\(\int \frac {128 x^2+32 x^3-2 x^5+e^4 (-64 x^2-32 x^3+4 x^4-2 x^5)+e^{14} (32+8 x+e^4 (-16-8 x+x^2))+e^7 (128 x+32 x^2-2 x^4+e^4 (-64 x-32 x^2+4 x^3-x^4))+(-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} (-16-8 x+x^2)+e^7 (-64 x-32 x^2+4 x^3-x^4)) \log (\frac {-32 x-16 x^2+2 x^3-x^4+e^7 (-16-8 x+x^2)}{e^7 x^2+2 x^3})}{-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} (-16-8 x+x^2)+e^7 (-64 x-32 x^2+4 x^3-x^4)} \, dx\) [463]

Optimal result

Integrand size = 267, antiderivative size = 33 \[ \int \frac {128 x^2+32 x^3-2 x^5+e^4 \left (-64 x^2-32 x^3+4 x^4-2 x^5\right )+e^{14} \left (32+8 x+e^4 \left (-16-8 x+x^2\right )\right )+e^7 \left (128 x+32 x^2-2 x^4+e^4 \left (-64 x-32 x^2+4 x^3-x^4\right )\right )+\left (-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )\right ) \log \left (\frac {-32 x-16 x^2+2 x^3-x^4+e^7 \left (-16-8 x+x^2\right )}{e^7 x^2+2 x^3}\right )}{-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )} \, dx=x \left (e^4+\log \left (2-\frac {x}{2+\frac {e^7}{x}}-\frac {(4+x)^2}{x^2}\right )\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {128 x^2+32 x^3-2 x^5+e^4 \left (-64 x^2-32 x^3+4 x^4-2 x^5\right )+e^{14} \left (32+8 x+e^4 \left (-16-8 x+x^2\right )\right )+e^7 \left (128 x+32 x^2-2 x^4+e^4 \left (-64 x-32 x^2+4 x^3-x^4\right )\right )+\left (-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )\right ) \log \left (\frac {-32 x-16 x^2+2 x^3-x^4+e^7 \left (-16-8 x+x^2\right )}{e^7 x^2+2 x^3}\right )}{-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )} \, dx=e^4 x+x \log \left (\frac {e^7 \left (-16-8 x+x^2\right )-x \left (32+16 x-2 x^2+x^3\right )}{x^2 \left (e^7+2 x\right )}\right ) \] Input:

Integrate[(128*x^2 + 32*x^3 - 2*x^5 + E^4*(-64*x^2 - 32*x^3 + 4*x^4 - 2*x^ 
5) + E^14*(32 + 8*x + E^4*(-16 - 8*x + x^2)) + E^7*(128*x + 32*x^2 - 2*x^4 
 + E^4*(-64*x - 32*x^2 + 4*x^3 - x^4)) + (-64*x^2 - 32*x^3 + 4*x^4 - 2*x^5 
 + E^14*(-16 - 8*x + x^2) + E^7*(-64*x - 32*x^2 + 4*x^3 - x^4))*Log[(-32*x 
 - 16*x^2 + 2*x^3 - x^4 + E^7*(-16 - 8*x + x^2))/(E^7*x^2 + 2*x^3)])/(-64* 
x^2 - 32*x^3 + 4*x^4 - 2*x^5 + E^14*(-16 - 8*x + x^2) + E^7*(-64*x - 32*x^ 
2 + 4*x^3 - x^4)),x]
 

Output:

E^4*x + x*Log[(E^7*(-16 - 8*x + x^2) - x*(32 + 16*x - 2*x^2 + x^3))/(x^2*( 
E^7 + 2*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.

\(\displaystyle \int \frac {-2 x^5+32 x^3+128 x^2+e^{14} \left (e^4 \left (x^2-8 x-16\right )+8 x+32\right )+e^7 \left (-2 x^4+32 x^2+e^4 \left (-x^4+4 x^3-32 x^2-64 x\right )+128 x\right )+e^4 \left (-2 x^5+4 x^4-32 x^3-64 x^2\right )+\left (-2 x^5+4 x^4-32 x^3-64 x^2+e^{14} \left (x^2-8 x-16\right )+e^7 \left (-x^4+4 x^3-32 x^2-64 x\right )\right ) \log \left (\frac {-x^4+2 x^3-16 x^2+e^7 \left (x^2-8 x-16\right )-32 x}{2 x^3+e^7 x^2}\right )}{-2 x^5+4 x^4-32 x^3-64 x^2+e^{14} \left (x^2-8 x-16\right )+e^7 \left (-x^4+4 x^3-32 x^2-64 x\right )} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {\left (8 x^3-4 \left (4+e^7\right ) x^2+2 \left (64+e^{14}\right ) x-e^{21}+256\right ) \left (-2 x^5+32 x^3+128 x^2+e^{14} \left (e^4 \left (x^2-8 x-16\right )+8 x+32\right )+e^7 \left (-2 x^4+32 x^2+e^4 \left (-x^4+4 x^3-32 x^2-64 x\right )+128 x\right )+e^4 \left (-2 x^5+4 x^4-32 x^3-64 x^2\right )+\left (-2 x^5+4 x^4-32 x^3-64 x^2+e^{14} \left (x^2-8 x-16\right )+e^7 \left (-x^4+4 x^3-32 x^2-64 x\right )\right ) \log \left (\frac {-x^4+2 x^3-16 x^2+e^7 \left (x^2-8 x-16\right )-32 x}{2 x^3+e^7 x^2}\right )\right )}{e^{28} \left (x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7\right )}-\frac {16 \left (-2 x^5+32 x^3+128 x^2+e^{14} \left (e^4 \left (x^2-8 x-16\right )+8 x+32\right )+e^7 \left (-2 x^4+32 x^2+e^4 \left (-x^4+4 x^3-32 x^2-64 x\right )+128 x\right )+e^4 \left (-2 x^5+4 x^4-32 x^3-64 x^2\right )+\left (-2 x^5+4 x^4-32 x^3-64 x^2+e^{14} \left (x^2-8 x-16\right )+e^7 \left (-x^4+4 x^3-32 x^2-64 x\right )\right ) \log \left (\frac {-x^4+2 x^3-16 x^2+e^7 \left (x^2-8 x-16\right )-32 x}{2 x^3+e^7 x^2}\right )\right )}{e^{28} \left (2 x+e^7\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 e^7 \left (x^3-16 x-64\right ) x-e^{18} \left (x^2-8 x-16\right )+2 \left (x^3-16 x-64\right ) x^2+2 e^4 \left (x^3-2 x^2+16 x+32\right ) x^2+e^{11} \left (x^3-4 x^2+32 x+64\right ) x-\left (e^{14} \left (x^2-8 x-16\right )-2 \left (x^3-2 x^2+16 x+32\right ) x^2-e^7 \left (x^3-4 x^2+32 x+64\right ) x\right ) \log \left (\frac {e^7 \left (x^2-8 x-16\right )-x \left (x^3-2 x^2+16 x+32\right )}{x^2 \left (2 x+e^7\right )}\right )-8 e^{14} (x+4)}{\left (2 x+e^7\right ) \left (x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 \left (x^3-16 x-64\right ) x^2}{\left (2 x+e^7\right ) \left (x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7\right )}+\frac {2 e^4 \left (x^3-2 x^2+16 x+32\right ) x^2}{\left (2 x+e^7\right ) \left (x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7\right )}+\frac {2 e^7 \left (x^3-16 x-64\right ) x}{\left (2 x+e^7\right ) \left (x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7\right )}+\frac {e^{11} \left (x^3-4 x^2+32 x+64\right ) x}{\left (2 x+e^7\right ) \left (x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7\right )}+\frac {e^{18} \left (-x^2+8 x+16\right )}{\left (2 x+e^7\right ) \left (x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7\right )}+\frac {8 e^{14} (-x-4)}{\left (2 x+e^7\right ) \left (x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7\right )}+\log \left (\frac {-x^4+2 x^3-\left (16-e^7\right ) x^2-8 \left (4+e^7\right ) x-16 e^7}{x^2 \left (2 x+e^7\right )}\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \log \left (-\frac {x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}{x^2 \left (2 x+e^7\right )}\right ) x+e^4 x+\frac {\left (512-64 e^7+e^{21}\right ) \log \left (2 x+e^7\right )}{2 e^{14}}+\frac {\left (256-64 e^7-4 e^{14}-e^{21}\right ) \log \left (2 x+e^7\right )}{2 e^{10}}-\frac {\left (512-128 e^7-8 e^{14}-e^{21}\right ) \log \left (2 x+e^7\right )}{2 e^{10}}+\frac {2 \left (64-16 e^7-e^{14}\right ) \log \left (2 x+e^7\right )}{e^{10}}-\frac {32 \left (8-e^7\right ) \log \left (2 x+e^7\right )}{e^{14}}-\frac {1}{2} e^7 \log \left (2 x+e^7\right )+\frac {\left (128-16 e^7+e^{14}\right ) \log \left (x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7\right )}{2 e^{14}}-\frac {8 \left (8-e^7\right ) \log \left (x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7\right )}{e^{14}}-\frac {1}{2} \log \left (x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7\right )-\frac {32 \left (384-80 e^7+4 e^{14}+e^{21}\right ) \int \frac {1}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{14}}-\frac {4 \left (768-256 e^7+4 e^{14}+e^{21}\right ) \int \frac {1}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{10}}+\frac {4 \left (1536-320 e^7+12 e^{14}-e^{21}\right ) \int \frac {1}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{14}}+\frac {8 \left (768-256 e^7+4 e^{14}-e^{21}\right ) \int \frac {1}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{10}}+\frac {32 \left (192-40 e^7+2 e^{14}-e^{21}\right ) \int \frac {1}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{14}}-\frac {4 \left (768-256 e^7+4 e^{14}-3 e^{21}\right ) \int \frac {1}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{10}}+4 \left (4+17 e^7\right ) \int \frac {1}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx-\frac {16 \left (256-16 e^7+6 e^{14}+e^{21}\right ) \int \frac {x}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{14}}+\frac {\left (2048-128 e^7-64 e^{14}+e^{21}\right ) \int \frac {x}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{14}}+\frac {8 \left (256-16 e^7+6 e^{14}-e^{21}\right ) \int \frac {x}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{14}}-\frac {\left (1024-192 e^7-e^{21}\right ) \int \frac {x}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{10}}+\frac {2 \left (1024-192 e^7-5 e^{21}\right ) \int \frac {x}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{10}}-\frac {\left (1024-192 e^7-9 e^{21}\right ) \int \frac {x}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{10}}+\left (112+23 e^7\right ) \int \frac {x}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx+\frac {2 \left (128+112 e^7-16 e^{14}+e^{21}\right ) \int \frac {x^2}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{14}}+\frac {\left (64+48 e^7-17 e^{14}-e^{21}\right ) \int \frac {x^2}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{10}}-\frac {\left (128+96 e^7-34 e^{14}-e^{21}\right ) \int \frac {x^2}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{10}}-\frac {\left (128+112 e^7+13 e^{14}\right ) \int \frac {x^2}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{14}}-\frac {16 \left (8+7 e^7-e^{14}\right ) \int \frac {x^2}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{14}}+\frac {\left (64+48 e^7-17 e^{14}\right ) \int \frac {x^2}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx}{e^{10}}+\left (29-2 e^7\right ) \int \frac {x^2}{x^4-2 x^3+\left (16-e^7\right ) x^2+8 \left (4+e^7\right ) x+16 e^7}dx\)

Input:

Int[(128*x^2 + 32*x^3 - 2*x^5 + E^4*(-64*x^2 - 32*x^3 + 4*x^4 - 2*x^5) + E 
^14*(32 + 8*x + E^4*(-16 - 8*x + x^2)) + E^7*(128*x + 32*x^2 - 2*x^4 + E^4 
*(-64*x - 32*x^2 + 4*x^3 - x^4)) + (-64*x^2 - 32*x^3 + 4*x^4 - 2*x^5 + E^1 
4*(-16 - 8*x + x^2) + E^7*(-64*x - 32*x^2 + 4*x^3 - x^4))*Log[(-32*x - 16* 
x^2 + 2*x^3 - x^4 + E^7*(-16 - 8*x + x^2))/(E^7*x^2 + 2*x^3)])/(-64*x^2 - 
32*x^3 + 4*x^4 - 2*x^5 + E^14*(-16 - 8*x + x^2) + E^7*(-64*x - 32*x^2 + 4* 
x^3 - x^4)),x]
 

Output:

$Aborted
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.08 (sec) , antiderivative size = 377, normalized size of antiderivative = 11.42

\[x \,{\mathrm e}^{4}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{5}+\left ({\mathrm e}^{7}-4\right ) \textit {\_Z}^{4}+\left (-4 \,{\mathrm e}^{7}+32\right ) \textit {\_Z}^{3}+\left (-{\mathrm e}^{14}+32 \,{\mathrm e}^{7}+64\right ) \textit {\_Z}^{2}+\left (8 \,{\mathrm e}^{14}+64 \,{\mathrm e}^{7}\right ) \textit {\_Z} +16 \,{\mathrm e}^{14}\right )}{\sum }\frac {\left (\left ({\mathrm e}^{7}+4\right ) \textit {\_R}^{4}+4 \left ({\mathrm e}^{7}-16\right ) \textit {\_R}^{3}+\left ({\mathrm e}^{14}-64 \,{\mathrm e}^{7}-192\right ) \textit {\_R}^{2}+16 \left (-{\mathrm e}^{14}-12 \,{\mathrm e}^{7}\right ) \textit {\_R} -48 \,{\mathrm e}^{14}\right ) \ln \left (x -\textit {\_R} \right )}{-2 \textit {\_R}^{3} {\mathrm e}^{7}-5 \textit {\_R}^{4}+{\mathrm e}^{14} \textit {\_R} +6 \textit {\_R}^{2} {\mathrm e}^{7}+8 \textit {\_R}^{3}-4 \,{\mathrm e}^{14}-32 \,{\mathrm e}^{7} \textit {\_R} -48 \textit {\_R}^{2}-32 \,{\mathrm e}^{7}-64 \textit {\_R}}\right )}{2}+x \ln \left (\frac {-x^{4}+x^{2} {\mathrm e}^{7}+2 x^{3}-8 \,{\mathrm e}^{7} x -16 x^{2}-16 \,{\mathrm e}^{7}-32 x}{x^{2} \left ({\mathrm e}^{7}+2 x \right )}\right )+{\mathrm e}^{-28} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\left (-{\mathrm e}^{7}+16\right ) \textit {\_Z}^{2}+\left (8 \,{\mathrm e}^{7}+32\right ) \textit {\_Z} +16 \,{\mathrm e}^{7}\right )}{\sum }\frac {\left ({\mathrm e}^{28} \textit {\_R}^{3}+\left (-16 \,{\mathrm e}^{28}+{\mathrm e}^{35}\right ) \textit {\_R}^{2}+12 \left (-4 \,{\mathrm e}^{28}-{\mathrm e}^{35}\right ) \textit {\_R} -32 \,{\mathrm e}^{35}\right ) \ln \left (x -\textit {\_R} \right )}{-16-2 \textit {\_R}^{3}+{\mathrm e}^{7} \textit {\_R} +3 \textit {\_R}^{2}-4 \,{\mathrm e}^{7}-16 \textit {\_R}}\right )-\frac {\left ({\mathrm e}^{35}-4 \,{\mathrm e}^{28}+4 \left ({\mathrm e}^{14}\right )^{2}-128 \,{\mathrm e}^{21}+128 \,{\mathrm e}^{7} {\mathrm e}^{14}\right ) {\mathrm e}^{-28} \ln \left ({\mathrm e}^{7}+2 x \right )}{2}\]

Input:

int((((x^2-8*x-16)*exp(7)^2+(-x^4+4*x^3-32*x^2-64*x)*exp(7)-2*x^5+4*x^4-32 
*x^3-64*x^2)*ln(((x^2-8*x-16)*exp(7)-x^4+2*x^3-16*x^2-32*x)/(x^2*exp(7)+2* 
x^3))+((x^2-8*x-16)*exp(4)+8*x+32)*exp(7)^2+((-x^4+4*x^3-32*x^2-64*x)*exp( 
4)-2*x^4+32*x^2+128*x)*exp(7)+(-2*x^5+4*x^4-32*x^3-64*x^2)*exp(4)-2*x^5+32 
*x^3+128*x^2)/((x^2-8*x-16)*exp(7)^2+(-x^4+4*x^3-32*x^2-64*x)*exp(7)-2*x^5 
+4*x^4-32*x^3-64*x^2),x)
 

Output:

x*exp(4)-1/2*sum(((exp(7)+4)*_R^4+4*(exp(7)-16)*_R^3+(exp(14)-64*exp(7)-19 
2)*_R^2+16*(-exp(14)-12*exp(7))*_R-48*exp(14))/(-2*_R^3*exp(7)-5*_R^4+exp( 
14)*_R+6*_R^2*exp(7)+8*_R^3-4*exp(14)-32*exp(7)*_R-48*_R^2-32*exp(7)-64*_R 
)*ln(x-_R),_R=RootOf(2*_Z^5+(exp(7)-4)*_Z^4+(-4*exp(7)+32)*_Z^3+(-exp(14)+ 
32*exp(7)+64)*_Z^2+(8*exp(14)+64*exp(7))*_Z+16*exp(14)))+x*ln((-x^4+x^2*ex 
p(7)+2*x^3-8*exp(7)*x-16*x^2-16*exp(7)-32*x)/x^2/(exp(7)+2*x))+exp(-28)*su 
m((exp(28)*_R^3+(-16*exp(28)+exp(35))*_R^2+12*(-4*exp(28)-exp(35))*_R-32*e 
xp(35))/(-16-2*_R^3+exp(7)*_R+3*_R^2-4*exp(7)-16*_R)*ln(x-_R),_R=RootOf(_Z 
^4-2*_Z^3+(-exp(7)+16)*_Z^2+(8*exp(7)+32)*_Z+16*exp(7)))-1/2*(exp(7)^5-4*e 
xp(7)^4+4*exp(7)^2*exp(14)-128*exp(7)^3+128*exp(7)*exp(14)+768*exp(7)^2-76 
8*exp(14))/exp(7)^4*ln(exp(7)+2*x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \[ \int \frac {128 x^2+32 x^3-2 x^5+e^4 \left (-64 x^2-32 x^3+4 x^4-2 x^5\right )+e^{14} \left (32+8 x+e^4 \left (-16-8 x+x^2\right )\right )+e^7 \left (128 x+32 x^2-2 x^4+e^4 \left (-64 x-32 x^2+4 x^3-x^4\right )\right )+\left (-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )\right ) \log \left (\frac {-32 x-16 x^2+2 x^3-x^4+e^7 \left (-16-8 x+x^2\right )}{e^7 x^2+2 x^3}\right )}{-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )} \, dx=x e^{4} + x \log \left (-\frac {x^{4} - 2 \, x^{3} + 16 \, x^{2} - {\left (x^{2} - 8 \, x - 16\right )} e^{7} + 32 \, x}{2 \, x^{3} + x^{2} e^{7}}\right ) \] Input:

integrate((((x^2-8*x-16)*exp(7)^2+(-x^4+4*x^3-32*x^2-64*x)*exp(7)-2*x^5+4* 
x^4-32*x^3-64*x^2)*log(((x^2-8*x-16)*exp(7)-x^4+2*x^3-16*x^2-32*x)/(x^2*ex 
p(7)+2*x^3))+((x^2-8*x-16)*exp(4)+8*x+32)*exp(7)^2+((-x^4+4*x^3-32*x^2-64* 
x)*exp(4)-2*x^4+32*x^2+128*x)*exp(7)+(-2*x^5+4*x^4-32*x^3-64*x^2)*exp(4)-2 
*x^5+32*x^3+128*x^2)/((x^2-8*x-16)*exp(7)^2+(-x^4+4*x^3-32*x^2-64*x)*exp(7 
)-2*x^5+4*x^4-32*x^3-64*x^2),x, algorithm="fricas")
 

Output:

x*e^4 + x*log(-(x^4 - 2*x^3 + 16*x^2 - (x^2 - 8*x - 16)*e^7 + 32*x)/(2*x^3 
 + x^2*e^7))
 

Sympy [A] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {128 x^2+32 x^3-2 x^5+e^4 \left (-64 x^2-32 x^3+4 x^4-2 x^5\right )+e^{14} \left (32+8 x+e^4 \left (-16-8 x+x^2\right )\right )+e^7 \left (128 x+32 x^2-2 x^4+e^4 \left (-64 x-32 x^2+4 x^3-x^4\right )\right )+\left (-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )\right ) \log \left (\frac {-32 x-16 x^2+2 x^3-x^4+e^7 \left (-16-8 x+x^2\right )}{e^7 x^2+2 x^3}\right )}{-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )} \, dx=x \log {\left (\frac {- x^{4} + 2 x^{3} - 16 x^{2} - 32 x + \left (x^{2} - 8 x - 16\right ) e^{7}}{2 x^{3} + x^{2} e^{7}} \right )} + x e^{4} \] Input:

integrate((((x**2-8*x-16)*exp(7)**2+(-x**4+4*x**3-32*x**2-64*x)*exp(7)-2*x 
**5+4*x**4-32*x**3-64*x**2)*ln(((x**2-8*x-16)*exp(7)-x**4+2*x**3-16*x**2-3 
2*x)/(x**2*exp(7)+2*x**3))+((x**2-8*x-16)*exp(4)+8*x+32)*exp(7)**2+((-x**4 
+4*x**3-32*x**2-64*x)*exp(4)-2*x**4+32*x**2+128*x)*exp(7)+(-2*x**5+4*x**4- 
32*x**3-64*x**2)*exp(4)-2*x**5+32*x**3+128*x**2)/((x**2-8*x-16)*exp(7)**2+ 
(-x**4+4*x**3-32*x**2-64*x)*exp(7)-2*x**5+4*x**4-32*x**3-64*x**2),x)
 

Output:

x*log((-x**4 + 2*x**3 - 16*x**2 - 32*x + (x**2 - 8*x - 16)*exp(7))/(2*x**3 
 + x**2*exp(7))) + x*exp(4)
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.61 \[ \int \frac {128 x^2+32 x^3-2 x^5+e^4 \left (-64 x^2-32 x^3+4 x^4-2 x^5\right )+e^{14} \left (32+8 x+e^4 \left (-16-8 x+x^2\right )\right )+e^7 \left (128 x+32 x^2-2 x^4+e^4 \left (-64 x-32 x^2+4 x^3-x^4\right )\right )+\left (-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )\right ) \log \left (\frac {-32 x-16 x^2+2 x^3-x^4+e^7 \left (-16-8 x+x^2\right )}{e^7 x^2+2 x^3}\right )}{-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )} \, dx=x e^{4} + x \log \left (-x^{4} + 2 \, x^{3} + x^{2} {\left (e^{7} - 16\right )} - 8 \, x {\left (e^{7} + 4\right )} - 16 \, e^{7}\right ) - x \log \left (2 \, x + e^{7}\right ) - 2 \, x \log \left (x\right ) \] Input:

integrate((((x^2-8*x-16)*exp(7)^2+(-x^4+4*x^3-32*x^2-64*x)*exp(7)-2*x^5+4* 
x^4-32*x^3-64*x^2)*log(((x^2-8*x-16)*exp(7)-x^4+2*x^3-16*x^2-32*x)/(x^2*ex 
p(7)+2*x^3))+((x^2-8*x-16)*exp(4)+8*x+32)*exp(7)^2+((-x^4+4*x^3-32*x^2-64* 
x)*exp(4)-2*x^4+32*x^2+128*x)*exp(7)+(-2*x^5+4*x^4-32*x^3-64*x^2)*exp(4)-2 
*x^5+32*x^3+128*x^2)/((x^2-8*x-16)*exp(7)^2+(-x^4+4*x^3-32*x^2-64*x)*exp(7 
)-2*x^5+4*x^4-32*x^3-64*x^2),x, algorithm="maxima")
 

Output:

x*e^4 + x*log(-x^4 + 2*x^3 + x^2*(e^7 - 16) - 8*x*(e^7 + 4) - 16*e^7) - x* 
log(2*x + e^7) - 2*x*log(x)
 

Giac [A] (verification not implemented)

Time = 1.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.73 \[ \int \frac {128 x^2+32 x^3-2 x^5+e^4 \left (-64 x^2-32 x^3+4 x^4-2 x^5\right )+e^{14} \left (32+8 x+e^4 \left (-16-8 x+x^2\right )\right )+e^7 \left (128 x+32 x^2-2 x^4+e^4 \left (-64 x-32 x^2+4 x^3-x^4\right )\right )+\left (-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )\right ) \log \left (\frac {-32 x-16 x^2+2 x^3-x^4+e^7 \left (-16-8 x+x^2\right )}{e^7 x^2+2 x^3}\right )}{-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )} \, dx=x e^{4} + x \log \left (-\frac {x^{4} - 2 \, x^{3} - x^{2} e^{7} + 16 \, x^{2} + 8 \, x e^{7} + 32 \, x + 16 \, e^{7}}{2 \, x^{3} + x^{2} e^{7}}\right ) \] Input:

integrate((((x^2-8*x-16)*exp(7)^2+(-x^4+4*x^3-32*x^2-64*x)*exp(7)-2*x^5+4* 
x^4-32*x^3-64*x^2)*log(((x^2-8*x-16)*exp(7)-x^4+2*x^3-16*x^2-32*x)/(x^2*ex 
p(7)+2*x^3))+((x^2-8*x-16)*exp(4)+8*x+32)*exp(7)^2+((-x^4+4*x^3-32*x^2-64* 
x)*exp(4)-2*x^4+32*x^2+128*x)*exp(7)+(-2*x^5+4*x^4-32*x^3-64*x^2)*exp(4)-2 
*x^5+32*x^3+128*x^2)/((x^2-8*x-16)*exp(7)^2+(-x^4+4*x^3-32*x^2-64*x)*exp(7 
)-2*x^5+4*x^4-32*x^3-64*x^2),x, algorithm="giac")
 

Output:

x*e^4 + x*log(-(x^4 - 2*x^3 - x^2*e^7 + 16*x^2 + 8*x*e^7 + 32*x + 16*e^7)/ 
(2*x^3 + x^2*e^7))
 

Mupad [B] (verification not implemented)

Time = 3.80 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.58 \[ \int \frac {128 x^2+32 x^3-2 x^5+e^4 \left (-64 x^2-32 x^3+4 x^4-2 x^5\right )+e^{14} \left (32+8 x+e^4 \left (-16-8 x+x^2\right )\right )+e^7 \left (128 x+32 x^2-2 x^4+e^4 \left (-64 x-32 x^2+4 x^3-x^4\right )\right )+\left (-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )\right ) \log \left (\frac {-32 x-16 x^2+2 x^3-x^4+e^7 \left (-16-8 x+x^2\right )}{e^7 x^2+2 x^3}\right )}{-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )} \, dx=x\,\left (\ln \left (-\frac {32\,x+{\mathrm {e}}^7\,\left (-x^2+8\,x+16\right )+16\,x^2-2\,x^3+x^4}{2\,x^3+{\mathrm {e}}^7\,x^2}\right )+{\mathrm {e}}^4\right ) \] Input:

int(-(exp(14)*(8*x - exp(4)*(8*x - x^2 + 16) + 32) - log(-(32*x + exp(7)*( 
8*x - x^2 + 16) + 16*x^2 - 2*x^3 + x^4)/(x^2*exp(7) + 2*x^3))*(exp(14)*(8* 
x - x^2 + 16) + exp(7)*(64*x + 32*x^2 - 4*x^3 + x^4) + 64*x^2 + 32*x^3 - 4 
*x^4 + 2*x^5) + exp(7)*(128*x - exp(4)*(64*x + 32*x^2 - 4*x^3 + x^4) + 32* 
x^2 - 2*x^4) + 128*x^2 + 32*x^3 - 2*x^5 - exp(4)*(64*x^2 + 32*x^3 - 4*x^4 
+ 2*x^5))/(exp(14)*(8*x - x^2 + 16) + exp(7)*(64*x + 32*x^2 - 4*x^3 + x^4) 
 + 64*x^2 + 32*x^3 - 4*x^4 + 2*x^5),x)
 

Output:

x*(log(-(32*x + exp(7)*(8*x - x^2 + 16) + 16*x^2 - 2*x^3 + x^4)/(x^2*exp(7 
) + 2*x^3)) + exp(4))
 

Reduce [F]

\[ \int \frac {128 x^2+32 x^3-2 x^5+e^4 \left (-64 x^2-32 x^3+4 x^4-2 x^5\right )+e^{14} \left (32+8 x+e^4 \left (-16-8 x+x^2\right )\right )+e^7 \left (128 x+32 x^2-2 x^4+e^4 \left (-64 x-32 x^2+4 x^3-x^4\right )\right )+\left (-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )\right ) \log \left (\frac {-32 x-16 x^2+2 x^3-x^4+e^7 \left (-16-8 x+x^2\right )}{e^7 x^2+2 x^3}\right )}{-64 x^2-32 x^3+4 x^4-2 x^5+e^{14} \left (-16-8 x+x^2\right )+e^7 \left (-64 x-32 x^2+4 x^3-x^4\right )} \, dx=\text {too large to display} \] Input:

int((((x^2-8*x-16)*exp(7)^2+(-x^4+4*x^3-32*x^2-64*x)*exp(7)-2*x^5+4*x^4-32 
*x^3-64*x^2)*log(((x^2-8*x-16)*exp(7)-x^4+2*x^3-16*x^2-32*x)/(x^2*exp(7)+2 
*x^3))+((x^2-8*x-16)*exp(4)+8*x+32)*exp(7)^2+((-x^4+4*x^3-32*x^2-64*x)*exp 
(4)-2*x^4+32*x^2+128*x)*exp(7)+(-2*x^5+4*x^4-32*x^3-64*x^2)*exp(4)-2*x^5+3 
2*x^3+128*x^2)/((x^2-8*x-16)*exp(7)^2+(-x^4+4*x^3-32*x^2-64*x)*exp(7)-2*x^ 
5+4*x^4-32*x^3-64*x^2),x)
 

Output:

(36*int(x**2/(3*e**28*x**2 - 24*e**28*x - 48*e**28 - 3*e**21*x**4 + 12*e** 
21*x**3 - 136*e**21*x**2 + 128*e**21*x + 640*e**21 - 6*e**14*x**5 + 52*e** 
14*x**4 - 256*e**14*x**3 + 1200*e**14*x**2 + 1664*e**14*x - 1792*e**14 + 8 
0*e**7*x**5 - 272*e**7*x**4 + 1728*e**7*x**3 - 1024*e**7*x**2 - 7168*e**7* 
x - 224*x**5 + 448*x**4 - 3584*x**3 - 7168*x**2),x)*e**49 - 96*int(x**2/(3 
*e**28*x**2 - 24*e**28*x - 48*e**28 - 3*e**21*x**4 + 12*e**21*x**3 - 136*e 
**21*x**2 + 128*e**21*x + 640*e**21 - 6*e**14*x**5 + 52*e**14*x**4 - 256*e 
**14*x**3 + 1200*e**14*x**2 + 1664*e**14*x - 1792*e**14 + 80*e**7*x**5 - 2 
72*e**7*x**4 + 1728*e**7*x**3 - 1024*e**7*x**2 - 7168*e**7*x - 224*x**5 + 
448*x**4 - 3584*x**3 - 7168*x**2),x)*e**42 + 13696*int(x**2/(3*e**28*x**2 
- 24*e**28*x - 48*e**28 - 3*e**21*x**4 + 12*e**21*x**3 - 136*e**21*x**2 + 
128*e**21*x + 640*e**21 - 6*e**14*x**5 + 52*e**14*x**4 - 256*e**14*x**3 + 
1200*e**14*x**2 + 1664*e**14*x - 1792*e**14 + 80*e**7*x**5 - 272*e**7*x**4 
 + 1728*e**7*x**3 - 1024*e**7*x**2 - 7168*e**7*x - 224*x**5 + 448*x**4 - 3 
584*x**3 - 7168*x**2),x)*e**35 - 610304*int(x**2/(3*e**28*x**2 - 24*e**28* 
x - 48*e**28 - 3*e**21*x**4 + 12*e**21*x**3 - 136*e**21*x**2 + 128*e**21*x 
 + 640*e**21 - 6*e**14*x**5 + 52*e**14*x**4 - 256*e**14*x**3 + 1200*e**14* 
x**2 + 1664*e**14*x - 1792*e**14 + 80*e**7*x**5 - 272*e**7*x**4 + 1728*e** 
7*x**3 - 1024*e**7*x**2 - 7168*e**7*x - 224*x**5 + 448*x**4 - 3584*x**3 - 
7168*x**2),x)*e**28 + 7422976*int(x**2/(3*e**28*x**2 - 24*e**28*x - 48*...