2)/(2 + x))^(-1 + x), x]/((8 + 5*x)*(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2)), x] - 4*Defer[Int]
[Defer[Int][((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x)/(2 + x), x], x] - 4*Defer[Int][Defer[Int
][((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x)/(2 + x), x]/x, x] + 4*Defer[Int][Defer[Int][((6 +
(11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x)/(2 + x), x]/(2 + x), x] - 20*Defer[Int][Defer[Int][((6 + (
11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x)/(2 + x), x]/(8 + 5*x), x] + 36*Defer[Int][Defer[Int][((6 +
(11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x)/(2 + x), x]/(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2), x] +
24*Defer[Int][Defer[Int][((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x)/(2 + x), x]/(x*(6 + 11*x +
8*E^x*x + 5*x^2 + 5*E^x*x^2)), x] + 44*Defer[Int][(x*Defer[Int][((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 +
x))^(-1 + x)/(2 + x), x])/(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2), x] + 20*Defer[Int][(x^2*Defer[Int][((6 + (
11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x)/(2 + x), x])/(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2), x] +
24*Defer[Int][Defer[Int][((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x)/(2 + x), x]/((8 + 5*x)*(6
+ 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2)), x] - 4*Defer[Int][Defer[Int][(E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x
^2)/(2 + x))^(-1 + x))/(2 + x), x], x] - 4*Defer[Int][Defer[Int][(E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/
(2 + x))^(-1 + x))/(2 + x), x]/x, x] + 4*Defer[Int][Defer[Int][(E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2
+ x))^(-1 + x))/(2 + x), x]/(2 + x), x] - 20*Defer[Int][Defer[Int][(E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^
2)/(2 + x))^(-1 + x))/(2 + x), x]/(8 + 5*x), x] + 36*Defer[Int][Defer[Int][(E^x*((6 + (11 + 8*E^x)*x + 5*(1 +
E^x)*x^2)/(2 + x))^(-1 + x))/(2 + x), x]/(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2), x] + 24*Defer[Int][Defer[In
t][(E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x))/(2 + x), x]/(x*(6 + 11*x + 8*E^x*x + 5*x^2
+ 5*E^x*x^2)), x] + 44*Defer[Int][(x*Defer[Int][(E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x)
)/(2 + x), x])/(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2), x] + 20*Defer[Int][(x^2*Defer[Int][(E^x*((6 + (11 + 8
*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x))/(2 + x), x])/(6 + 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2), x] + 24*D
efer[Int][Defer[Int][(E^x*((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^(-1 + x))/(2 + x), x]/((8 + 5*x)*(6
+ 11*x + 8*E^x*x + 5*x^2 + 5*E^x*x^2)), x]

Rubi steps Aborted

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx=\left (\frac {6+\left (11+8 e^x\right ) x+5 \left (1+e^x\right ) x^2}{2+x}\right )^x \]

[In]

Integrate[(((6 + 11*x + 5*x^2 + E^x*(8*x + 5*x^2))/(2 + x))^x*(16*x + 20*x^2 + 5*x^3 + E^x*(16*x + 36*x^2 + 23
*x^3 + 5*x^4) + (12 + 28*x + 21*x^2 + 5*x^3 + E^x*(16*x + 18*x^2 + 5*x^3))*Log[(6 + 11*x + 5*x^2 + E^x*(8*x +
5*x^2))/(2 + x)]))/(12 + 28*x + 21*x^2 + 5*x^3 + E^x*(16*x + 18*x^2 + 5*x^3)),x]

[Out]

((6 + (11 + 8*E^x)*x + 5*(1 + E^x)*x^2)/(2 + x))^x

Maple [A] (verified)

Time = 21.55 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57

method	result	size
parallelrisch	\({\mathrm e}^{x \ln \left (\frac {\left (5 x^{2}+8 x \right ) {\mathrm e}^{x}+5 x^{2}+11 x +6}{2+x}\right )}\)	\(33\)
risch	\(5^{x} \left (2+x \right )^{-x} {\left (\frac {6}{5}+\left ({\mathrm e}^{x}+1\right ) x^{2}+\left (\frac {8 \,{\mathrm e}^{x}}{5}+\frac {11}{5}\right ) x \right )}^{x} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (\frac {6}{5}+\left ({\mathrm e}^{x}+1\right ) x^{2}+\left (\frac {8 \,{\mathrm e}^{x}}{5}+\frac {11}{5}\right ) x \right )}{2+x}\right ) x \left (-\operatorname {csgn}\left (\frac {i \left (\frac {6}{5}+\left ({\mathrm e}^{x}+1\right ) x^{2}+\left (\frac {8 \,{\mathrm e}^{x}}{5}+\frac {11}{5}\right ) x \right )}{2+x}\right )+\operatorname {csgn}\left (\frac {i}{2+x}\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (\frac {6}{5}+\left ({\mathrm e}^{x}+1\right ) x^{2}+\left (\frac {8 \,{\mathrm e}^{x}}{5}+\frac {11}{5}\right ) x \right )}{2+x}\right )+\operatorname {csgn}\left (i \left (\frac {6}{5}+\left ({\mathrm e}^{x}+1\right ) x^{2}+\left (\frac {8 \,{\mathrm e}^{x}}{5}+\frac {11}{5}\right ) x \right )\right )\right )}{2}}\)	\(156\)

[In]

int((((5*x^3+18*x^2+16*x)*exp(x)+5*x^3+21*x^2+28*x+12)*ln(((5*x^2+8*x)*exp(x)+5*x^2+11*x+6)/(2+x))+(5*x^4+23*x
^3+36*x^2+16*x)*exp(x)+5*x^3+20*x^2+16*x)*exp(x*ln(((5*x^2+8*x)*exp(x)+5*x^2+11*x+6)/(2+x)))/((5*x^3+18*x^2+16
*x)*exp(x)+5*x^3+21*x^2+28*x+12),x,method=_RETURNVERBOSE)

[Out]

exp(x*ln(((5*x^2+8*x)*exp(x)+5*x^2+11*x+6)/(2+x)))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx=\left (\frac {5 \, x^{2} + {\left (5 \, x^{2} + 8 \, x\right )} e^{x} + 11 \, x + 6}{x + 2}\right )^{x} \]

[In]

integrate((((5*x^3+18*x^2+16*x)*exp(x)+5*x^3+21*x^2+28*x+12)*log(((5*x^2+8*x)*exp(x)+5*x^2+11*x+6)/(2+x))+(5*x
^4+23*x^3+36*x^2+16*x)*exp(x)+5*x^3+20*x^2+16*x)*exp(x*log(((5*x^2+8*x)*exp(x)+5*x^2+11*x+6)/(2+x)))/((5*x^3+1
8*x^2+16*x)*exp(x)+5*x^3+21*x^2+28*x+12),x, algorithm="fricas")

[Out]

((5*x^2 + (5*x^2 + 8*x)*e^x + 11*x + 6)/(x + 2))^x

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx=\text {Timed out} \]

[In]

integrate((((5*x**3+18*x**2+16*x)*exp(x)+5*x**3+21*x**2+28*x+12)*ln(((5*x**2+8*x)*exp(x)+5*x**2+11*x+6)/(2+x))
+(5*x**4+23*x**3+36*x**2+16*x)*exp(x)+5*x**3+20*x**2+16*x)*exp(x*ln(((5*x**2+8*x)*exp(x)+5*x**2+11*x+6)/(2+x))
)/((5*x**3+18*x**2+16*x)*exp(x)+5*x**3+21*x**2+28*x+12),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx=e^{\left (x \log \left (5 \, x^{2} + {\left (5 \, x^{2} + 8 \, x\right )} e^{x} + 11 \, x + 6\right ) - x \log \left (x + 2\right )\right )} \]

[In]

integrate((((5*x^3+18*x^2+16*x)*exp(x)+5*x^3+21*x^2+28*x+12)*log(((5*x^2+8*x)*exp(x)+5*x^2+11*x+6)/(2+x))+(5*x
^4+23*x^3+36*x^2+16*x)*exp(x)+5*x^3+20*x^2+16*x)*exp(x*log(((5*x^2+8*x)*exp(x)+5*x^2+11*x+6)/(2+x)))/((5*x^3+1
8*x^2+16*x)*exp(x)+5*x^3+21*x^2+28*x+12),x, algorithm="maxima")

[Out]

e^(x*log(5*x^2 + (5*x^2 + 8*x)*e^x + 11*x + 6) - x*log(x + 2))

Giac [F]

\[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx=\int { \frac {{\left (5 \, x^{3} + 20 \, x^{2} + {\left (5 \, x^{4} + 23 \, x^{3} + 36 \, x^{2} + 16 \, x\right )} e^{x} + {\left (5 \, x^{3} + 21 \, x^{2} + {\left (5 \, x^{3} + 18 \, x^{2} + 16 \, x\right )} e^{x} + 28 \, x + 12\right )} \log \left (\frac {5 \, x^{2} + {\left (5 \, x^{2} + 8 \, x\right )} e^{x} + 11 \, x + 6}{x + 2}\right ) + 16 \, x\right )} \left (\frac {5 \, x^{2} + {\left (5 \, x^{2} + 8 \, x\right )} e^{x} + 11 \, x + 6}{x + 2}\right )^{x}}{5 \, x^{3} + 21 \, x^{2} + {\left (5 \, x^{3} + 18 \, x^{2} + 16 \, x\right )} e^{x} + 28 \, x + 12} \,d x } \]

[In]

integrate((((5*x^3+18*x^2+16*x)*exp(x)+5*x^3+21*x^2+28*x+12)*log(((5*x^2+8*x)*exp(x)+5*x^2+11*x+6)/(2+x))+(5*x
^4+23*x^3+36*x^2+16*x)*exp(x)+5*x^3+20*x^2+16*x)*exp(x*log(((5*x^2+8*x)*exp(x)+5*x^2+11*x+6)/(2+x)))/((5*x^3+1
8*x^2+16*x)*exp(x)+5*x^3+21*x^2+28*x+12),x, algorithm="giac")

[Out]

integrate((5*x^3 + 20*x^2 + (5*x^4 + 23*x^3 + 36*x^2 + 16*x)*e^x + (5*x^3 + 21*x^2 + (5*x^3 + 18*x^2 + 16*x)*e
^x + 28*x + 12)*log((5*x^2 + (5*x^2 + 8*x)*e^x + 11*x + 6)/(x + 2)) + 16*x)*((5*x^2 + (5*x^2 + 8*x)*e^x + 11*x
+ 6)/(x + 2))^x/(5*x^3 + 21*x^2 + (5*x^3 + 18*x^2 + 16*x)*e^x + 28*x + 12), x)

Mupad [B] (verification not implemented)

Time = 13.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {\left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )^x \left (16 x+20 x^2+5 x^3+e^x \left (16 x+36 x^2+23 x^3+5 x^4\right )+\left (12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )\right ) \log \left (\frac {6+11 x+5 x^2+e^x \left (8 x+5 x^2\right )}{2+x}\right )\right )}{12+28 x+21 x^2+5 x^3+e^x \left (16 x+18 x^2+5 x^3\right )} \, dx={\left (\frac {11\,x+5\,x^2\,{\mathrm {e}}^x+8\,x\,{\mathrm {e}}^x+5\,x^2+6}{x+2}\right )}^x \]

[In]

int((exp(x*log((11*x + exp(x)*(8*x + 5*x^2) + 5*x^2 + 6)/(x + 2)))*(16*x + log((11*x + exp(x)*(8*x + 5*x^2) +
5*x^2 + 6)/(x + 2))*(28*x + 21*x^2 + 5*x^3 + exp(x)*(16*x + 18*x^2 + 5*x^3) + 12) + exp(x)*(16*x + 36*x^2 + 23
*x^3 + 5*x^4) + 20*x^2 + 5*x^3))/(28*x + 21*x^2 + 5*x^3 + exp(x)*(16*x + 18*x^2 + 5*x^3) + 12),x)

[Out]

((11*x + 5*x^2*exp(x) + 8*x*exp(x) + 5*x^2 + 6)/(x + 2))^x