\(\int \frac {70+40 x-30 x^2+e^x (15 x-5 x^2)+e^x (30+30 x-5 x^2-5 x^3) \log (2+2 x)}{49 x^3+21 x^4-24 x^5+4 x^6+e^x (42 x^3+16 x^4-22 x^5+4 x^6) \log (2+2 x)+e^{2 x} (9 x^3+3 x^4-5 x^5+x^6) \log ^2(2+2 x)} \, dx\) [2071]

Optimal result

Integrand size = 136, antiderivative size = 28 \[ \int \frac {70+40 x-30 x^2+e^x \left (15 x-5 x^2\right )+e^x \left (30+30 x-5 x^2-5 x^3\right ) \log (2+2 x)}{49 x^3+21 x^4-24 x^5+4 x^6+e^x \left (42 x^3+16 x^4-22 x^5+4 x^6\right ) \log (2+2 x)+e^{2 x} \left (9 x^3+3 x^4-5 x^5+x^6\right ) \log ^2(2+2 x)} \, dx=\frac {5}{x \left (-x+(-3+x) x \left (2+e^x \log (2+2 x)\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 3.83 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {70+40 x-30 x^2+e^x \left (15 x-5 x^2\right )+e^x \left (30+30 x-5 x^2-5 x^3\right ) \log (2+2 x)}{49 x^3+21 x^4-24 x^5+4 x^6+e^x \left (42 x^3+16 x^4-22 x^5+4 x^6\right ) \log (2+2 x)+e^{2 x} \left (9 x^3+3 x^4-5 x^5+x^6\right ) \log ^2(2+2 x)} \, dx=\frac {5}{x^2 \left (-7+2 x+e^x (-3+x) \log (2 (1+x))\right )} \] Input:

Integrate[(70 + 40*x - 30*x^2 + E^x*(15*x - 5*x^2) + E^x*(30 + 30*x - 5*x^ 
2 - 5*x^3)*Log[2 + 2*x])/(49*x^3 + 21*x^4 - 24*x^5 + 4*x^6 + E^x*(42*x^3 + 
 16*x^4 - 22*x^5 + 4*x^6)*Log[2 + 2*x] + E^(2*x)*(9*x^3 + 3*x^4 - 5*x^5 + 
x^6)*Log[2 + 2*x]^2),x]
 

Output:

5/(x^2*(-7 + 2*x + E^x*(-3 + x)*Log[2*(1 + x)]))
 

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.

\(\displaystyle \int \frac {-30 x^2+e^x \left (15 x-5 x^2\right )+e^x \left (-5 x^3-5 x^2+30 x+30\right ) \log (2 x+2)+40 x+70}{4 x^6-24 x^5+21 x^4+49 x^3+e^{2 x} \left (x^6-5 x^5+3 x^4+9 x^3\right ) \log ^2(2 x+2)+e^x \left (4 x^6-22 x^5+16 x^4+42 x^3\right ) \log (2 x+2)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-5 \left (e^x+6\right ) x^2-5 e^x \left (x^3+x^2-6 x-6\right ) \log (2 (x+1))+5 \left (3 e^x+8\right ) x+70}{x^3 (x+1) \left (-2 x-e^x (x-3) \log (2 (x+1))+7\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {5 \left (-2 x^3 \log (2 (x+1))-2 x^2+11 x^2 \log (2 (x+1))+13 x-7 x \log (2 (x+1))-20 \log (2 (x+1))-21\right )}{(3-x) x^2 (x+1) \left (-2 x-e^x x \log (2 (x+1))+3 e^x \log (2 (x+1))+7\right )^2 \log (2 x+2)}+\frac {5 \left (x^3 (-\log (2 (x+1)))-x^2-x^2 \log (2 (x+1))+3 x+6 x \log (2 (x+1))+6 \log (2 (x+1))\right )}{(3-x) x^3 (x+1) \left (-2 x-e^x x \log (2 (x+1))+3 e^x \log (2 (x+1))+7\right ) \log (2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {5 \left (-\left (\left (e^x+6\right ) x^2\right )-e^x \left (x^3+x^2-6 x-6\right ) \log (2 (x+1))+\left (3 e^x+8\right ) x+14\right )}{x^3 (x+1) \left (-2 x-e^x (x-3) \log (2 (x+1))+7\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 5 \int \frac {-\left (\left (6+e^x\right ) x^2\right )+\left (8+3 e^x\right ) x+e^x \left (-x^3-x^2+6 x+6\right ) \log (2 (x+1))+14}{x^3 (x+1) \left (-2 x+e^x (3-x) \log (2 (x+1))+7\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 5 \int \left (\frac {-2 \log (2 (x+1)) x^3+11 \log (2 (x+1)) x^2-2 x^2-7 \log (2 (x+1)) x+13 x-20 \log (2 (x+1))-21}{(3-x) x^2 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right )^2 \log (2 x+2)}+\frac {-\log (2 (x+1)) x^3-\log (2 (x+1)) x^2-x^2+6 \log (2 (x+1)) x+3 x+6 \log (2 (x+1))}{(3-x) x^3 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right ) \log (2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 5 \int \frac {-\left (\left (6+e^x\right ) x^2\right )+\left (8+3 e^x\right ) x-e^x \left (x^3+x^2-6 x-6\right ) \log (2 (x+1))+14}{x^3 (x+1) \left (-2 x-e^x (x-3) \log (2 (x+1))+7\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 5 \int \left (\frac {-2 \log (2 (x+1)) x^3+11 \log (2 (x+1)) x^2-2 x^2-7 \log (2 (x+1)) x+13 x-20 \log (2 (x+1))-21}{(3-x) x^2 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right )^2 \log (2 x+2)}+\frac {-\log (2 (x+1)) x^3-\log (2 (x+1)) x^2-x^2+6 \log (2 (x+1)) x+3 x+6 \log (2 (x+1))}{(3-x) x^3 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right ) \log (2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 5 \int \frac {-\left (\left (6+e^x\right ) x^2\right )+\left (8+3 e^x\right ) x-e^x \left (x^3+x^2-6 x-6\right ) \log (2 (x+1))+14}{x^3 (x+1) \left (-2 x-e^x (x-3) \log (2 (x+1))+7\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 5 \int \left (\frac {-2 \log (2 (x+1)) x^3+11 \log (2 (x+1)) x^2-2 x^2-7 \log (2 (x+1)) x+13 x-20 \log (2 (x+1))-21}{(3-x) x^2 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right )^2 \log (2 x+2)}+\frac {-\log (2 (x+1)) x^3-\log (2 (x+1)) x^2-x^2+6 \log (2 (x+1)) x+3 x+6 \log (2 (x+1))}{(3-x) x^3 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right ) \log (2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 5 \int \frac {-\left (\left (6+e^x\right ) x^2\right )+\left (8+3 e^x\right ) x-e^x \left (x^3+x^2-6 x-6\right ) \log (2 (x+1))+14}{x^3 (x+1) \left (-2 x-e^x (x-3) \log (2 (x+1))+7\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 5 \int \left (\frac {-2 \log (2 (x+1)) x^3+11 \log (2 (x+1)) x^2-2 x^2-7 \log (2 (x+1)) x+13 x-20 \log (2 (x+1))-21}{(3-x) x^2 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right )^2 \log (2 x+2)}+\frac {-\log (2 (x+1)) x^3-\log (2 (x+1)) x^2-x^2+6 \log (2 (x+1)) x+3 x+6 \log (2 (x+1))}{(3-x) x^3 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right ) \log (2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 5 \int \frac {-\left (\left (6+e^x\right ) x^2\right )+\left (8+3 e^x\right ) x-e^x \left (x^3+x^2-6 x-6\right ) \log (2 (x+1))+14}{x^3 (x+1) \left (-2 x-e^x (x-3) \log (2 (x+1))+7\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 5 \int \left (\frac {-2 \log (2 (x+1)) x^3+11 \log (2 (x+1)) x^2-2 x^2-7 \log (2 (x+1)) x+13 x-20 \log (2 (x+1))-21}{(3-x) x^2 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right )^2 \log (2 x+2)}+\frac {-\log (2 (x+1)) x^3-\log (2 (x+1)) x^2-x^2+6 \log (2 (x+1)) x+3 x+6 \log (2 (x+1))}{(3-x) x^3 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right ) \log (2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 5 \int \frac {-\left (\left (6+e^x\right ) x^2\right )+\left (8+3 e^x\right ) x-e^x \left (x^3+x^2-6 x-6\right ) \log (2 (x+1))+14}{x^3 (x+1) \left (-2 x-e^x (x-3) \log (2 (x+1))+7\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 5 \int \left (\frac {-2 \log (2 (x+1)) x^3+11 \log (2 (x+1)) x^2-2 x^2-7 \log (2 (x+1)) x+13 x-20 \log (2 (x+1))-21}{(3-x) x^2 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right )^2 \log (2 x+2)}+\frac {-\log (2 (x+1)) x^3-\log (2 (x+1)) x^2-x^2+6 \log (2 (x+1)) x+3 x+6 \log (2 (x+1))}{(3-x) x^3 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right ) \log (2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 5 \int \frac {-\left (\left (6+e^x\right ) x^2\right )+\left (8+3 e^x\right ) x-e^x \left (x^3+x^2-6 x-6\right ) \log (2 (x+1))+14}{x^3 (x+1) \left (-2 x-e^x (x-3) \log (2 (x+1))+7\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 5 \int \left (\frac {-2 \log (2 (x+1)) x^3+11 \log (2 (x+1)) x^2-2 x^2-7 \log (2 (x+1)) x+13 x-20 \log (2 (x+1))-21}{(3-x) x^2 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right )^2 \log (2 x+2)}+\frac {-\log (2 (x+1)) x^3-\log (2 (x+1)) x^2-x^2+6 \log (2 (x+1)) x+3 x+6 \log (2 (x+1))}{(3-x) x^3 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right ) \log (2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 5 \int \frac {-\left (\left (6+e^x\right ) x^2\right )+\left (8+3 e^x\right ) x-e^x \left (x^3+x^2-6 x-6\right ) \log (2 (x+1))+14}{x^3 (x+1) \left (-2 x-e^x (x-3) \log (2 (x+1))+7\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 5 \int \left (\frac {-2 \log (2 (x+1)) x^3+11 \log (2 (x+1)) x^2-2 x^2-7 \log (2 (x+1)) x+13 x-20 \log (2 (x+1))-21}{(3-x) x^2 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right )^2 \log (2 x+2)}+\frac {-\log (2 (x+1)) x^3-\log (2 (x+1)) x^2-x^2+6 \log (2 (x+1)) x+3 x+6 \log (2 (x+1))}{(3-x) x^3 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right ) \log (2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 5 \int \frac {-\left (\left (6+e^x\right ) x^2\right )+\left (8+3 e^x\right ) x-e^x \left (x^3+x^2-6 x-6\right ) \log (2 (x+1))+14}{x^3 (x+1) \left (-2 x-e^x (x-3) \log (2 (x+1))+7\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 5 \int \left (\frac {-2 \log (2 (x+1)) x^3+11 \log (2 (x+1)) x^2-2 x^2-7 \log (2 (x+1)) x+13 x-20 \log (2 (x+1))-21}{(3-x) x^2 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right )^2 \log (2 x+2)}+\frac {-\log (2 (x+1)) x^3-\log (2 (x+1)) x^2-x^2+6 \log (2 (x+1)) x+3 x+6 \log (2 (x+1))}{(3-x) x^3 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right ) \log (2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 5 \int \frac {-\left (\left (6+e^x\right ) x^2\right )+\left (8+3 e^x\right ) x-e^x \left (x^3+x^2-6 x-6\right ) \log (2 (x+1))+14}{x^3 (x+1) \left (-2 x-e^x (x-3) \log (2 (x+1))+7\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 5 \int \left (\frac {-2 \log (2 (x+1)) x^3+11 \log (2 (x+1)) x^2-2 x^2-7 \log (2 (x+1)) x+13 x-20 \log (2 (x+1))-21}{(3-x) x^2 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right )^2 \log (2 x+2)}+\frac {-\log (2 (x+1)) x^3-\log (2 (x+1)) x^2-x^2+6 \log (2 (x+1)) x+3 x+6 \log (2 (x+1))}{(3-x) x^3 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right ) \log (2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 5 \int \frac {-\left (\left (6+e^x\right ) x^2\right )+\left (8+3 e^x\right ) x-e^x \left (x^3+x^2-6 x-6\right ) \log (2 (x+1))+14}{x^3 (x+1) \left (-2 x-e^x (x-3) \log (2 (x+1))+7\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 5 \int \left (\frac {-2 \log (2 (x+1)) x^3+11 \log (2 (x+1)) x^2-2 x^2-7 \log (2 (x+1)) x+13 x-20 \log (2 (x+1))-21}{(3-x) x^2 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right )^2 \log (2 x+2)}+\frac {-\log (2 (x+1)) x^3-\log (2 (x+1)) x^2-x^2+6 \log (2 (x+1)) x+3 x+6 \log (2 (x+1))}{(3-x) x^3 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right ) \log (2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 5 \int \frac {-\left (\left (6+e^x\right ) x^2\right )+\left (8+3 e^x\right ) x-e^x \left (x^3+x^2-6 x-6\right ) \log (2 (x+1))+14}{x^3 (x+1) \left (-2 x-e^x (x-3) \log (2 (x+1))+7\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 5 \int \left (\frac {-2 \log (2 (x+1)) x^3+11 \log (2 (x+1)) x^2-2 x^2-7 \log (2 (x+1)) x+13 x-20 \log (2 (x+1))-21}{(3-x) x^2 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right )^2 \log (2 x+2)}+\frac {-\log (2 (x+1)) x^3-\log (2 (x+1)) x^2-x^2+6 \log (2 (x+1)) x+3 x+6 \log (2 (x+1))}{(3-x) x^3 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right ) \log (2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 5 \int \frac {-\left (\left (6+e^x\right ) x^2\right )+\left (8+3 e^x\right ) x-e^x \left (x^3+x^2-6 x-6\right ) \log (2 (x+1))+14}{x^3 (x+1) \left (-2 x-e^x (x-3) \log (2 (x+1))+7\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 5 \int \left (\frac {-2 \log (2 (x+1)) x^3+11 \log (2 (x+1)) x^2-2 x^2-7 \log (2 (x+1)) x+13 x-20 \log (2 (x+1))-21}{(3-x) x^2 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right )^2 \log (2 x+2)}+\frac {-\log (2 (x+1)) x^3-\log (2 (x+1)) x^2-x^2+6 \log (2 (x+1)) x+3 x+6 \log (2 (x+1))}{(3-x) x^3 (x+1) \left (-e^x \log (2 (x+1)) x-2 x+3 e^x \log (2 (x+1))+7\right ) \log (2 x+2)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle 5 \int \frac {-\left (\left (6+e^x\right ) x^2\right )+\left (8+3 e^x\right ) x-e^x \left (x^3+x^2-6 x-6\right ) \log (2 (x+1))+14}{x^3 (x+1) \left (-2 x-e^x (x-3) \log (2 (x+1))+7\right )^2}dx\)

Input:

Int[(70 + 40*x - 30*x^2 + E^x*(15*x - 5*x^2) + E^x*(30 + 30*x - 5*x^2 - 5* 
x^3)*Log[2 + 2*x])/(49*x^3 + 21*x^4 - 24*x^5 + 4*x^6 + E^x*(42*x^3 + 16*x^ 
4 - 22*x^5 + 4*x^6)*Log[2 + 2*x] + E^(2*x)*(9*x^3 + 3*x^4 - 5*x^5 + x^6)*L 
og[2 + 2*x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18

method	result	size
risch	\(\frac {5}{x^{2} \left (x \,{\mathrm e}^{x} \ln \left (2+2 x \right )-3 \ln \left (2+2 x \right ) {\mathrm e}^{x}+2 x -7\right )}\)	\(33\)
parallelrisch	\(\frac {5}{x^{2} \left (x \,{\mathrm e}^{x} \ln \left (2+2 x \right )-3 \ln \left (2+2 x \right ) {\mathrm e}^{x}+2 x -7\right )}\)	\(33\)

Input:

int(((-5*x^3-5*x^2+30*x+30)*exp(x)*ln(2+2*x)+(-5*x^2+15*x)*exp(x)-30*x^2+4 
0*x+70)/((x^6-5*x^5+3*x^4+9*x^3)*exp(x)^2*ln(2+2*x)^2+(4*x^6-22*x^5+16*x^4 
+42*x^3)*exp(x)*ln(2+2*x)+4*x^6-24*x^5+21*x^4+49*x^3),x,method=_RETURNVERB 
OSE)
 

Output:

5/x^2/(x*exp(x)*ln(2+2*x)-3*ln(2+2*x)*exp(x)+2*x-7)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {70+40 x-30 x^2+e^x \left (15 x-5 x^2\right )+e^x \left (30+30 x-5 x^2-5 x^3\right ) \log (2+2 x)}{49 x^3+21 x^4-24 x^5+4 x^6+e^x \left (42 x^3+16 x^4-22 x^5+4 x^6\right ) \log (2+2 x)+e^{2 x} \left (9 x^3+3 x^4-5 x^5+x^6\right ) \log ^2(2+2 x)} \, dx=\frac {5}{2 \, x^{3} + {\left (x^{3} - 3 \, x^{2}\right )} e^{x} \log \left (2 \, x + 2\right ) - 7 \, x^{2}} \] Input:

integrate(((-5*x^3-5*x^2+30*x+30)*exp(x)*log(2+2*x)+(-5*x^2+15*x)*exp(x)-3 
0*x^2+40*x+70)/((x^6-5*x^5+3*x^4+9*x^3)*exp(x)^2*log(2+2*x)^2+(4*x^6-22*x^ 
5+16*x^4+42*x^3)*exp(x)*log(2+2*x)+4*x^6-24*x^5+21*x^4+49*x^3),x, algorith 
m="fricas")
 

Output:

5/(2*x^3 + (x^3 - 3*x^2)*e^x*log(2*x + 2) - 7*x^2)
 

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {70+40 x-30 x^2+e^x \left (15 x-5 x^2\right )+e^x \left (30+30 x-5 x^2-5 x^3\right ) \log (2+2 x)}{49 x^3+21 x^4-24 x^5+4 x^6+e^x \left (42 x^3+16 x^4-22 x^5+4 x^6\right ) \log (2+2 x)+e^{2 x} \left (9 x^3+3 x^4-5 x^5+x^6\right ) \log ^2(2+2 x)} \, dx=\frac {5}{2 x^{3} - 7 x^{2} + \left (x^{3} \log {\left (2 x + 2 \right )} - 3 x^{2} \log {\left (2 x + 2 \right )}\right ) e^{x}} \] Input:

integrate(((-5*x**3-5*x**2+30*x+30)*exp(x)*ln(2+2*x)+(-5*x**2+15*x)*exp(x) 
-30*x**2+40*x+70)/((x**6-5*x**5+3*x**4+9*x**3)*exp(x)**2*ln(2+2*x)**2+(4*x 
**6-22*x**5+16*x**4+42*x**3)*exp(x)*ln(2+2*x)+4*x**6-24*x**5+21*x**4+49*x* 
*3),x)
 

Output:

5/(2*x**3 - 7*x**2 + (x**3*log(2*x + 2) - 3*x**2*log(2*x + 2))*exp(x))
 

Maxima [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {70+40 x-30 x^2+e^x \left (15 x-5 x^2\right )+e^x \left (30+30 x-5 x^2-5 x^3\right ) \log (2+2 x)}{49 x^3+21 x^4-24 x^5+4 x^6+e^x \left (42 x^3+16 x^4-22 x^5+4 x^6\right ) \log (2+2 x)+e^{2 x} \left (9 x^3+3 x^4-5 x^5+x^6\right ) \log ^2(2+2 x)} \, dx=\frac {5}{2 \, x^{3} + {\left (x^{3} - 3 \, x^{2}\right )} e^{x} \log \left (x + 1\right ) - 7 \, x^{2} + {\left (x^{3} \log \left (2\right ) - 3 \, x^{2} \log \left (2\right )\right )} e^{x}} \] Input:

integrate(((-5*x^3-5*x^2+30*x+30)*exp(x)*log(2+2*x)+(-5*x^2+15*x)*exp(x)-3 
0*x^2+40*x+70)/((x^6-5*x^5+3*x^4+9*x^3)*exp(x)^2*log(2+2*x)^2+(4*x^6-22*x^ 
5+16*x^4+42*x^3)*exp(x)*log(2+2*x)+4*x^6-24*x^5+21*x^4+49*x^3),x, algorith 
m="maxima")
 

Output:

5/(2*x^3 + (x^3 - 3*x^2)*e^x*log(x + 1) - 7*x^2 + (x^3*log(2) - 3*x^2*log( 
2))*e^x)
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {70+40 x-30 x^2+e^x \left (15 x-5 x^2\right )+e^x \left (30+30 x-5 x^2-5 x^3\right ) \log (2+2 x)}{49 x^3+21 x^4-24 x^5+4 x^6+e^x \left (42 x^3+16 x^4-22 x^5+4 x^6\right ) \log (2+2 x)+e^{2 x} \left (9 x^3+3 x^4-5 x^5+x^6\right ) \log ^2(2+2 x)} \, dx=\frac {5}{x^{3} e^{x} \log \left (2 \, x + 2\right ) - 3 \, x^{2} e^{x} \log \left (2 \, x + 2\right ) + 2 \, x^{3} - 7 \, x^{2}} \] Input:

integrate(((-5*x^3-5*x^2+30*x+30)*exp(x)*log(2+2*x)+(-5*x^2+15*x)*exp(x)-3 
0*x^2+40*x+70)/((x^6-5*x^5+3*x^4+9*x^3)*exp(x)^2*log(2+2*x)^2+(4*x^6-22*x^ 
5+16*x^4+42*x^3)*exp(x)*log(2+2*x)+4*x^6-24*x^5+21*x^4+49*x^3),x, algorith 
m="giac")
 

Output:

5/(x^3*e^x*log(2*x + 2) - 3*x^2*e^x*log(2*x + 2) + 2*x^3 - 7*x^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {70+40 x-30 x^2+e^x \left (15 x-5 x^2\right )+e^x \left (30+30 x-5 x^2-5 x^3\right ) \log (2+2 x)}{49 x^3+21 x^4-24 x^5+4 x^6+e^x \left (42 x^3+16 x^4-22 x^5+4 x^6\right ) \log (2+2 x)+e^{2 x} \left (9 x^3+3 x^4-5 x^5+x^6\right ) \log ^2(2+2 x)} \, dx=\int \frac {40\,x+{\mathrm {e}}^x\,\left (15\,x-5\,x^2\right )-30\,x^2+{\mathrm {e}}^x\,\ln \left (2\,x+2\right )\,\left (-5\,x^3-5\,x^2+30\,x+30\right )+70}{49\,x^3+21\,x^4-24\,x^5+4\,x^6+{\mathrm {e}}^x\,\ln \left (2\,x+2\right )\,\left (4\,x^6-22\,x^5+16\,x^4+42\,x^3\right )+{\mathrm {e}}^{2\,x}\,{\ln \left (2\,x+2\right )}^2\,\left (x^6-5\,x^5+3\,x^4+9\,x^3\right )} \,d x \] Input:

int((40*x + exp(x)*(15*x - 5*x^2) - 30*x^2 + exp(x)*log(2*x + 2)*(30*x - 5 
*x^2 - 5*x^3 + 30) + 70)/(49*x^3 + 21*x^4 - 24*x^5 + 4*x^6 + exp(x)*log(2* 
x + 2)*(42*x^3 + 16*x^4 - 22*x^5 + 4*x^6) + exp(2*x)*log(2*x + 2)^2*(9*x^3 
 + 3*x^4 - 5*x^5 + x^6)),x)
 

Output:

int((40*x + exp(x)*(15*x - 5*x^2) - 30*x^2 + exp(x)*log(2*x + 2)*(30*x - 5 
*x^2 - 5*x^3 + 30) + 70)/(49*x^3 + 21*x^4 - 24*x^5 + 4*x^6 + exp(x)*log(2* 
x + 2)*(42*x^3 + 16*x^4 - 22*x^5 + 4*x^6) + exp(2*x)*log(2*x + 2)^2*(9*x^3 
 + 3*x^4 - 5*x^5 + x^6)), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {70+40 x-30 x^2+e^x \left (15 x-5 x^2\right )+e^x \left (30+30 x-5 x^2-5 x^3\right ) \log (2+2 x)}{49 x^3+21 x^4-24 x^5+4 x^6+e^x \left (42 x^3+16 x^4-22 x^5+4 x^6\right ) \log (2+2 x)+e^{2 x} \left (9 x^3+3 x^4-5 x^5+x^6\right ) \log ^2(2+2 x)} \, dx=\frac {5}{x^{2} \left (e^{x} \mathrm {log}\left (2 x +2\right ) x -3 e^{x} \mathrm {log}\left (2 x +2\right )+2 x -7\right )} \] Input:

int(((-5*x^3-5*x^2+30*x+30)*exp(x)*log(2+2*x)+(-5*x^2+15*x)*exp(x)-30*x^2+ 
40*x+70)/((x^6-5*x^5+3*x^4+9*x^3)*exp(x)^2*log(2+2*x)^2+(4*x^6-22*x^5+16*x 
^4+42*x^3)*exp(x)*log(2+2*x)+4*x^6-24*x^5+21*x^4+49*x^3),x)
 

Output:

5/(x**2*(e**x*log(2*x + 2)*x - 3*e**x*log(2*x + 2) + 2*x - 7))