\(\int \frac {-5 x+25 x^2+5 e^4 x^2+20 x^3+45 x^4-30 x^5-10 x^6+5 x^7+(35+5 e^4+5 x+5 x^2-5 x^3-10 x^4+5 x^5) \log (7+e^4+x+x^2-x^3-2 x^4+x^5)}{7 x^2+e^4 x^2+x^3+x^4-x^5-2 x^6+x^7} \, dx\) [2530]

Optimal result

Integrand size = 131, antiderivative size = 31 \[ \int \frac {-5 x+25 x^2+5 e^4 x^2+20 x^3+45 x^4-30 x^5-10 x^6+5 x^7+\left (35+5 e^4+5 x+5 x^2-5 x^3-10 x^4+5 x^5\right ) \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{7 x^2+e^4 x^2+x^3+x^4-x^5-2 x^6+x^7} \, dx=5 \left (x-\frac {\log \left (7+e^4+x \left (-x+\left (1+x-x^2\right )^2\right )\right )}{x}\right ) \] Output:

5*x-5*ln(7+x*((-x^2+x+1)^2-x)+exp(4))/x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-5 x+25 x^2+5 e^4 x^2+20 x^3+45 x^4-30 x^5-10 x^6+5 x^7+\left (35+5 e^4+5 x+5 x^2-5 x^3-10 x^4+5 x^5\right ) \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{7 x^2+e^4 x^2+x^3+x^4-x^5-2 x^6+x^7} \, dx=5 \left (x-\frac {\log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x}\right ) \] Input:

Integrate[(-5*x + 25*x^2 + 5*E^4*x^2 + 20*x^3 + 45*x^4 - 30*x^5 - 10*x^6 + 
 5*x^7 + (35 + 5*E^4 + 5*x + 5*x^2 - 5*x^3 - 10*x^4 + 5*x^5)*Log[7 + E^4 + 
 x + x^2 - x^3 - 2*x^4 + x^5])/(7*x^2 + E^4*x^2 + x^3 + x^4 - x^5 - 2*x^6 
+ x^7),x]
 

Output:

5*(x - Log[7 + E^4 + x + x^2 - x^3 - 2*x^4 + x^5]/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.

\(\displaystyle \int \frac {5 x^7-10 x^6-30 x^5+45 x^4+20 x^3+5 e^4 x^2+25 x^2+\left (5 x^5-10 x^4-5 x^3+5 x^2+5 x+5 e^4+35\right ) \log \left (x^5-2 x^4-x^3+x^2+x+e^4+7\right )-5 x}{x^7-2 x^6-x^5+x^4+x^3+e^4 x^2+7 x^2} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {5 x^7-10 x^6-30 x^5+45 x^4+20 x^3+5 e^4 x^2+25 x^2+\left (5 x^5-10 x^4-5 x^3+5 x^2+5 x+5 e^4+35\right ) \log \left (x^5-2 x^4-x^3+x^2+x+e^4+7\right )-5 x}{x^7-2 x^6-x^5+x^4+x^3+\left (7+e^4\right ) x^2}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {5 x^7-10 x^6-30 x^5+45 x^4+20 x^3+\left (25+5 e^4\right ) x^2+\left (5 x^5-10 x^4-5 x^3+5 x^2+5 x+5 e^4+35\right ) \log \left (x^5-2 x^4-x^3+x^2+x+e^4+7\right )-5 x}{x^7-2 x^6-x^5+x^4+x^3+\left (7+e^4\right ) x^2}dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {5 x^7-10 x^6-30 x^5+45 x^4+20 x^3+\left (25+5 e^4\right ) x^2+\left (5 x^5-10 x^4-5 x^3+5 x^2+5 x+5 e^4+35\right ) \log \left (x^5-2 x^4-x^3+x^2+x+e^4+7\right )-5 x}{x^2 \left (x^5-2 x^4-x^3+x^2+x+e^4+7\right )}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {5 x^7-10 x^6-30 x^5+45 x^4+20 x^3+\left (25+5 e^4\right ) x^2+\left (5 x^5-10 x^4-5 x^3+5 x^2+5 x+5 e^4+35\right ) \log \left (x^5-2 x^4-x^3+x^2+x+e^4+7\right )-5 x}{x^2 \left (x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {5 x^5}{x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )}+\frac {10 x^4}{-x^5+2 x^4+x^3-x^2-x-7 \left (1+\frac {e^4}{7}\right )}+\frac {30 x^3}{-x^5+2 x^4+x^3-x^2-x-7 \left (1+\frac {e^4}{7}\right )}+\frac {45 x^2}{x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )}+\frac {20 x}{x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )}+\frac {5 \left (5+e^4\right )}{x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )}+\frac {5}{\left (-x^5+2 x^4+x^3-x^2-x-7 \left (1+\frac {e^4}{7}\right )\right ) x}+\frac {5 \log \left (x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 5 x-2 \log \left (-x^5+2 x^4+x^3-x^2-x-e^4-7\right )+2 \log \left (x^5-2 x^4-x^3+x^2+x+e^4+7\right )-2 \int \frac {1}{-x^5+2 x^4+x^3-x^2-x-7 \left (1+\frac {e^4}{7}\right )}dx-4 \int \frac {x}{-x^5+2 x^4+x^3-x^2-x-7 \left (1+\frac {e^4}{7}\right )}dx+\frac {2 \int \frac {x^2}{-x^5+2 x^4+x^3-x^2-x-7 \left (1+\frac {e^4}{7}\right )}dx}{7+e^4}-\int \frac {x^2}{-x^5+2 x^4+x^3-x^2-x-7 \left (1+\frac {e^4}{7}\right )}dx+\frac {2 \int \frac {x^3}{-x^5+2 x^4+x^3-x^2-x-7 \left (1+\frac {e^4}{7}\right )}dx}{7+e^4}+9 \int \frac {x^3}{-x^5+2 x^4+x^3-x^2-x-7 \left (1+\frac {e^4}{7}\right )}dx-\left (37+5 e^4\right ) \int \frac {1}{x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )}dx+\frac {2 \left (33+5 e^4\right ) \int \frac {1}{x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )}dx}{7+e^4}+\frac {4 \int \frac {1}{x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )}dx}{7+e^4}+5 \left (5+e^4\right ) \int \frac {1}{x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )}dx-\frac {3 \left (36+5 e^4\right ) \int \frac {x}{x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )}dx}{7+e^4}+\frac {3 \int \frac {x}{x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )}dx}{7+e^4}+11 \int \frac {x}{x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )}dx-\frac {2 \left (139+20 e^4\right ) \int \frac {x^2}{x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )}dx}{7+e^4}+39 \int \frac {x^2}{x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )}dx+\frac {\left (177+25 e^4\right ) \int \frac {x^3}{x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )}dx}{7+e^4}-16 \int \frac {x^3}{x^5-2 x^4-x^3+x^2+x+7 \left (1+\frac {e^4}{7}\right )}dx-\frac {5 \log \left (x^5-2 x^4-x^3+x^2+x+e^4+7\right )}{x}\)

Input:

Int[(-5*x + 25*x^2 + 5*E^4*x^2 + 20*x^3 + 45*x^4 - 30*x^5 - 10*x^6 + 5*x^7 
 + (35 + 5*E^4 + 5*x + 5*x^2 - 5*x^3 - 10*x^4 + 5*x^5)*Log[7 + E^4 + x + x 
^2 - x^3 - 2*x^4 + x^5])/(7*x^2 + E^4*x^2 + x^3 + x^4 - x^5 - 2*x^6 + x^7) 
,x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 39.56 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03

method	result	size
risch	\(-\frac {5 \ln \left ({\mathrm e}^{4}+x^{5}-2 x^{4}-x^{3}+x^{2}+x +7\right )}{x}+5 x\)	\(32\)
norman	\(\frac {5 x^{2}-5 \ln \left ({\mathrm e}^{4}+x^{5}-2 x^{4}-x^{3}+x^{2}+x +7\right )}{x}\)	\(35\)
parallelrisch	\(\frac {5 x^{2}+20 x -5 \ln \left ({\mathrm e}^{4}+x^{5}-2 x^{4}-x^{3}+x^{2}+x +7\right )}{x}\)	\(38\)
default	\(-\frac {5 \ln \left ({\mathrm e}^{4}+x^{5}-2 x^{4}-x^{3}+x^{2}+x +7\right )}{x}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left ({\mathrm e}^{4}+\textit {\_Z}^{5}-2 \textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +7\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+\left (5 \,{\mathrm e}^{4}+37\right ) \textit {\_R}^{3}+\left (-8 \,{\mathrm e}^{4}-55\right ) \textit {\_R}^{2}+\left (-3 \,{\mathrm e}^{4}-22\right ) \textit {\_R} +2 \,{\mathrm e}^{4}+13\right ) \ln \left (x -\textit {\_R} \right )}{5 \textit {\_R}^{4}-8 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} +1}\right )}{{\mathrm e}^{4}+7}+5 x +\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left ({\mathrm e}^{4}+\textit {\_Z}^{5}-2 \textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +7\right )}{\sum }\frac {\left (\textit {\_R}^{4}+\left (-5 \,{\mathrm e}^{4}-37\right ) \textit {\_R}^{3}+\left (8 \,{\mathrm e}^{4}+55\right ) \textit {\_R}^{2}+\left (3 \,{\mathrm e}^{4}+22\right ) \textit {\_R} -2 \,{\mathrm e}^{4}-13\right ) \ln \left (x -\textit {\_R} \right )}{5 \textit {\_R}^{4}-8 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} +1}\right )}{{\mathrm e}^{4}+7}\)	\(232\)
parts	\(-\frac {5 \ln \left ({\mathrm e}^{4}+x^{5}-2 x^{4}-x^{3}+x^{2}+x +7\right )}{x}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left ({\mathrm e}^{4}+\textit {\_Z}^{5}-2 \textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +7\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+\left (5 \,{\mathrm e}^{4}+37\right ) \textit {\_R}^{3}+\left (-8 \,{\mathrm e}^{4}-55\right ) \textit {\_R}^{2}+\left (-3 \,{\mathrm e}^{4}-22\right ) \textit {\_R} +2 \,{\mathrm e}^{4}+13\right ) \ln \left (x -\textit {\_R} \right )}{5 \textit {\_R}^{4}-8 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} +1}\right )}{{\mathrm e}^{4}+7}+5 x +\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left ({\mathrm e}^{4}+\textit {\_Z}^{5}-2 \textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +7\right )}{\sum }\frac {\left (\textit {\_R}^{4}+\left (-5 \,{\mathrm e}^{4}-37\right ) \textit {\_R}^{3}+\left (8 \,{\mathrm e}^{4}+55\right ) \textit {\_R}^{2}+\left (3 \,{\mathrm e}^{4}+22\right ) \textit {\_R} -2 \,{\mathrm e}^{4}-13\right ) \ln \left (x -\textit {\_R} \right )}{5 \textit {\_R}^{4}-8 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} +1}\right )}{{\mathrm e}^{4}+7}\)	\(232\)

Input:

int(((5*exp(4)+5*x^5-10*x^4-5*x^3+5*x^2+5*x+35)*ln(exp(4)+x^5-2*x^4-x^3+x^ 
2+x+7)+5*x^2*exp(4)+5*x^7-10*x^6-30*x^5+45*x^4+20*x^3+25*x^2-5*x)/(x^2*exp 
(4)+x^7-2*x^6-x^5+x^4+x^3+7*x^2),x,method=_RETURNVERBOSE)
 

Output:

-5/x*ln(exp(4)+x^5-2*x^4-x^3+x^2+x+7)+5*x
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {-5 x+25 x^2+5 e^4 x^2+20 x^3+45 x^4-30 x^5-10 x^6+5 x^7+\left (35+5 e^4+5 x+5 x^2-5 x^3-10 x^4+5 x^5\right ) \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{7 x^2+e^4 x^2+x^3+x^4-x^5-2 x^6+x^7} \, dx=\frac {5 \, {\left (x^{2} - \log \left (x^{5} - 2 \, x^{4} - x^{3} + x^{2} + x + e^{4} + 7\right )\right )}}{x} \] Input:

integrate(((5*exp(4)+5*x^5-10*x^4-5*x^3+5*x^2+5*x+35)*log(exp(4)+x^5-2*x^4 
-x^3+x^2+x+7)+5*x^2*exp(4)+5*x^7-10*x^6-30*x^5+45*x^4+20*x^3+25*x^2-5*x)/( 
x^2*exp(4)+x^7-2*x^6-x^5+x^4+x^3+7*x^2),x, algorithm="fricas")
 

Output:

5*(x^2 - log(x^5 - 2*x^4 - x^3 + x^2 + x + e^4 + 7))/x
 

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {-5 x+25 x^2+5 e^4 x^2+20 x^3+45 x^4-30 x^5-10 x^6+5 x^7+\left (35+5 e^4+5 x+5 x^2-5 x^3-10 x^4+5 x^5\right ) \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{7 x^2+e^4 x^2+x^3+x^4-x^5-2 x^6+x^7} \, dx=5 x - \frac {5 \log {\left (x^{5} - 2 x^{4} - x^{3} + x^{2} + x + 7 + e^{4} \right )}}{x} \] Input:

integrate(((5*exp(4)+5*x**5-10*x**4-5*x**3+5*x**2+5*x+35)*ln(exp(4)+x**5-2 
*x**4-x**3+x**2+x+7)+5*x**2*exp(4)+5*x**7-10*x**6-30*x**5+45*x**4+20*x**3+ 
25*x**2-5*x)/(x**2*exp(4)+x**7-2*x**6-x**5+x**4+x**3+7*x**2),x)
 

Output:

5*x - 5*log(x**5 - 2*x**4 - x**3 + x**2 + x + 7 + exp(4))/x
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {-5 x+25 x^2+5 e^4 x^2+20 x^3+45 x^4-30 x^5-10 x^6+5 x^7+\left (35+5 e^4+5 x+5 x^2-5 x^3-10 x^4+5 x^5\right ) \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{7 x^2+e^4 x^2+x^3+x^4-x^5-2 x^6+x^7} \, dx=\frac {5 \, {\left (x^{2} - \log \left (x^{5} - 2 \, x^{4} - x^{3} + x^{2} + x + e^{4} + 7\right )\right )}}{x} \] Input:

integrate(((5*exp(4)+5*x^5-10*x^4-5*x^3+5*x^2+5*x+35)*log(exp(4)+x^5-2*x^4 
-x^3+x^2+x+7)+5*x^2*exp(4)+5*x^7-10*x^6-30*x^5+45*x^4+20*x^3+25*x^2-5*x)/( 
x^2*exp(4)+x^7-2*x^6-x^5+x^4+x^3+7*x^2),x, algorithm="maxima")
 

Output:

5*(x^2 - log(x^5 - 2*x^4 - x^3 + x^2 + x + e^4 + 7))/x
 

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {-5 x+25 x^2+5 e^4 x^2+20 x^3+45 x^4-30 x^5-10 x^6+5 x^7+\left (35+5 e^4+5 x+5 x^2-5 x^3-10 x^4+5 x^5\right ) \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{7 x^2+e^4 x^2+x^3+x^4-x^5-2 x^6+x^7} \, dx=\frac {5 \, {\left (x^{2} - \log \left (x^{5} - 2 \, x^{4} - x^{3} + x^{2} + x + e^{4} + 7\right )\right )}}{x} \] Input:

integrate(((5*exp(4)+5*x^5-10*x^4-5*x^3+5*x^2+5*x+35)*log(exp(4)+x^5-2*x^4 
-x^3+x^2+x+7)+5*x^2*exp(4)+5*x^7-10*x^6-30*x^5+45*x^4+20*x^3+25*x^2-5*x)/( 
x^2*exp(4)+x^7-2*x^6-x^5+x^4+x^3+7*x^2),x, algorithm="giac")
 

Output:

5*(x^2 - log(x^5 - 2*x^4 - x^3 + x^2 + x + e^4 + 7))/x
 

Mupad [B] (verification not implemented)

Time = 3.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {-5 x+25 x^2+5 e^4 x^2+20 x^3+45 x^4-30 x^5-10 x^6+5 x^7+\left (35+5 e^4+5 x+5 x^2-5 x^3-10 x^4+5 x^5\right ) \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{7 x^2+e^4 x^2+x^3+x^4-x^5-2 x^6+x^7} \, dx=-\frac {5\,\left (\ln \left (x^5-2\,x^4-x^3+x^2+x+{\mathrm {e}}^4+7\right )-x^2\right )}{x} \] Input:

int((5*x^2*exp(4) - 5*x + 25*x^2 + 20*x^3 + 45*x^4 - 30*x^5 - 10*x^6 + 5*x 
^7 + log(x + exp(4) + x^2 - x^3 - 2*x^4 + x^5 + 7)*(5*x + 5*exp(4) + 5*x^2 
 - 5*x^3 - 10*x^4 + 5*x^5 + 35))/(x^2*exp(4) + 7*x^2 + x^3 + x^4 - x^5 - 2 
*x^6 + x^7),x)
 

Output:

-(5*(log(x + exp(4) + x^2 - x^3 - 2*x^4 + x^5 + 7) - x^2))/x
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {-5 x+25 x^2+5 e^4 x^2+20 x^3+45 x^4-30 x^5-10 x^6+5 x^7+\left (35+5 e^4+5 x+5 x^2-5 x^3-10 x^4+5 x^5\right ) \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{7 x^2+e^4 x^2+x^3+x^4-x^5-2 x^6+x^7} \, dx=\frac {-5 \,\mathrm {log}\left (x^{5}+e^{4}-2 x^{4}-x^{3}+x^{2}+x +7\right )+5 x^{2}}{x} \] Input:

int(((5*exp(4)+5*x^5-10*x^4-5*x^3+5*x^2+5*x+35)*log(exp(4)+x^5-2*x^4-x^3+x 
^2+x+7)+5*x^2*exp(4)+5*x^7-10*x^6-30*x^5+45*x^4+20*x^3+25*x^2-5*x)/(x^2*ex 
p(4)+x^7-2*x^6-x^5+x^4+x^3+7*x^2),x)
 

Output:

(5*( - log(e**4 + x**5 - 2*x**4 - x**3 + x**2 + x + 7) + x**2))/x