\(\int \frac {-320+480 x^2+320 x^3+60 x^4+e^{\frac {4}{5} (40+3 x)} (-20+48 x)+e^{\frac {3}{5} (40+3 x)} (160-288 x-144 x^2)+e^{\frac {2}{5} (40+3 x)} (-480+576 x+696 x^2+144 x^3)+e^{\frac {1}{5} (40+3 x)} (640-384 x-1056 x^2-448 x^3-48 x^4)+(160 x+10 e^{\frac {4}{5} (40+3 x)} x+320 x^2+240 x^3+80 x^4+10 x^5+e^{\frac {3}{5} (40+3 x)} (-80 x-40 x^2)+e^{\frac {2}{5} (40+3 x)} (240 x+240 x^2+60 x^3)+e^{\frac {1}{5} (40+3 x)} (-320 x-480 x^2-240 x^3-40 x^4)) \log (x)+(160 x^2+12 e^{\frac {4}{5} (40+3 x)} x^2+240 x^3+120 x^4+20 x^5+e^{\frac {3}{5} (40+3 x)} (-92 x^2-36 x^3)+e^{\frac {2}{5} (40+3 x)} (264 x^2+204 x^3+36 x^4)+e^{\frac {1}{5} (40+3 x)} (-336 x^2-384 x^3-132 x^4-12 x^5)) \log ^2(x)}{5 x^2} \, dx\) [1969]

Optimal result

Integrand size = 367, antiderivative size = 27 \[ \int \frac {-320+480 x^2+320 x^3+60 x^4+e^{\frac {4}{5} (40+3 x)} (-20+48 x)+e^{\frac {3}{5} (40+3 x)} \left (160-288 x-144 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (-480+576 x+696 x^2+144 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (640-384 x-1056 x^2-448 x^3-48 x^4\right )+\left (160 x+10 e^{\frac {4}{5} (40+3 x)} x+320 x^2+240 x^3+80 x^4+10 x^5+e^{\frac {3}{5} (40+3 x)} \left (-80 x-40 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (240 x+240 x^2+60 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (-320 x-480 x^2-240 x^3-40 x^4\right )\right ) \log (x)+\left (160 x^2+12 e^{\frac {4}{5} (40+3 x)} x^2+240 x^3+120 x^4+20 x^5+e^{\frac {3}{5} (40+3 x)} \left (-92 x^2-36 x^3\right )+e^{\frac {2}{5} (40+3 x)} \left (264 x^2+204 x^3+36 x^4\right )+e^{\frac {1}{5} (40+3 x)} \left (-336 x^2-384 x^3-132 x^4-12 x^5\right )\right ) \log ^2(x)}{5 x^2} \, dx=\left (2-e^{8+\frac {3 x}{5}}+x\right )^4 \left (\frac {4}{x}+\log ^2(x)\right ) \] Output:

(2-exp(3/5*x+8)+x)^4*(ln(x)^2+4/x)
                                                                                    
                                                                                    
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(104\) vs. \(2(27)=54\).

Time = 0.14 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.85 \[ \int \frac {-320+480 x^2+320 x^3+60 x^4+e^{\frac {4}{5} (40+3 x)} (-20+48 x)+e^{\frac {3}{5} (40+3 x)} \left (160-288 x-144 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (-480+576 x+696 x^2+144 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (640-384 x-1056 x^2-448 x^3-48 x^4\right )+\left (160 x+10 e^{\frac {4}{5} (40+3 x)} x+320 x^2+240 x^3+80 x^4+10 x^5+e^{\frac {3}{5} (40+3 x)} \left (-80 x-40 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (240 x+240 x^2+60 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (-320 x-480 x^2-240 x^3-40 x^4\right )\right ) \log (x)+\left (160 x^2+12 e^{\frac {4}{5} (40+3 x)} x^2+240 x^3+120 x^4+20 x^5+e^{\frac {3}{5} (40+3 x)} \left (-92 x^2-36 x^3\right )+e^{\frac {2}{5} (40+3 x)} \left (264 x^2+204 x^3+36 x^4\right )+e^{\frac {1}{5} (40+3 x)} \left (-336 x^2-384 x^3-132 x^4-12 x^5\right )\right ) \log ^2(x)}{5 x^2} \, dx=\frac {2}{5} \left (\frac {10 \left (16+e^{32+\frac {12 x}{5}}+24 x^2+8 x^3+x^4-4 e^{24+\frac {9 x}{5}} (2+x)+6 e^{16+\frac {6 x}{5}} (2+x)^2-4 e^{8+\frac {3 x}{5}} (2+x)^3\right )}{x}+\frac {5}{2} \left (2-e^{8+\frac {3 x}{5}}+x\right )^4 \log ^2(x)\right ) \] Input:

Integrate[(-320 + 480*x^2 + 320*x^3 + 60*x^4 + E^((4*(40 + 3*x))/5)*(-20 + 
 48*x) + E^((3*(40 + 3*x))/5)*(160 - 288*x - 144*x^2) + E^((2*(40 + 3*x))/ 
5)*(-480 + 576*x + 696*x^2 + 144*x^3) + E^((40 + 3*x)/5)*(640 - 384*x - 10 
56*x^2 - 448*x^3 - 48*x^4) + (160*x + 10*E^((4*(40 + 3*x))/5)*x + 320*x^2 
+ 240*x^3 + 80*x^4 + 10*x^5 + E^((3*(40 + 3*x))/5)*(-80*x - 40*x^2) + E^(( 
2*(40 + 3*x))/5)*(240*x + 240*x^2 + 60*x^3) + E^((40 + 3*x)/5)*(-320*x - 4 
80*x^2 - 240*x^3 - 40*x^4))*Log[x] + (160*x^2 + 12*E^((4*(40 + 3*x))/5)*x^ 
2 + 240*x^3 + 120*x^4 + 20*x^5 + E^((3*(40 + 3*x))/5)*(-92*x^2 - 36*x^3) + 
 E^((2*(40 + 3*x))/5)*(264*x^2 + 204*x^3 + 36*x^4) + E^((40 + 3*x)/5)*(-33 
6*x^2 - 384*x^3 - 132*x^4 - 12*x^5))*Log[x]^2)/(5*x^2),x]
 

Output:

(2*((10*(16 + E^(32 + (12*x)/5) + 24*x^2 + 8*x^3 + x^4 - 4*E^(24 + (9*x)/5 
)*(2 + x) + 6*E^(16 + (6*x)/5)*(2 + x)^2 - 4*E^(8 + (3*x)/5)*(2 + x)^3))/x 
 + (5*(2 - E^(8 + (3*x)/5) + x)^4*Log[x]^2)/2))/5
 

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.

\(\displaystyle \int \frac {60 x^4+320 x^3+480 x^2+e^{\frac {3}{5} (3 x+40)} \left (-144 x^2-288 x+160\right )+e^{\frac {2}{5} (3 x+40)} \left (144 x^3+696 x^2+576 x-480\right )+e^{\frac {1}{5} (3 x+40)} \left (-48 x^4-448 x^3-1056 x^2-384 x+640\right )+\left (20 x^5+120 x^4+240 x^3+12 e^{\frac {4}{5} (3 x+40)} x^2+160 x^2+e^{\frac {3}{5} (3 x+40)} \left (-36 x^3-92 x^2\right )+e^{\frac {2}{5} (3 x+40)} \left (36 x^4+204 x^3+264 x^2\right )+e^{\frac {1}{5} (3 x+40)} \left (-12 x^5-132 x^4-384 x^3-336 x^2\right )\right ) \log ^2(x)+\left (10 x^5+80 x^4+240 x^3+320 x^2+e^{\frac {3}{5} (3 x+40)} \left (-40 x^2-80 x\right )+e^{\frac {2}{5} (3 x+40)} \left (60 x^3+240 x^2+240 x\right )+e^{\frac {1}{5} (3 x+40)} \left (-40 x^4-240 x^3-480 x^2-320 x\right )+10 e^{\frac {4}{5} (3 x+40)} x+160 x\right ) \log (x)+e^{\frac {4}{5} (3 x+40)} (48 x-20)-320}{5 x^2} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \int -\frac {2 \left (-30 x^4-160 x^3-240 x^2-2 \left (5 x^5+30 x^4+60 x^3+3 e^{\frac {4}{5} (3 x+40)} x^2+40 x^2-e^{\frac {3}{5} (3 x+40)} \left (9 x^3+23 x^2\right )+3 e^{\frac {2}{5} (3 x+40)} \left (3 x^4+17 x^3+22 x^2\right )-3 e^{\frac {1}{5} (3 x+40)} \left (x^5+11 x^4+32 x^3+28 x^2\right )\right ) \log ^2(x)+2 e^{\frac {4}{5} (3 x+40)} (5-12 x)-8 e^{\frac {3}{5} (3 x+40)} \left (-9 x^2-18 x+10\right )+12 e^{\frac {2}{5} (3 x+40)} \left (-6 x^3-29 x^2-24 x+20\right )-8 e^{\frac {1}{5} (3 x+40)} \left (-3 x^4-28 x^3-66 x^2-24 x+40\right )-5 \left (x^5+8 x^4+24 x^3+32 x^2+e^{\frac {4}{5} (3 x+40)} x+16 x-4 e^{\frac {3}{5} (3 x+40)} \left (x^2+2 x\right )+6 e^{\frac {2}{5} (3 x+40)} \left (x^3+4 x^2+4 x\right )-4 e^{\frac {1}{5} (3 x+40)} \left (x^4+6 x^3+12 x^2+8 x\right )\right ) \log (x)+160\right )}{x^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{5} \int \frac {-30 x^4-160 x^3-240 x^2-2 \left (5 x^5+30 x^4+60 x^3+3 e^{\frac {4}{5} (3 x+40)} x^2+40 x^2-e^{\frac {3}{5} (3 x+40)} \left (9 x^3+23 x^2\right )+3 e^{\frac {2}{5} (3 x+40)} \left (3 x^4+17 x^3+22 x^2\right )-3 e^{\frac {1}{5} (3 x+40)} \left (x^5+11 x^4+32 x^3+28 x^2\right )\right ) \log ^2(x)+2 e^{\frac {4}{5} (3 x+40)} (5-12 x)-8 e^{\frac {3}{5} (3 x+40)} \left (-9 x^2-18 x+10\right )+12 e^{\frac {2}{5} (3 x+40)} \left (-6 x^3-29 x^2-24 x+20\right )-8 e^{\frac {1}{5} (3 x+40)} \left (-3 x^4-28 x^3-66 x^2-24 x+40\right )-5 \left (x^5+8 x^4+24 x^3+32 x^2+e^{\frac {4}{5} (3 x+40)} x+16 x-4 e^{\frac {3}{5} (3 x+40)} \left (x^2+2 x\right )+6 e^{\frac {2}{5} (3 x+40)} \left (x^3+4 x^2+4 x\right )-4 e^{\frac {1}{5} (3 x+40)} \left (x^4+6 x^3+12 x^2+8 x\right )\right ) \log (x)+160}{x^2}dx\)

\(\Big \downarrow \) 2010

\(\displaystyle -\frac {2}{5} \int \left (-\frac {5 \left (2 \log ^2(x) x^2+\log (x) x^2+2 \log (x) x+6 x-4\right ) (x+2)^3}{x^2}+\frac {2 e^{\frac {3 x}{5}+8} \left (3 \log ^2(x) x^3+21 \log ^2(x) x^2+10 \log (x) x^2+12 x^2+20 \log (x) x+64 x-40\right ) (x+2)^2}{x^2}-\frac {6 e^{\frac {6 x}{5}+16} \left (3 \log ^2(x) x^3+11 \log ^2(x) x^2+5 \log (x) x^2+12 x^2+10 \log (x) x+34 x-20\right ) (x+2)}{x^2}-\frac {e^{\frac {12 x}{5}+32} \left (6 x^2 \log ^2(x)+5 x \log (x)+24 x-10\right )}{x^2}+\frac {2 e^{\frac {9 x}{5}+24} \left (9 \log ^2(x) x^3+23 \log ^2(x) x^2+10 \log (x) x^2+36 x^2+20 \log (x) x+72 x-40\right )}{x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2}{5} \left (840 \text {Subst}\left (\int e^{3 x+8} \log ^2(5 x)dx,x,\frac {x}{5}\right )-660 \text {Subst}\left (\int e^{6 x+16} \log ^2(5 x)dx,x,\frac {x}{5}\right )+230 \text {Subst}\left (\int e^{9 x+24} \log ^2(5 x)dx,x,\frac {x}{5}\right )+6 \int e^{\frac {3 x}{5}+8} x^3 \log ^2(x)dx+66 \int e^{\frac {3 x}{5}+8} x^2 \log ^2(x)dx-18 \int e^{\frac {6 x}{5}+16} x^2 \log ^2(x)dx+192 \int e^{\frac {3 x}{5}+8} x \log ^2(x)dx-102 \int e^{\frac {6 x}{5}+16} x \log ^2(x)dx+18 \int e^{\frac {9 x}{5}+24} x \log ^2(x)dx-96 e^8 x \, _3F_3\left (1,1,1;2,2,2;\frac {3 x}{5}\right )+144 e^{16} x \, _3F_3\left (1,1,1;2,2,2;\frac {6 x}{5}\right )-72 e^{24} x \, _3F_3\left (1,1,1;2,2,2;\frac {9 x}{5}\right )-160 e^8 \log (x) \left (\operatorname {ExpIntegralE}\left (1,-\frac {3 x}{5}\right )+\operatorname {ExpIntegralEi}\left (\frac {3 x}{5}\right )\right )+120 e^{16} \log (x) \left (\operatorname {ExpIntegralE}\left (1,-\frac {6 x}{5}\right )+\operatorname {ExpIntegralEi}\left (\frac {6 x}{5}\right )\right )-40 e^{24} \log (x) \left (\operatorname {ExpIntegralE}\left (1,-\frac {9 x}{5}\right )+\operatorname {ExpIntegralEi}\left (\frac {9 x}{5}\right )\right )-\frac {6800}{27} e^8 \operatorname {ExpIntegralEi}\left (\frac {3 x}{5}\right )+\frac {475}{6} e^{16} \operatorname {ExpIntegralEi}\left (\frac {6 x}{5}\right )-\frac {100}{9} e^{24} \operatorname {ExpIntegralEi}\left (\frac {9 x}{5}\right )+160 e^8 \operatorname {ExpIntegralEi}\left (\frac {3 x}{5}\right ) \log (x)-120 e^{16} \operatorname {ExpIntegralEi}\left (\frac {6 x}{5}\right ) \log (x)+40 e^{24} \operatorname {ExpIntegralEi}\left (\frac {9 x}{5}\right ) \log (x)+40 e^{\frac {3 x}{5}+8} x^2-\frac {5 e^{\frac {12 x}{5}+32} \left (x^2 \log ^2(x)+4 x\right )}{2 x^2}+\frac {100}{3} e^{\frac {3 x}{5}+8} x^2 \log (x)-\frac {10 (x+2)^4}{x}+\frac {3820}{9} e^{\frac {3 x}{5}+8}-\frac {1315}{6} e^{\frac {6 x}{5}+16}+40 e^{\frac {9 x}{5}+24}+\frac {1660}{9} e^{\frac {3 x}{5}+8} x-60 e^{\frac {6 x}{5}+16} x+\frac {320 e^{\frac {3 x}{5}+8}}{x}-\frac {240 e^{\frac {6 x}{5}+16}}{x}+\frac {80 e^{\frac {9 x}{5}+24}}{x}-\frac {5}{2} (x+2)^4 \log ^2(x)-20 e^{24} \log ^2\left (-\frac {9 x}{5}\right )+60 e^{16} \log ^2\left (-\frac {6 x}{5}\right )-80 e^8 \log ^2\left (-\frac {3 x}{5}\right )+\frac {6800}{27} e^{\frac {3 x}{5}+8} \log (x)-\frac {475}{6} e^{\frac {6 x}{5}+16} \log (x)+\frac {100}{9} e^{\frac {9 x}{5}+24} \log (x)+\frac {800}{9} e^{\frac {3 x}{5}+8} x \log (x)-25 e^{\frac {6 x}{5}+16} x \log (x)-40 e^{24} \gamma \log (x)+120 e^{16} \gamma \log (x)-160 e^8 \gamma \log (x)\right )\)

Input:

Int[(-320 + 480*x^2 + 320*x^3 + 60*x^4 + E^((4*(40 + 3*x))/5)*(-20 + 48*x) 
 + E^((3*(40 + 3*x))/5)*(160 - 288*x - 144*x^2) + E^((2*(40 + 3*x))/5)*(-4 
80 + 576*x + 696*x^2 + 144*x^3) + E^((40 + 3*x)/5)*(640 - 384*x - 1056*x^2 
 - 448*x^3 - 48*x^4) + (160*x + 10*E^((4*(40 + 3*x))/5)*x + 320*x^2 + 240* 
x^3 + 80*x^4 + 10*x^5 + E^((3*(40 + 3*x))/5)*(-80*x - 40*x^2) + E^((2*(40 
+ 3*x))/5)*(240*x + 240*x^2 + 60*x^3) + E^((40 + 3*x)/5)*(-320*x - 480*x^2 
 - 240*x^3 - 40*x^4))*Log[x] + (160*x^2 + 12*E^((4*(40 + 3*x))/5)*x^2 + 24 
0*x^3 + 120*x^4 + 20*x^5 + E^((3*(40 + 3*x))/5)*(-92*x^2 - 36*x^3) + E^((2 
*(40 + 3*x))/5)*(264*x^2 + 204*x^3 + 36*x^4) + E^((40 + 3*x)/5)*(-336*x^2 
- 384*x^3 - 132*x^4 - 12*x^5))*Log[x]^2)/(5*x^2),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(229\) vs. \(2(24)=48\).

Time = 0.07 (sec) , antiderivative size = 230, normalized size of antiderivative = 8.52

\[\frac {\left (5 x^{4}-20 \,{\mathrm e}^{\frac {3 x}{5}+8} x^{3}+30 \,{\mathrm e}^{\frac {6 x}{5}+16} x^{2}-20 \,{\mathrm e}^{\frac {9 x}{5}+24} x +5 \,{\mathrm e}^{\frac {12 x}{5}+32}+40 x^{3}-120 \,{\mathrm e}^{\frac {3 x}{5}+8} x^{2}+120 \,{\mathrm e}^{\frac {6 x}{5}+16} x -40 \,{\mathrm e}^{\frac {9 x}{5}+24}+120 x^{2}-240 \,{\mathrm e}^{\frac {3 x}{5}+8} x +120 \,{\mathrm e}^{\frac {6 x}{5}+16}+160 x -160 \,{\mathrm e}^{\frac {3 x}{5}+8}+80\right ) \ln \left (x \right )^{2}}{5}+\frac {4 x^{4}-16 \,{\mathrm e}^{\frac {3 x}{5}+8} x^{3}+24 \,{\mathrm e}^{\frac {6 x}{5}+16} x^{2}-16 \,{\mathrm e}^{\frac {9 x}{5}+24} x +4 \,{\mathrm e}^{\frac {12 x}{5}+32}+32 x^{3}-96 \,{\mathrm e}^{\frac {3 x}{5}+8} x^{2}+96 \,{\mathrm e}^{\frac {6 x}{5}+16} x -32 \,{\mathrm e}^{\frac {9 x}{5}+24}+96 x^{2}-192 \,{\mathrm e}^{\frac {3 x}{5}+8} x +96 \,{\mathrm e}^{\frac {6 x}{5}+16}-128 \,{\mathrm e}^{\frac {3 x}{5}+8}+64}{x}\]

Input:

int(1/5*((12*x^2*exp(3/5*x+8)^4+(-36*x^3-92*x^2)*exp(3/5*x+8)^3+(36*x^4+20 
4*x^3+264*x^2)*exp(3/5*x+8)^2+(-12*x^5-132*x^4-384*x^3-336*x^2)*exp(3/5*x+ 
8)+20*x^5+120*x^4+240*x^3+160*x^2)*ln(x)^2+(10*x*exp(3/5*x+8)^4+(-40*x^2-8 
0*x)*exp(3/5*x+8)^3+(60*x^3+240*x^2+240*x)*exp(3/5*x+8)^2+(-40*x^4-240*x^3 
-480*x^2-320*x)*exp(3/5*x+8)+10*x^5+80*x^4+240*x^3+320*x^2+160*x)*ln(x)+(4 
8*x-20)*exp(3/5*x+8)^4+(-144*x^2-288*x+160)*exp(3/5*x+8)^3+(144*x^3+696*x^ 
2+576*x-480)*exp(3/5*x+8)^2+(-48*x^4-448*x^3-1056*x^2-384*x+640)*exp(3/5*x 
+8)+60*x^4+320*x^3+480*x^2-320)/x^2,x)
 

Output:

1/5*(5*x^4-20*exp(3/5*x+8)*x^3+30*exp(6/5*x+16)*x^2-20*exp(9/5*x+24)*x+5*e 
xp(12/5*x+32)+40*x^3-120*exp(3/5*x+8)*x^2+120*exp(6/5*x+16)*x-40*exp(9/5*x 
+24)+120*x^2-240*exp(3/5*x+8)*x+120*exp(6/5*x+16)+160*x-160*exp(3/5*x+8)+8 
0)*ln(x)^2+4*(x^4-4*exp(3/5*x+8)*x^3+6*exp(6/5*x+16)*x^2-4*exp(9/5*x+24)*x 
+exp(12/5*x+32)+8*x^3-24*exp(3/5*x+8)*x^2+24*exp(6/5*x+16)*x-8*exp(9/5*x+2 
4)+24*x^2-48*exp(3/5*x+8)*x+24*exp(6/5*x+16)-32*exp(3/5*x+8)+16)/x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (24) = 48\).

Time = 0.10 (sec) , antiderivative size = 172, normalized size of antiderivative = 6.37 \[ \int \frac {-320+480 x^2+320 x^3+60 x^4+e^{\frac {4}{5} (40+3 x)} (-20+48 x)+e^{\frac {3}{5} (40+3 x)} \left (160-288 x-144 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (-480+576 x+696 x^2+144 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (640-384 x-1056 x^2-448 x^3-48 x^4\right )+\left (160 x+10 e^{\frac {4}{5} (40+3 x)} x+320 x^2+240 x^3+80 x^4+10 x^5+e^{\frac {3}{5} (40+3 x)} \left (-80 x-40 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (240 x+240 x^2+60 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (-320 x-480 x^2-240 x^3-40 x^4\right )\right ) \log (x)+\left (160 x^2+12 e^{\frac {4}{5} (40+3 x)} x^2+240 x^3+120 x^4+20 x^5+e^{\frac {3}{5} (40+3 x)} \left (-92 x^2-36 x^3\right )+e^{\frac {2}{5} (40+3 x)} \left (264 x^2+204 x^3+36 x^4\right )+e^{\frac {1}{5} (40+3 x)} \left (-336 x^2-384 x^3-132 x^4-12 x^5\right )\right ) \log ^2(x)}{5 x^2} \, dx=\frac {4 \, x^{4} + 32 \, x^{3} + {\left (x^{5} + 8 \, x^{4} + 24 \, x^{3} + 32 \, x^{2} + x e^{\left (\frac {12}{5} \, x + 32\right )} - 4 \, {\left (x^{2} + 2 \, x\right )} e^{\left (\frac {9}{5} \, x + 24\right )} + 6 \, {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} e^{\left (\frac {6}{5} \, x + 16\right )} - 4 \, {\left (x^{4} + 6 \, x^{3} + 12 \, x^{2} + 8 \, x\right )} e^{\left (\frac {3}{5} \, x + 8\right )} + 16 \, x\right )} \log \left (x\right )^{2} + 96 \, x^{2} - 16 \, {\left (x + 2\right )} e^{\left (\frac {9}{5} \, x + 24\right )} + 24 \, {\left (x^{2} + 4 \, x + 4\right )} e^{\left (\frac {6}{5} \, x + 16\right )} - 16 \, {\left (x^{3} + 6 \, x^{2} + 12 \, x + 8\right )} e^{\left (\frac {3}{5} \, x + 8\right )} + 4 \, e^{\left (\frac {12}{5} \, x + 32\right )} + 64}{x} \] Input:

integrate(1/5*((12*x^2*exp(3/5*x+8)^4+(-36*x^3-92*x^2)*exp(3/5*x+8)^3+(36* 
x^4+204*x^3+264*x^2)*exp(3/5*x+8)^2+(-12*x^5-132*x^4-384*x^3-336*x^2)*exp( 
3/5*x+8)+20*x^5+120*x^4+240*x^3+160*x^2)*log(x)^2+(10*x*exp(3/5*x+8)^4+(-4 
0*x^2-80*x)*exp(3/5*x+8)^3+(60*x^3+240*x^2+240*x)*exp(3/5*x+8)^2+(-40*x^4- 
240*x^3-480*x^2-320*x)*exp(3/5*x+8)+10*x^5+80*x^4+240*x^3+320*x^2+160*x)*l 
og(x)+(48*x-20)*exp(3/5*x+8)^4+(-144*x^2-288*x+160)*exp(3/5*x+8)^3+(144*x^ 
3+696*x^2+576*x-480)*exp(3/5*x+8)^2+(-48*x^4-448*x^3-1056*x^2-384*x+640)*e 
xp(3/5*x+8)+60*x^4+320*x^3+480*x^2-320)/x^2,x, algorithm="fricas")
 

Output:

(4*x^4 + 32*x^3 + (x^5 + 8*x^4 + 24*x^3 + 32*x^2 + x*e^(12/5*x + 32) - 4*( 
x^2 + 2*x)*e^(9/5*x + 24) + 6*(x^3 + 4*x^2 + 4*x)*e^(6/5*x + 16) - 4*(x^4 
+ 6*x^3 + 12*x^2 + 8*x)*e^(3/5*x + 8) + 16*x)*log(x)^2 + 96*x^2 - 16*(x + 
2)*e^(9/5*x + 24) + 24*(x^2 + 4*x + 4)*e^(6/5*x + 16) - 16*(x^3 + 6*x^2 + 
12*x + 8)*e^(3/5*x + 8) + 4*e^(12/5*x + 32) + 64)/x
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (20) = 40\).

Time = 0.37 (sec) , antiderivative size = 231, normalized size of antiderivative = 8.56 \[ \int \frac {-320+480 x^2+320 x^3+60 x^4+e^{\frac {4}{5} (40+3 x)} (-20+48 x)+e^{\frac {3}{5} (40+3 x)} \left (160-288 x-144 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (-480+576 x+696 x^2+144 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (640-384 x-1056 x^2-448 x^3-48 x^4\right )+\left (160 x+10 e^{\frac {4}{5} (40+3 x)} x+320 x^2+240 x^3+80 x^4+10 x^5+e^{\frac {3}{5} (40+3 x)} \left (-80 x-40 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (240 x+240 x^2+60 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (-320 x-480 x^2-240 x^3-40 x^4\right )\right ) \log (x)+\left (160 x^2+12 e^{\frac {4}{5} (40+3 x)} x^2+240 x^3+120 x^4+20 x^5+e^{\frac {3}{5} (40+3 x)} \left (-92 x^2-36 x^3\right )+e^{\frac {2}{5} (40+3 x)} \left (264 x^2+204 x^3+36 x^4\right )+e^{\frac {1}{5} (40+3 x)} \left (-336 x^2-384 x^3-132 x^4-12 x^5\right )\right ) \log ^2(x)}{5 x^2} \, dx=4 x^{3} + 32 x^{2} + 96 x + \left (x^{4} + 8 x^{3} + 24 x^{2} + 32 x + 16\right ) \log {\left (x \right )}^{2} + \frac {64}{x} + \frac {\left (x^{4} \log {\left (x \right )}^{2} + 4 x^{3}\right ) e^{\frac {12 x}{5} + 32} + \left (- 4 x^{5} \log {\left (x \right )}^{2} - 8 x^{4} \log {\left (x \right )}^{2} - 16 x^{4} - 32 x^{3}\right ) e^{\frac {9 x}{5} + 24} + \left (6 x^{6} \log {\left (x \right )}^{2} + 24 x^{5} \log {\left (x \right )}^{2} + 24 x^{5} + 24 x^{4} \log {\left (x \right )}^{2} + 96 x^{4} + 96 x^{3}\right ) e^{\frac {6 x}{5} + 16} + \left (- 4 x^{7} \log {\left (x \right )}^{2} - 24 x^{6} \log {\left (x \right )}^{2} - 16 x^{6} - 48 x^{5} \log {\left (x \right )}^{2} - 96 x^{5} - 32 x^{4} \log {\left (x \right )}^{2} - 192 x^{4} - 128 x^{3}\right ) e^{\frac {3 x}{5} + 8}}{x^{4}} \] Input:

integrate(1/5*((12*x**2*exp(3/5*x+8)**4+(-36*x**3-92*x**2)*exp(3/5*x+8)**3 
+(36*x**4+204*x**3+264*x**2)*exp(3/5*x+8)**2+(-12*x**5-132*x**4-384*x**3-3 
36*x**2)*exp(3/5*x+8)+20*x**5+120*x**4+240*x**3+160*x**2)*ln(x)**2+(10*x*e 
xp(3/5*x+8)**4+(-40*x**2-80*x)*exp(3/5*x+8)**3+(60*x**3+240*x**2+240*x)*ex 
p(3/5*x+8)**2+(-40*x**4-240*x**3-480*x**2-320*x)*exp(3/5*x+8)+10*x**5+80*x 
**4+240*x**3+320*x**2+160*x)*ln(x)+(48*x-20)*exp(3/5*x+8)**4+(-144*x**2-28 
8*x+160)*exp(3/5*x+8)**3+(144*x**3+696*x**2+576*x-480)*exp(3/5*x+8)**2+(-4 
8*x**4-448*x**3-1056*x**2-384*x+640)*exp(3/5*x+8)+60*x**4+320*x**3+480*x** 
2-320)/x**2,x)
 

Output:

4*x**3 + 32*x**2 + 96*x + (x**4 + 8*x**3 + 24*x**2 + 32*x + 16)*log(x)**2 
+ 64/x + ((x**4*log(x)**2 + 4*x**3)*exp(12*x/5 + 32) + (-4*x**5*log(x)**2 
- 8*x**4*log(x)**2 - 16*x**4 - 32*x**3)*exp(9*x/5 + 24) + (6*x**6*log(x)** 
2 + 24*x**5*log(x)**2 + 24*x**5 + 24*x**4*log(x)**2 + 96*x**4 + 96*x**3)*e 
xp(6*x/5 + 16) + (-4*x**7*log(x)**2 - 24*x**6*log(x)**2 - 16*x**6 - 48*x** 
5*log(x)**2 - 96*x**5 - 32*x**4*log(x)**2 - 192*x**4 - 128*x**3)*exp(3*x/5 
 + 8))/x**4
 

Maxima [F]

\[ \int \frac {-320+480 x^2+320 x^3+60 x^4+e^{\frac {4}{5} (40+3 x)} (-20+48 x)+e^{\frac {3}{5} (40+3 x)} \left (160-288 x-144 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (-480+576 x+696 x^2+144 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (640-384 x-1056 x^2-448 x^3-48 x^4\right )+\left (160 x+10 e^{\frac {4}{5} (40+3 x)} x+320 x^2+240 x^3+80 x^4+10 x^5+e^{\frac {3}{5} (40+3 x)} \left (-80 x-40 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (240 x+240 x^2+60 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (-320 x-480 x^2-240 x^3-40 x^4\right )\right ) \log (x)+\left (160 x^2+12 e^{\frac {4}{5} (40+3 x)} x^2+240 x^3+120 x^4+20 x^5+e^{\frac {3}{5} (40+3 x)} \left (-92 x^2-36 x^3\right )+e^{\frac {2}{5} (40+3 x)} \left (264 x^2+204 x^3+36 x^4\right )+e^{\frac {1}{5} (40+3 x)} \left (-336 x^2-384 x^3-132 x^4-12 x^5\right )\right ) \log ^2(x)}{5 x^2} \, dx=\int { \frac {2 \, {\left (30 \, x^{4} + 160 \, x^{3} + 2 \, {\left (5 \, x^{5} + 30 \, x^{4} + 60 \, x^{3} + 3 \, x^{2} e^{\left (\frac {12}{5} \, x + 32\right )} + 40 \, x^{2} - {\left (9 \, x^{3} + 23 \, x^{2}\right )} e^{\left (\frac {9}{5} \, x + 24\right )} + 3 \, {\left (3 \, x^{4} + 17 \, x^{3} + 22 \, x^{2}\right )} e^{\left (\frac {6}{5} \, x + 16\right )} - 3 \, {\left (x^{5} + 11 \, x^{4} + 32 \, x^{3} + 28 \, x^{2}\right )} e^{\left (\frac {3}{5} \, x + 8\right )}\right )} \log \left (x\right )^{2} + 240 \, x^{2} + 2 \, {\left (12 \, x - 5\right )} e^{\left (\frac {12}{5} \, x + 32\right )} - 8 \, {\left (9 \, x^{2} + 18 \, x - 10\right )} e^{\left (\frac {9}{5} \, x + 24\right )} + 12 \, {\left (6 \, x^{3} + 29 \, x^{2} + 24 \, x - 20\right )} e^{\left (\frac {6}{5} \, x + 16\right )} - 8 \, {\left (3 \, x^{4} + 28 \, x^{3} + 66 \, x^{2} + 24 \, x - 40\right )} e^{\left (\frac {3}{5} \, x + 8\right )} + 5 \, {\left (x^{5} + 8 \, x^{4} + 24 \, x^{3} + 32 \, x^{2} + x e^{\left (\frac {12}{5} \, x + 32\right )} - 4 \, {\left (x^{2} + 2 \, x\right )} e^{\left (\frac {9}{5} \, x + 24\right )} + 6 \, {\left (x^{3} + 4 \, x^{2} + 4 \, x\right )} e^{\left (\frac {6}{5} \, x + 16\right )} - 4 \, {\left (x^{4} + 6 \, x^{3} + 12 \, x^{2} + 8 \, x\right )} e^{\left (\frac {3}{5} \, x + 8\right )} + 16 \, x\right )} \log \left (x\right ) - 160\right )}}{5 \, x^{2}} \,d x } \] Input:

integrate(1/5*((12*x^2*exp(3/5*x+8)^4+(-36*x^3-92*x^2)*exp(3/5*x+8)^3+(36* 
x^4+204*x^3+264*x^2)*exp(3/5*x+8)^2+(-12*x^5-132*x^4-384*x^3-336*x^2)*exp( 
3/5*x+8)+20*x^5+120*x^4+240*x^3+160*x^2)*log(x)^2+(10*x*exp(3/5*x+8)^4+(-4 
0*x^2-80*x)*exp(3/5*x+8)^3+(60*x^3+240*x^2+240*x)*exp(3/5*x+8)^2+(-40*x^4- 
240*x^3-480*x^2-320*x)*exp(3/5*x+8)+10*x^5+80*x^4+240*x^3+320*x^2+160*x)*l 
og(x)+(48*x-20)*exp(3/5*x+8)^4+(-144*x^2-288*x+160)*exp(3/5*x+8)^3+(144*x^ 
3+696*x^2+576*x-480)*exp(3/5*x+8)^2+(-48*x^4-448*x^3-1056*x^2-384*x+640)*e 
xp(3/5*x+8)+60*x^4+320*x^3+480*x^2-320)/x^2,x, algorithm="maxima")
 

Output:

1/2*x^4*log(x) + 16/3*x^3*log(x) - 4*(x*e^24 + 2*e^24)*e^(9/5*x)*log(x)^2 
+ 6*(x^2*e^16 + 4*x*e^16 + 4*e^16)*e^(6/5*x)*log(x)^2 + 4*x^3 + 24*x^2*log 
(x) + (x^4 + 8*x^3 + 24*x^2 + 32*x)*log(x)^2 + e^(12/5*x + 32)*log(x)^2 + 
32*x^2 + 48/5*Ei(12/5*x)*e^32 - 288/5*Ei(9/5*x)*e^24 + 576/5*Ei(6/5*x)*e^1 
6 + 416/5*Ei(3/5*x)*e^8 + 4*(6*x*e^16 - 5*e^16)*e^(6/5*x) - 16/9*(9*x^2*e^ 
8 - 30*x*e^8 + 50*e^8)*e^(3/5*x) - 4*((x^3*e^8 + 6*x^2*e^8 + 12*x*e^8 + 8* 
e^8)*log(x)^2 - 40*e^8*log(x))*e^(3/5*x) - 448/9*(3*x*e^8 - 5*e^8)*e^(3/5* 
x) + 384/5*e^8*gamma(-1, -3/5*x) - 576/5*e^16*gamma(-1, -6/5*x) + 288/5*e^ 
24*gamma(-1, -9/5*x) - 48/5*e^32*gamma(-1, -12/5*x) - 1/6*(3*x^4 + 32*x^3 
+ 144*x^2 + 384*x)*log(x) + 64*x*log(x) - 160*e^(3/5*x + 8)*log(x) + 16*lo 
g(x)^2 + 96*x + 64/x - 16*e^(9/5*x + 24) + 116*e^(6/5*x + 16) - 352*e^(3/5 
*x + 8) - 160*integrate(e^(3/5*x + 8)/x, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 717 vs. \(2 (24) = 48\).

Time = 0.15 (sec) , antiderivative size = 717, normalized size of antiderivative = 26.56 \[ \int \frac {-320+480 x^2+320 x^3+60 x^4+e^{\frac {4}{5} (40+3 x)} (-20+48 x)+e^{\frac {3}{5} (40+3 x)} \left (160-288 x-144 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (-480+576 x+696 x^2+144 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (640-384 x-1056 x^2-448 x^3-48 x^4\right )+\left (160 x+10 e^{\frac {4}{5} (40+3 x)} x+320 x^2+240 x^3+80 x^4+10 x^5+e^{\frac {3}{5} (40+3 x)} \left (-80 x-40 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (240 x+240 x^2+60 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (-320 x-480 x^2-240 x^3-40 x^4\right )\right ) \log (x)+\left (160 x^2+12 e^{\frac {4}{5} (40+3 x)} x^2+240 x^3+120 x^4+20 x^5+e^{\frac {3}{5} (40+3 x)} \left (-92 x^2-36 x^3\right )+e^{\frac {2}{5} (40+3 x)} \left (264 x^2+204 x^3+36 x^4\right )+e^{\frac {1}{5} (40+3 x)} \left (-336 x^2-384 x^3-132 x^4-12 x^5\right )\right ) \log ^2(x)}{5 x^2} \, dx=\text {Too large to display} \] Input:

integrate(1/5*((12*x^2*exp(3/5*x+8)^4+(-36*x^3-92*x^2)*exp(3/5*x+8)^3+(36* 
x^4+204*x^3+264*x^2)*exp(3/5*x+8)^2+(-12*x^5-132*x^4-384*x^3-336*x^2)*exp( 
3/5*x+8)+20*x^5+120*x^4+240*x^3+160*x^2)*log(x)^2+(10*x*exp(3/5*x+8)^4+(-4 
0*x^2-80*x)*exp(3/5*x+8)^3+(60*x^3+240*x^2+240*x)*exp(3/5*x+8)^2+(-40*x^4- 
240*x^3-480*x^2-320*x)*exp(3/5*x+8)+10*x^5+80*x^4+240*x^3+320*x^2+160*x)*l 
og(x)+(48*x-20)*exp(3/5*x+8)^4+(-144*x^2-288*x+160)*exp(3/5*x+8)^3+(144*x^ 
3+696*x^2+576*x-480)*exp(3/5*x+8)^2+(-48*x^4-448*x^3-1056*x^2-384*x+640)*e 
xp(3/5*x+8)+60*x^4+320*x^3+480*x^2-320)/x^2,x, algorithm="giac")
 

Output:

(x^5*log(5)^2 - 4*x^4*e^(3/5*x + 8)*log(5)^2 + 2*x^5*log(5)*log(1/5*x) - 8 
*x^4*e^(3/5*x + 8)*log(5)*log(1/5*x) + x^5*log(1/5*x)^2 - 4*x^4*e^(3/5*x + 
 8)*log(1/5*x)^2 + 8*x^4*log(5)^2 + 6*x^3*e^(6/5*x + 16)*log(5)^2 - 24*x^3 
*e^(3/5*x + 8)*log(5)^2 + 16*x^4*log(5)*log(1/5*x) + 12*x^3*e^(6/5*x + 16) 
*log(5)*log(1/5*x) - 48*x^3*e^(3/5*x + 8)*log(5)*log(1/5*x) + 8*x^4*log(1/ 
5*x)^2 + 6*x^3*e^(6/5*x + 16)*log(1/5*x)^2 - 24*x^3*e^(3/5*x + 8)*log(1/5* 
x)^2 + 24*x^3*log(5)^2 - 4*x^2*e^(9/5*x + 24)*log(5)^2 + 24*x^2*e^(6/5*x + 
 16)*log(5)^2 - 48*x^2*e^(3/5*x + 8)*log(5)^2 + 48*x^3*log(5)*log(1/5*x) - 
 8*x^2*e^(9/5*x + 24)*log(5)*log(1/5*x) + 48*x^2*e^(6/5*x + 16)*log(5)*log 
(1/5*x) - 96*x^2*e^(3/5*x + 8)*log(5)*log(1/5*x) + 24*x^3*log(1/5*x)^2 - 4 
*x^2*e^(9/5*x + 24)*log(1/5*x)^2 + 24*x^2*e^(6/5*x + 16)*log(1/5*x)^2 - 48 
*x^2*e^(3/5*x + 8)*log(1/5*x)^2 + 4*x^4 - 16*x^3*e^(3/5*x + 8) + 32*x^2*lo 
g(5)^2 + x*e^(12/5*x + 32)*log(5)^2 - 8*x*e^(9/5*x + 24)*log(5)^2 + 24*x*e 
^(6/5*x + 16)*log(5)^2 - 32*x*e^(3/5*x + 8)*log(5)^2 + 64*x^2*log(5)*log(1 
/5*x) + 2*x*e^(12/5*x + 32)*log(5)*log(1/5*x) - 16*x*e^(9/5*x + 24)*log(5) 
*log(1/5*x) + 48*x*e^(6/5*x + 16)*log(5)*log(1/5*x) - 64*x*e^(3/5*x + 8)*l 
og(5)*log(1/5*x) + 32*x^2*log(1/5*x)^2 + x*e^(12/5*x + 32)*log(1/5*x)^2 - 
8*x*e^(9/5*x + 24)*log(1/5*x)^2 + 24*x*e^(6/5*x + 16)*log(1/5*x)^2 - 32*x* 
e^(3/5*x + 8)*log(1/5*x)^2 + 32*x^3 + 24*x^2*e^(6/5*x + 16) - 96*x^2*e^(3/ 
5*x + 8) + 32*x*log(5)*log(1/5*x) + 16*x*log(1/5*x)^2 + 96*x^2 - 16*x*e...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-320+480 x^2+320 x^3+60 x^4+e^{\frac {4}{5} (40+3 x)} (-20+48 x)+e^{\frac {3}{5} (40+3 x)} \left (160-288 x-144 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (-480+576 x+696 x^2+144 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (640-384 x-1056 x^2-448 x^3-48 x^4\right )+\left (160 x+10 e^{\frac {4}{5} (40+3 x)} x+320 x^2+240 x^3+80 x^4+10 x^5+e^{\frac {3}{5} (40+3 x)} \left (-80 x-40 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (240 x+240 x^2+60 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (-320 x-480 x^2-240 x^3-40 x^4\right )\right ) \log (x)+\left (160 x^2+12 e^{\frac {4}{5} (40+3 x)} x^2+240 x^3+120 x^4+20 x^5+e^{\frac {3}{5} (40+3 x)} \left (-92 x^2-36 x^3\right )+e^{\frac {2}{5} (40+3 x)} \left (264 x^2+204 x^3+36 x^4\right )+e^{\frac {1}{5} (40+3 x)} \left (-336 x^2-384 x^3-132 x^4-12 x^5\right )\right ) \log ^2(x)}{5 x^2} \, dx=\int \frac {\frac {{\mathrm {e}}^{\frac {12\,x}{5}+32}\,\left (48\,x-20\right )}{5}-\frac {{\mathrm {e}}^{\frac {9\,x}{5}+24}\,\left (144\,x^2+288\,x-160\right )}{5}+\frac {{\mathrm {e}}^{\frac {6\,x}{5}+16}\,\left (144\,x^3+696\,x^2+576\,x-480\right )}{5}+\frac {\ln \left (x\right )\,\left (160\,x-{\mathrm {e}}^{\frac {9\,x}{5}+24}\,\left (40\,x^2+80\,x\right )+10\,x\,{\mathrm {e}}^{\frac {12\,x}{5}+32}+{\mathrm {e}}^{\frac {6\,x}{5}+16}\,\left (60\,x^3+240\,x^2+240\,x\right )-{\mathrm {e}}^{\frac {3\,x}{5}+8}\,\left (40\,x^4+240\,x^3+480\,x^2+320\,x\right )+320\,x^2+240\,x^3+80\,x^4+10\,x^5\right )}{5}+\frac {{\ln \left (x\right )}^2\,\left (12\,x^2\,{\mathrm {e}}^{\frac {12\,x}{5}+32}-{\mathrm {e}}^{\frac {9\,x}{5}+24}\,\left (36\,x^3+92\,x^2\right )-{\mathrm {e}}^{\frac {3\,x}{5}+8}\,\left (12\,x^5+132\,x^4+384\,x^3+336\,x^2\right )+160\,x^2+240\,x^3+120\,x^4+20\,x^5+{\mathrm {e}}^{\frac {6\,x}{5}+16}\,\left (36\,x^4+204\,x^3+264\,x^2\right )\right )}{5}-\frac {{\mathrm {e}}^{\frac {3\,x}{5}+8}\,\left (48\,x^4+448\,x^3+1056\,x^2+384\,x-640\right )}{5}+96\,x^2+64\,x^3+12\,x^4-64}{x^2} \,d x \] Input:

int(((exp((12*x)/5 + 32)*(48*x - 20))/5 - (exp((9*x)/5 + 24)*(288*x + 144* 
x^2 - 160))/5 + (exp((6*x)/5 + 16)*(576*x + 696*x^2 + 144*x^3 - 480))/5 + 
(log(x)*(160*x - exp((9*x)/5 + 24)*(80*x + 40*x^2) + 10*x*exp((12*x)/5 + 3 
2) + exp((6*x)/5 + 16)*(240*x + 240*x^2 + 60*x^3) - exp((3*x)/5 + 8)*(320* 
x + 480*x^2 + 240*x^3 + 40*x^4) + 320*x^2 + 240*x^3 + 80*x^4 + 10*x^5))/5 
+ (log(x)^2*(12*x^2*exp((12*x)/5 + 32) - exp((9*x)/5 + 24)*(92*x^2 + 36*x^ 
3) - exp((3*x)/5 + 8)*(336*x^2 + 384*x^3 + 132*x^4 + 12*x^5) + 160*x^2 + 2 
40*x^3 + 120*x^4 + 20*x^5 + exp((6*x)/5 + 16)*(264*x^2 + 204*x^3 + 36*x^4) 
))/5 - (exp((3*x)/5 + 8)*(384*x + 1056*x^2 + 448*x^3 + 48*x^4 - 640))/5 + 
96*x^2 + 64*x^3 + 12*x^4 - 64)/x^2,x)
 

Output:

int(((exp((12*x)/5 + 32)*(48*x - 20))/5 - (exp((9*x)/5 + 24)*(288*x + 144* 
x^2 - 160))/5 + (exp((6*x)/5 + 16)*(576*x + 696*x^2 + 144*x^3 - 480))/5 + 
(log(x)*(160*x - exp((9*x)/5 + 24)*(80*x + 40*x^2) + 10*x*exp((12*x)/5 + 3 
2) + exp((6*x)/5 + 16)*(240*x + 240*x^2 + 60*x^3) - exp((3*x)/5 + 8)*(320* 
x + 480*x^2 + 240*x^3 + 40*x^4) + 320*x^2 + 240*x^3 + 80*x^4 + 10*x^5))/5 
+ (log(x)^2*(12*x^2*exp((12*x)/5 + 32) - exp((9*x)/5 + 24)*(92*x^2 + 36*x^ 
3) - exp((3*x)/5 + 8)*(336*x^2 + 384*x^3 + 132*x^4 + 12*x^5) + 160*x^2 + 2 
40*x^3 + 120*x^4 + 20*x^5 + exp((6*x)/5 + 16)*(264*x^2 + 204*x^3 + 36*x^4) 
))/5 - (exp((3*x)/5 + 8)*(384*x + 1056*x^2 + 448*x^3 + 48*x^4 - 640))/5 + 
96*x^2 + 64*x^3 + 12*x^4 - 64)/x^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 336, normalized size of antiderivative = 12.44 \[ \int \frac {-320+480 x^2+320 x^3+60 x^4+e^{\frac {4}{5} (40+3 x)} (-20+48 x)+e^{\frac {3}{5} (40+3 x)} \left (160-288 x-144 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (-480+576 x+696 x^2+144 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (640-384 x-1056 x^2-448 x^3-48 x^4\right )+\left (160 x+10 e^{\frac {4}{5} (40+3 x)} x+320 x^2+240 x^3+80 x^4+10 x^5+e^{\frac {3}{5} (40+3 x)} \left (-80 x-40 x^2\right )+e^{\frac {2}{5} (40+3 x)} \left (240 x+240 x^2+60 x^3\right )+e^{\frac {1}{5} (40+3 x)} \left (-320 x-480 x^2-240 x^3-40 x^4\right )\right ) \log (x)+\left (160 x^2+12 e^{\frac {4}{5} (40+3 x)} x^2+240 x^3+120 x^4+20 x^5+e^{\frac {3}{5} (40+3 x)} \left (-92 x^2-36 x^3\right )+e^{\frac {2}{5} (40+3 x)} \left (264 x^2+204 x^3+36 x^4\right )+e^{\frac {1}{5} (40+3 x)} \left (-336 x^2-384 x^3-132 x^4-12 x^5\right )\right ) \log ^2(x)}{5 x^2} \, dx=\frac {64+8 \mathrm {log}\left (x \right )^{2} x^{4}+24 \mathrm {log}\left (x \right )^{2} x^{3}+\mathrm {log}\left (x \right )^{2} x^{5}+4 e^{\frac {12 x}{5}} e^{32}-32 e^{\frac {9 x}{5}} e^{24}+96 e^{\frac {6 x}{5}} e^{16}-128 e^{\frac {3 x}{5}} e^{8}+32 \mathrm {log}\left (x \right )^{2} x^{2}-4 e^{\frac {9 x}{5}} \mathrm {log}\left (x \right )^{2} e^{24} x^{2}-8 e^{\frac {9 x}{5}} \mathrm {log}\left (x \right )^{2} e^{24} x +6 e^{\frac {6 x}{5}} \mathrm {log}\left (x \right )^{2} e^{16} x^{3}+24 e^{\frac {6 x}{5}} \mathrm {log}\left (x \right )^{2} e^{16} x^{2}+24 e^{\frac {6 x}{5}} \mathrm {log}\left (x \right )^{2} e^{16} x -4 e^{\frac {3 x}{5}} \mathrm {log}\left (x \right )^{2} e^{8} x^{4}-24 e^{\frac {3 x}{5}} \mathrm {log}\left (x \right )^{2} e^{8} x^{3}-48 e^{\frac {3 x}{5}} \mathrm {log}\left (x \right )^{2} e^{8} x^{2}-32 e^{\frac {3 x}{5}} \mathrm {log}\left (x \right )^{2} e^{8} x +96 x^{2}-16 e^{\frac {9 x}{5}} e^{24} x +24 e^{\frac {6 x}{5}} e^{16} x^{2}+96 e^{\frac {6 x}{5}} e^{16} x -16 e^{\frac {3 x}{5}} e^{8} x^{3}-96 e^{\frac {3 x}{5}} e^{8} x^{2}-192 e^{\frac {3 x}{5}} e^{8} x +32 x^{3}+e^{\frac {12 x}{5}} \mathrm {log}\left (x \right )^{2} e^{32} x +4 x^{4}+16 \mathrm {log}\left (x \right )^{2} x}{x} \] Input:

int(1/5*((12*x^2*exp(3/5*x+8)^4+(-36*x^3-92*x^2)*exp(3/5*x+8)^3+(36*x^4+20 
4*x^3+264*x^2)*exp(3/5*x+8)^2+(-12*x^5-132*x^4-384*x^3-336*x^2)*exp(3/5*x+ 
8)+20*x^5+120*x^4+240*x^3+160*x^2)*log(x)^2+(10*x*exp(3/5*x+8)^4+(-40*x^2- 
80*x)*exp(3/5*x+8)^3+(60*x^3+240*x^2+240*x)*exp(3/5*x+8)^2+(-40*x^4-240*x^ 
3-480*x^2-320*x)*exp(3/5*x+8)+10*x^5+80*x^4+240*x^3+320*x^2+160*x)*log(x)+ 
(48*x-20)*exp(3/5*x+8)^4+(-144*x^2-288*x+160)*exp(3/5*x+8)^3+(144*x^3+696* 
x^2+576*x-480)*exp(3/5*x+8)^2+(-48*x^4-448*x^3-1056*x^2-384*x+640)*exp(3/5 
*x+8)+60*x^4+320*x^3+480*x^2-320)/x^2,x)
 

Output:

(e**((12*x)/5)*log(x)**2*e**32*x + 4*e**((12*x)/5)*e**32 - 4*e**((9*x)/5)* 
log(x)**2*e**24*x**2 - 8*e**((9*x)/5)*log(x)**2*e**24*x - 16*e**((9*x)/5)* 
e**24*x - 32*e**((9*x)/5)*e**24 + 6*e**((6*x)/5)*log(x)**2*e**16*x**3 + 24 
*e**((6*x)/5)*log(x)**2*e**16*x**2 + 24*e**((6*x)/5)*log(x)**2*e**16*x + 2 
4*e**((6*x)/5)*e**16*x**2 + 96*e**((6*x)/5)*e**16*x + 96*e**((6*x)/5)*e**1 
6 - 4*e**((3*x)/5)*log(x)**2*e**8*x**4 - 24*e**((3*x)/5)*log(x)**2*e**8*x* 
*3 - 48*e**((3*x)/5)*log(x)**2*e**8*x**2 - 32*e**((3*x)/5)*log(x)**2*e**8* 
x - 16*e**((3*x)/5)*e**8*x**3 - 96*e**((3*x)/5)*e**8*x**2 - 192*e**((3*x)/ 
5)*e**8*x - 128*e**((3*x)/5)*e**8 + log(x)**2*x**5 + 8*log(x)**2*x**4 + 24 
*log(x)**2*x**3 + 32*log(x)**2*x**2 + 16*log(x)**2*x + 4*x**4 + 32*x**3 + 
96*x**2 + 64)/x