\(\int \frac {19+52 x+73 x^2+58 x^3+28 x^4+8 x^5+x^6+e^x (25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6)}{25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6} \, dx\) [1463]

Optimal result

Integrand size = 91, antiderivative size = 18 \[ \int \frac {19+52 x+73 x^2+58 x^3+28 x^4+8 x^5+x^6+e^x \left (25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6\right )}{25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6} \, dx=4+e^x+x+\frac {x}{-1+x+(1+x)^4} \] Output:

4+x+exp(x)+x/((1+x)^4+x-1)
 

Mathematica [A] (verified)

Time = 1.79 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {19+52 x+73 x^2+58 x^3+28 x^4+8 x^5+x^6+e^x \left (25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6\right )}{25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6} \, dx=e^x+x+\frac {1}{5+6 x+4 x^2+x^3} \] Input:

Integrate[(19 + 52*x + 73*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6 + E^x*(25 + 
60*x + 76*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6))/(25 + 60*x + 76*x^2 + 58*x 
^3 + 28*x^4 + 8*x^5 + x^6),x]
 

Output:

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

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.

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^6+8 x^5+28 x^4+58 x^3+76 x^2+60 x+25\right )+52 x+19}{x^6+8 x^5+28 x^4+58 x^3+76 x^2+60 x+25} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^6+8 x^5+28 x^4+58 x^3+76 x^2+60 x+25\right )+52 x+19}{\left (x^3+4 x^2+6 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {58 x^3}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {73 x^2}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {52 x}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {19}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {x^6}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {8 x^5}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {28 x^4}{\left (x^3+4 x^2+6 x+5\right )^2}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^3+4 x^2+6 x+5\right )^2+52 x+19}{\left (x^3+4 x^2+6 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {58 x^3}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {73 x^2}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {52 x}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {19}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {x^6}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {8 x^5}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {28 x^4}{\left (x^3+4 x^2+6 x+5\right )^2}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^3+4 x^2+6 x+5\right )^2+52 x+19}{\left (x^3+4 x^2+6 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {58 x^3}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {73 x^2}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {52 x}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {19}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {x^6}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {8 x^5}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {28 x^4}{\left (x^3+4 x^2+6 x+5\right )^2}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^3+4 x^2+6 x+5\right )^2+52 x+19}{\left (x^3+4 x^2+6 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {58 x^3}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {73 x^2}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {52 x}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {19}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {x^6}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {8 x^5}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {28 x^4}{\left (x^3+4 x^2+6 x+5\right )^2}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^3+4 x^2+6 x+5\right )^2+52 x+19}{\left (x^3+4 x^2+6 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {58 x^3}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {73 x^2}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {52 x}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {19}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {x^6}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {8 x^5}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {28 x^4}{\left (x^3+4 x^2+6 x+5\right )^2}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^3+4 x^2+6 x+5\right )^2+52 x+19}{\left (x^3+4 x^2+6 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {58 x^3}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {73 x^2}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {52 x}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {19}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {x^6}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {8 x^5}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {28 x^4}{\left (x^3+4 x^2+6 x+5\right )^2}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^3+4 x^2+6 x+5\right )^2+52 x+19}{\left (x^3+4 x^2+6 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {58 x^3}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {73 x^2}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {52 x}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {19}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {x^6}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {8 x^5}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {28 x^4}{\left (x^3+4 x^2+6 x+5\right )^2}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^3+4 x^2+6 x+5\right )^2+52 x+19}{\left (x^3+4 x^2+6 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {58 x^3}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {73 x^2}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {52 x}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {19}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {x^6}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {8 x^5}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {28 x^4}{\left (x^3+4 x^2+6 x+5\right )^2}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^3+4 x^2+6 x+5\right )^2+52 x+19}{\left (x^3+4 x^2+6 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {58 x^3}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {73 x^2}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {52 x}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {19}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {x^6}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {8 x^5}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {28 x^4}{\left (x^3+4 x^2+6 x+5\right )^2}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^3+4 x^2+6 x+5\right )^2+52 x+19}{\left (x^3+4 x^2+6 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {58 x^3}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {73 x^2}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {52 x}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {19}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {x^6}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {8 x^5}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {28 x^4}{\left (x^3+4 x^2+6 x+5\right )^2}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^3+4 x^2+6 x+5\right )^2+52 x+19}{\left (x^3+4 x^2+6 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {58 x^3}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {73 x^2}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {52 x}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {19}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {x^6}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {8 x^5}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {28 x^4}{\left (x^3+4 x^2+6 x+5\right )^2}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^3+4 x^2+6 x+5\right )^2+52 x+19}{\left (x^3+4 x^2+6 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {58 x^3}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {73 x^2}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {52 x}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {19}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {x^6}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {8 x^5}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {28 x^4}{\left (x^3+4 x^2+6 x+5\right )^2}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^3+4 x^2+6 x+5\right )^2+52 x+19}{\left (x^3+4 x^2+6 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {58 x^3}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {73 x^2}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {52 x}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {19}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {x^6}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {8 x^5}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {28 x^4}{\left (x^3+4 x^2+6 x+5\right )^2}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^3+4 x^2+6 x+5\right )^2+52 x+19}{\left (x^3+4 x^2+6 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {58 x^3}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {73 x^2}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {52 x}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {19}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {x^6}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {8 x^5}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {28 x^4}{\left (x^3+4 x^2+6 x+5\right )^2}+e^x\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x^6+8 x^5+28 x^4+58 x^3+73 x^2+e^x \left (x^3+4 x^2+6 x+5\right )^2+52 x+19}{\left (x^3+4 x^2+6 x+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {58 x^3}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {73 x^2}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {52 x}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {19}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {x^6}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {8 x^5}{\left (x^3+4 x^2+6 x+5\right )^2}+\frac {28 x^4}{\left (x^3+4 x^2+6 x+5\right )^2}+e^x\right )dx\)

Input:

Int[(19 + 52*x + 73*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6 + E^x*(25 + 60*x + 
 76*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6))/(25 + 60*x + 76*x^2 + 58*x^3 + 2 
8*x^4 + 8*x^5 + x^6),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11

method	result	size
risch	\(x +\frac {1}{x^{3}+4 x^{2}+6 x +5}+{\mathrm e}^{x}\)	\(20\)
parts	\(x +\frac {1}{x^{3}+4 x^{2}+6 x +5}+{\mathrm e}^{x}\)	\(20\)
norman	\(\frac {x^{4}-19 x -10 x^{2}+{\mathrm e}^{x} x^{3}+6 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{x} x^{2}+5 \,{\mathrm e}^{x}-19}{x^{3}+4 x^{2}+6 x +5}\)	\(52\)
parallelrisch	\(\frac {x^{4}-19 x -10 x^{2}+{\mathrm e}^{x} x^{3}+6 \,{\mathrm e}^{x} x +4 \,{\mathrm e}^{x} x^{2}+5 \,{\mathrm e}^{x}-19}{x^{3}+4 x^{2}+6 x +5}\)	\(52\)
orering	\(\frac {\left (x^{10}+5 x^{9}-18 x^{8}-239 x^{7}-1044 x^{6}-2752 x^{5}-4937 x^{4}-6243 x^{3}-5486 x^{2}-3079 x -882\right ) \left (\left (x^{6}+8 x^{5}+28 x^{4}+58 x^{3}+76 x^{2}+60 x +25\right ) {\mathrm e}^{x}+x^{6}+8 x^{5}+28 x^{4}+58 x^{3}+73 x^{2}+52 x +19\right )}{\left (x^{9}+12 x^{8}+66 x^{7}+223 x^{6}+513 x^{5}+820 x^{4}+891 x^{3}+621 x^{2}+260 x +63\right ) \left (x^{6}+8 x^{5}+28 x^{4}+58 x^{3}+76 x^{2}+60 x +25\right )}-\frac {\left (x^{7}-36 x^{5}-166 x^{4}-387 x^{3}-541 x^{2}-441 x -189\right ) \left (x^{3}+4 x^{2}+6 x +5\right ) \left (\frac {\left (6 x^{5}+40 x^{4}+112 x^{3}+174 x^{2}+152 x +60\right ) {\mathrm e}^{x}+\left (x^{6}+8 x^{5}+28 x^{4}+58 x^{3}+76 x^{2}+60 x +25\right ) {\mathrm e}^{x}+6 x^{5}+40 x^{4}+112 x^{3}+174 x^{2}+146 x +52}{x^{6}+8 x^{5}+28 x^{4}+58 x^{3}+76 x^{2}+60 x +25}-\frac {\left (\left (x^{6}+8 x^{5}+28 x^{4}+58 x^{3}+76 x^{2}+60 x +25\right ) {\mathrm e}^{x}+x^{6}+8 x^{5}+28 x^{4}+58 x^{3}+73 x^{2}+52 x +19\right ) \left (6 x^{5}+40 x^{4}+112 x^{3}+174 x^{2}+152 x +60\right )}{\left (x^{6}+8 x^{5}+28 x^{4}+58 x^{3}+76 x^{2}+60 x +25\right )^{2}}\right )}{x^{9}+12 x^{8}+66 x^{7}+223 x^{6}+513 x^{5}+820 x^{4}+891 x^{3}+621 x^{2}+260 x +63}\)	\(505\)
default	\(\text {Expression too large to display}\)	\(1065\)

Input:

int(((x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25)*exp(x)+x^6+8*x^5+28*x^4+58*x 
^3+73*x^2+52*x+19)/(x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25),x,method=_RETU 
RNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (17) = 34\).

Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.78 \[ \int \frac {19+52 x+73 x^2+58 x^3+28 x^4+8 x^5+x^6+e^x \left (25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6\right )}{25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6} \, dx=\frac {x^{4} + 4 \, x^{3} + 6 \, x^{2} + {\left (x^{3} + 4 \, x^{2} + 6 \, x + 5\right )} e^{x} + 5 \, x + 1}{x^{3} + 4 \, x^{2} + 6 \, x + 5} \] Input:

integrate(((x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25)*exp(x)+x^6+8*x^5+28*x^ 
4+58*x^3+73*x^2+52*x+19)/(x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25),x, algor 
ithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {19+52 x+73 x^2+58 x^3+28 x^4+8 x^5+x^6+e^x \left (25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6\right )}{25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6} \, dx=x + e^{x} + \frac {1}{x^{3} + 4 x^{2} + 6 x + 5} \] Input:

integrate(((x**6+8*x**5+28*x**4+58*x**3+76*x**2+60*x+25)*exp(x)+x**6+8*x** 
5+28*x**4+58*x**3+73*x**2+52*x+19)/(x**6+8*x**5+28*x**4+58*x**3+76*x**2+60 
*x+25),x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (17) = 34\).

Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.78 \[ \int \frac {19+52 x+73 x^2+58 x^3+28 x^4+8 x^5+x^6+e^x \left (25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6\right )}{25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6} \, dx=\frac {x^{4} + 4 \, x^{3} + 6 \, x^{2} + {\left (x^{3} + 4 \, x^{2} + 6 \, x + 5\right )} e^{x} + 5 \, x + 1}{x^{3} + 4 \, x^{2} + 6 \, x + 5} \] Input:

integrate(((x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25)*exp(x)+x^6+8*x^5+28*x^ 
4+58*x^3+73*x^2+52*x+19)/(x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25),x, algor 
ithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (17) = 34\).

Time = 0.12 (sec) , antiderivative size = 56, normalized size of antiderivative = 3.11 \[ \int \frac {19+52 x+73 x^2+58 x^3+28 x^4+8 x^5+x^6+e^x \left (25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6\right )}{25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6} \, dx=\frac {x^{4} + x^{3} e^{x} + 4 \, x^{3} + 4 \, x^{2} e^{x} + 6 \, x^{2} + 6 \, x e^{x} + 5 \, x + 5 \, e^{x} + 1}{x^{3} + 4 \, x^{2} + 6 \, x + 5} \] Input:

integrate(((x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25)*exp(x)+x^6+8*x^5+28*x^ 
4+58*x^3+73*x^2+52*x+19)/(x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25),x, algor 
ithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {19+52 x+73 x^2+58 x^3+28 x^4+8 x^5+x^6+e^x \left (25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6\right )}{25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6} \, dx=x+{\mathrm {e}}^x+\frac {1}{x^3+4\,x^2+6\,x+5} \] Input:

int((52*x + exp(x)*(60*x + 76*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6 + 25) + 
73*x^2 + 58*x^3 + 28*x^4 + 8*x^5 + x^6 + 19)/(60*x + 76*x^2 + 58*x^3 + 28* 
x^4 + 8*x^5 + x^6 + 25),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.33 \[ \int \frac {19+52 x+73 x^2+58 x^3+28 x^4+8 x^5+x^6+e^x \left (25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6\right )}{25+60 x+76 x^2+58 x^3+28 x^4+8 x^5+x^6} \, dx=\frac {2 e^{x} x^{3}+8 e^{x} x^{2}+12 e^{x} x +10 e^{x}+2 x^{4}+5 x^{3}-8 x -13}{2 x^{3}+8 x^{2}+12 x +10} \] Input:

int(((x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25)*exp(x)+x^6+8*x^5+28*x^4+58*x 
^3+73*x^2+52*x+19)/(x^6+8*x^5+28*x^4+58*x^3+76*x^2+60*x+25),x)
 

Output:

(2*e**x*x**3 + 8*e**x*x**2 + 12*e**x*x + 10*e**x + 2*x**4 + 5*x**3 - 8*x - 
 13)/(2*(x**3 + 4*x**2 + 6*x + 5))