3.10.37 \(\int \frac {(-12 x^4-x^5+x^6+e^x (6 x^4+2 x^5)) \log ^2(3+x)+\log (e^{2 x}+e^x (-5+x)) ((-10 x^4+2 e^x x^4+2 x^5) \log (3+x)+(-60 x^3-8 x^4+4 x^5+e^x (12 x^3+4 x^4)) \log ^2(3+x))}{-75-10 x+5 x^2+e^x (15+5 x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {1}{5} x^4 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \]

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Rubi [F]  time = 129.88, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-12 x^4-x^5+x^6+e^x \left (6 x^4+2 x^5\right )\right ) \log ^2(3+x)+\log \left (e^{2 x}+e^x (-5+x)\right ) \left (\left (-10 x^4+2 e^x x^4+2 x^5\right ) \log (3+x)+\left (-60 x^3-8 x^4+4 x^5+e^x \left (12 x^3+4 x^4\right )\right ) \log ^2(3+x)\right )}{-75-10 x+5 x^2+e^x (15+5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-12*x^4 - x^5 + x^6 + E^x*(6*x^4 + 2*x^5))*Log[3 + x]^2 + Log[E^(2*x) + E^x*(-5 + x)]*((-10*x^4 + 2*E^x*
x^4 + 2*x^5)*Log[3 + x] + (-60*x^3 - 8*x^4 + 4*x^5 + E^x*(12*x^3 + 4*x^4))*Log[3 + x]^2))/(-75 - 10*x + 5*x^2
+ E^x*(15 + 5*x)),x]

[Out]

(2916*x)/25 - (1557*x^2)/50 + (101*x^3)/50 - (59*x^4)/300 + (9*x^5)/500 - (54*(3 + x)^2)/5 + (8*(3 + x)^3)/5 -
(3*(3 + x)^4)/20 + (4*(3 + x)^5)/625 + (45*x*Log[-(E^x*(5 - E^x - x))])/2 - (39*x^2*Log[-(E^x*(5 - E^x - x))]
)/20 + (7*x^3*Log[-(E^x*(5 - E^x - x))])/30 - (x^4*Log[-(E^x*(5 - E^x - x))])/40 - (3888*Log[3 + x])/25 + (54*
x^2*Log[3 + x])/5 - (6*x^3*Log[3 + x])/5 + (x^4*Log[3 + x])/5 - (x^5*Log[3 + x])/25 - (54*x*Log[-(E^x*(5 - E^x
- x))]*Log[3 + x])/5 + (9*x^2*Log[-(E^x*(5 - E^x - x))]*Log[3 + x])/5 - (2*x^3*Log[-(E^x*(5 - E^x - x))]*Log[
3 + x])/5 + (x^4*Log[-(E^x*(5 - E^x - x))]*Log[3 + x])/10 - ((8100*(3 + x) - 2700*(3 + x)^2 + 600*(3 + x)^3 -
75*(3 + x)^4 + 4*(3 + x)^5 - 4860*Log[3 + x])*Log[3 + x])/125 - (486*Log[3 + x]^2)/25 + (2*x^5*Log[3 + x]^2)/2
5 - 135*Defer[Int][x/(-5 + E^x + x), x] + (324*Log[3 + x]*Defer[Int][x/(-5 + E^x + x), x])/5 + (171*Defer[Int]
[x^2/(-5 + E^x + x), x])/5 - (108*Log[3 + x]*Defer[Int][x^2/(-5 + E^x + x), x])/5 - (67*Defer[Int][x^3/(-5 + E
^x + x), x])/20 + (21*Log[3 + x]*Defer[Int][x^3/(-5 + E^x + x), x])/5 + (23*Defer[Int][x^4/(-5 + E^x + x), x])
/60 - Log[3 + x]*Defer[Int][x^4/(-5 + E^x + x), x] - Defer[Int][x^5/(-5 + E^x + x), x]/40 + (Log[3 + x]*Defer[
Int][x^5/(-5 + E^x + x), x])/10 + (6*Defer[Int][(x^4*Log[3 + x]^2)/(-5 + E^x + x), x])/5 - Defer[Int][(x^5*Log
[3 + x]^2)/(-5 + E^x + x), x]/5 - (135*Defer[Int][Log[E^x*(-5 + E^x + x)]/(3 + x), x])/2 + (162*Defer[Int][(Lo
g[3 + x]*Log[E^x*(-5 + E^x + x)])/(3 + x), x])/5 + (4*Defer[Int][x^3*Log[3 + x]^2*Log[E^x*(-5 + E^x + x)], x])
/5 - (324*Defer[Int][Defer[Int][x/(-5 + E^x + x), x]/(3 + x), x])/5 + (108*Defer[Int][Defer[Int][x^2/(-5 + E^x
+ x), x]/(3 + x), x])/5 - (21*Defer[Int][Defer[Int][x^3/(-5 + E^x + x), x]/(3 + x), x])/5 + Defer[Int][Defer[
Int][x^4/(-5 + E^x + x), x]/(3 + x), x] - Defer[Int][Defer[Int][x^5/(-5 + E^x + x), x]/(3 + x), x]/10

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1}{5} x^3 \log (3+x) \left (\frac {2 x \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}+\log (3+x) \left (\frac {x \left (-4+2 e^x+x\right )}{-5+e^x+x}+4 \log \left (e^x \left (-5+e^x+x\right )\right )\right )\right ) \, dx\\ &=\frac {1}{5} \int x^3 \log (3+x) \left (\frac {2 x \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}+\log (3+x) \left (\frac {x \left (-4+2 e^x+x\right )}{-5+e^x+x}+4 \log \left (e^x \left (-5+e^x+x\right )\right )\right )\right ) \, dx\\ &=\frac {1}{5} \int \left (-\frac {(-6+x) x^4 \log ^2(3+x)}{-5+e^x+x}+\frac {2 x^3 \log (3+x) \left (3 x \log (3+x)+x^2 \log (3+x)+x \log \left (e^x \left (-5+e^x+x\right )\right )+6 \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+2 x \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )\right )}{3+x}\right ) \, dx\\ &=-\left (\frac {1}{5} \int \frac {(-6+x) x^4 \log ^2(3+x)}{-5+e^x+x} \, dx\right )+\frac {2}{5} \int \frac {x^3 \log (3+x) \left (3 x \log (3+x)+x^2 \log (3+x)+x \log \left (e^x \left (-5+e^x+x\right )\right )+6 \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+2 x \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )\right )}{3+x} \, dx\\ &=-\left (\frac {1}{5} \int \left (-\frac {6 x^4 \log ^2(3+x)}{-5+e^x+x}+\frac {x^5 \log ^2(3+x)}{-5+e^x+x}\right ) \, dx\right )+\frac {2}{5} \int \frac {x^3 \log (3+x) \left (x \log \left (e^x \left (-5+e^x+x\right )\right )+(3+x) \log (3+x) \left (x+2 \log \left (e^x \left (-5+e^x+x\right )\right )\right )\right )}{3+x} \, dx\\ &=-\left (\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx\right )+\frac {2}{5} \int \left (x^4 \log ^2(3+x)+\frac {x^3 \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}\right ) \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx\\ &=-\left (\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx\right )+\frac {2}{5} \int x^4 \log ^2(3+x) \, dx+\frac {2}{5} \int \frac {x^3 \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx\\ &=\frac {2}{25} x^5 \log ^2(3+x)-\frac {4}{25} \int \frac {x^5 \log (3+x)}{3+x} \, dx-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {2}{5} \int \frac {x^3 \log (3+x) (x+2 (3+x) \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx\\ &=\frac {2}{25} x^5 \log ^2(3+x)-\frac {4}{25} \operatorname {Subst}\left (\int \frac {(-3+x)^5 \log (x)}{x} \, dx,x,3+x\right )-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {2}{5} \int \left (9 \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )-3 x \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )+x^2 \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )-\frac {27 \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}\right ) \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx\\ &=-\frac {1}{125} \left (8100 (3+x)-2700 (3+x)^2+600 (3+x)^3-75 (3+x)^4+4 (3+x)^5-4860 \log (3+x)\right ) \log (3+x)+\frac {2}{25} x^5 \log ^2(3+x)+\frac {4}{25} \operatorname {Subst}\left (\int \left (405-135 x+30 x^2-\frac {15 x^3}{4}+\frac {x^4}{5}-\frac {243 \log (x)}{x}\right ) \, dx,x,3+x\right )-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {2}{5} \int x^2 \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx-\frac {6}{5} \int x \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {18}{5} \int \log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx-\frac {54}{5} \int \frac {\log (3+x) (x+6 \log (3+x)+2 x \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx\\ &=\frac {324 x}{5}-\frac {54}{5} (3+x)^2+\frac {8}{5} (3+x)^3-\frac {3}{20} (3+x)^4+\frac {4}{625} (3+x)^5-\frac {1}{125} \left (8100 (3+x)-2700 (3+x)^2+600 (3+x)^3-75 (3+x)^4+4 (3+x)^5-4860 \log (3+x)\right ) \log (3+x)+\frac {2}{25} x^5 \log ^2(3+x)-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {2}{5} \int x^2 \log (3+x) (x+2 (3+x) \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx-\frac {6}{5} \int x \log (3+x) (x+2 (3+x) \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {18}{5} \int \log (3+x) (x+2 (3+x) \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx-\frac {54}{5} \int \frac {\log (3+x) (x+2 (3+x) \log (3+x)) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx-\frac {972}{25} \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,3+x\right )\\ &=\frac {324 x}{5}-\frac {54}{5} (3+x)^2+\frac {8}{5} (3+x)^3-\frac {3}{20} (3+x)^4+\frac {4}{625} (3+x)^5-\frac {1}{125} \left (8100 (3+x)-2700 (3+x)^2+600 (3+x)^3-75 (3+x)^4+4 (3+x)^5-4860 \log (3+x)\right ) \log (3+x)-\frac {486}{25} \log ^2(3+x)+\frac {2}{25} x^5 \log ^2(3+x)-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {2}{5} \int \left (x^3 \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+6 x^2 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+2 x^3 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )\right ) \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx-\frac {6}{5} \int \left (x^2 \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+6 x \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+2 x^2 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )\right ) \, dx+\frac {18}{5} \int \left (x \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+6 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )+2 x \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )\right ) \, dx-\frac {54}{5} \int \left (\frac {x \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}+\frac {6 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}+\frac {2 x \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}\right ) \, dx\\ &=\frac {324 x}{5}-\frac {54}{5} (3+x)^2+\frac {8}{5} (3+x)^3-\frac {3}{20} (3+x)^4+\frac {4}{625} (3+x)^5-\frac {1}{125} \left (8100 (3+x)-2700 (3+x)^2+600 (3+x)^3-75 (3+x)^4+4 (3+x)^5-4860 \log (3+x)\right ) \log (3+x)-\frac {486}{25} \log ^2(3+x)+\frac {2}{25} x^5 \log ^2(3+x)-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {2}{5} \int x^3 \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {4}{5} \int x^3 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx-\frac {6}{5} \int x^2 \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {18}{5} \int x \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx-\frac {54}{5} \int \frac {x \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx+\frac {108}{5} \int \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx-\frac {108}{5} \int \frac {x \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx-\frac {324}{5} \int \frac {\log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx\\ &=\frac {324 x}{5}-\frac {54}{5} (3+x)^2+\frac {8}{5} (3+x)^3-\frac {3}{20} (3+x)^4+\frac {4}{625} (3+x)^5+\frac {9}{5} x^2 \log \left (-e^x \left (5-e^x-x\right )\right ) \log (3+x)-\frac {2}{5} x^3 \log \left (-e^x \left (5-e^x-x\right )\right ) \log (3+x)+\frac {1}{10} x^4 \log \left (-e^x \left (5-e^x-x\right )\right ) \log (3+x)-\frac {1}{125} \left (8100 (3+x)-2700 (3+x)^2+600 (3+x)^3-75 (3+x)^4+4 (3+x)^5-4860 \log (3+x)\right ) \log (3+x)-\frac {486}{25} \log ^2(3+x)+\frac {2}{25} x^5 \log ^2(3+x)-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx-\frac {2}{5} \int \frac {\left (4-2 e^x-x\right ) x^4 \log (3+x)}{4 \left (5-e^x-x\right )} \, dx-\frac {2}{5} \int \frac {x^4 \log \left (e^x \left (-5+e^x+x\right )\right )}{4 (3+x)} \, dx+\frac {4}{5} \int x^3 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {6}{5} \int \frac {\left (4-2 e^x-x\right ) x^3 \log (3+x)}{3 \left (5-e^x-x\right )} \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {6}{5} \int \frac {x^3 \log \left (e^x \left (-5+e^x+x\right )\right )}{9+3 x} \, dx-\frac {18}{5} \int \frac {\left (4-2 e^x-x\right ) x^2 \log (3+x)}{2 \left (5-e^x-x\right )} \, dx-\frac {18}{5} \int \frac {x^2 \log \left (e^x \left (-5+e^x+x\right )\right )}{6+2 x} \, dx-\frac {54}{5} \int \left (\log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )-\frac {3 \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}\right ) \, dx+\frac {108}{5} \int \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx-\frac {108}{5} \int \left (\log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )-\frac {3 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}\right ) \, dx-\frac {324}{5} \int \frac {\log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx\\ &=\frac {324 x}{5}-\frac {54}{5} (3+x)^2+\frac {8}{5} (3+x)^3-\frac {3}{20} (3+x)^4+\frac {4}{625} (3+x)^5+\frac {9}{5} x^2 \log \left (-e^x \left (5-e^x-x\right )\right ) \log (3+x)-\frac {2}{5} x^3 \log \left (-e^x \left (5-e^x-x\right )\right ) \log (3+x)+\frac {1}{10} x^4 \log \left (-e^x \left (5-e^x-x\right )\right ) \log (3+x)-\frac {1}{125} \left (8100 (3+x)-2700 (3+x)^2+600 (3+x)^3-75 (3+x)^4+4 (3+x)^5-4860 \log (3+x)\right ) \log (3+x)-\frac {486}{25} \log ^2(3+x)+\frac {2}{25} x^5 \log ^2(3+x)-\frac {1}{10} \int \frac {\left (4-2 e^x-x\right ) x^4 \log (3+x)}{5-e^x-x} \, dx-\frac {1}{10} \int \frac {x^4 \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx-\frac {1}{5} \int \frac {x^5 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {2}{5} \int \frac {\left (4-2 e^x-x\right ) x^3 \log (3+x)}{5-e^x-x} \, dx+\frac {4}{5} \int x^3 \log ^2(3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {6}{5} \int \frac {x^4 \log ^2(3+x)}{-5+e^x+x} \, dx+\frac {6}{5} \int \left (3 \log \left (e^x \left (-5+e^x+x\right )\right )-x \log \left (e^x \left (-5+e^x+x\right )\right )+\frac {1}{3} x^2 \log \left (e^x \left (-5+e^x+x\right )\right )-\frac {9 \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x}\right ) \, dx-\frac {9}{5} \int \frac {\left (4-2 e^x-x\right ) x^2 \log (3+x)}{5-e^x-x} \, dx-\frac {18}{5} \int \left (-\frac {3}{2} \log \left (e^x \left (-5+e^x+x\right )\right )+\frac {1}{2} x \log \left (e^x \left (-5+e^x+x\right )\right )+\frac {9 \log \left (e^x \left (-5+e^x+x\right )\right )}{2 (3+x)}\right ) \, dx-\frac {54}{5} \int \log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right ) \, dx+\frac {162}{5} \int \frac {\log (3+x) \log \left (e^x \left (-5+e^x+x\right )\right )}{3+x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.63, size = 43, normalized size = 1.79 \begin {gather*} \frac {1}{5} \left (-162 (3+x)-162 \log \left (-5+e^x+x\right )+\left (162+x^4 \log ^2(3+x)\right ) \log \left (e^x \left (-5+e^x+x\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-12*x^4 - x^5 + x^6 + E^x*(6*x^4 + 2*x^5))*Log[3 + x]^2 + Log[E^(2*x) + E^x*(-5 + x)]*((-10*x^4 +
2*E^x*x^4 + 2*x^5)*Log[3 + x] + (-60*x^3 - 8*x^4 + 4*x^5 + E^x*(12*x^3 + 4*x^4))*Log[3 + x]^2))/(-75 - 10*x +
5*x^2 + E^x*(15 + 5*x)),x]

[Out]

(-162*(3 + x) - 162*Log[-5 + E^x + x] + (162 + x^4*Log[3 + x]^2)*Log[E^x*(-5 + E^x + x)])/5

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fricas [A]  time = 0.85, size = 23, normalized size = 0.96 \begin {gather*} \frac {1}{5} \, x^{4} \log \left ({\left (x - 5\right )} e^{x} + e^{\left (2 \, x\right )}\right ) \log \left (x + 3\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((4*x^4+12*x^3)*exp(x)+4*x^5-8*x^4-60*x^3)*log(3+x)^2+(2*exp(x)*x^4+2*x^5-10*x^4)*log(3+x))*log(ex
p(x)^2+(x-5)*exp(x))+((2*x^5+6*x^4)*exp(x)+x^6-x^5-12*x^4)*log(3+x)^2)/((5*x+15)*exp(x)+5*x^2-10*x-75),x, algo
rithm="fricas")

[Out]

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

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giac [A]  time = 0.46, size = 29, normalized size = 1.21 \begin {gather*} \frac {1}{5} \, x^{5} \log \left (x + 3\right )^{2} + \frac {1}{5} \, x^{4} \log \left (x + e^{x} - 5\right ) \log \left (x + 3\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((4*x^4+12*x^3)*exp(x)+4*x^5-8*x^4-60*x^3)*log(3+x)^2+(2*exp(x)*x^4+2*x^5-10*x^4)*log(3+x))*log(ex
p(x)^2+(x-5)*exp(x))+((2*x^5+6*x^4)*exp(x)+x^6-x^5-12*x^4)*log(3+x)^2)/((5*x+15)*exp(x)+5*x^2-10*x-75),x, algo
rithm="giac")

[Out]

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

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maple [C]  time = 0.14, size = 165, normalized size = 6.88

method	result	size
risch	\(\frac {x^{4} \ln \left (3+x \right )^{2} \ln \left ({\mathrm e}^{x}\right )}{5}+\frac {x^{4} \ln \left (3+x \right )^{2} \ln \left ({\mathrm e}^{x}+x -5\right )}{5}-\frac {i \pi \,x^{4} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x -5\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x} \left ({\mathrm e}^{x}+x -5\right )\right ) \ln \left (3+x \right )^{2}}{10}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (i \left ({\mathrm e}^{x}+x -5\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{x} \left ({\mathrm e}^{x}+x -5\right )\right )^{2} \ln \left (3+x \right )^{2}}{10}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x} \left ({\mathrm e}^{x}+x -5\right )\right )^{2} \ln \left (3+x \right )^{2}}{10}-\frac {i \pi \,x^{4} \mathrm {csgn}\left (i {\mathrm e}^{x} \left ({\mathrm e}^{x}+x -5\right )\right )^{3} \ln \left (3+x \right )^{2}}{10}\)	\(165\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((4*x^4+12*x^3)*exp(x)+4*x^5-8*x^4-60*x^3)*ln(3+x)^2+(2*exp(x)*x^4+2*x^5-10*x^4)*ln(3+x))*ln(exp(x)^2+(x
-5)*exp(x))+((2*x^5+6*x^4)*exp(x)+x^6-x^5-12*x^4)*ln(3+x)^2)/((5*x+15)*exp(x)+5*x^2-10*x-75),x,method=_RETURNV
ERBOSE)

[Out]

1/5*x^4*ln(3+x)^2*ln(exp(x))+1/5*x^4*ln(3+x)^2*ln(exp(x)+x-5)-1/10*I*Pi*x^4*csgn(I*(exp(x)+x-5))*csgn(I*exp(x)
)*csgn(I*exp(x)*(exp(x)+x-5))*ln(3+x)^2+1/10*I*Pi*x^4*csgn(I*(exp(x)+x-5))*csgn(I*exp(x)*(exp(x)+x-5))^2*ln(3+
x)^2+1/10*I*Pi*x^4*csgn(I*exp(x))*csgn(I*exp(x)*(exp(x)+x-5))^2*ln(3+x)^2-1/10*I*Pi*x^4*csgn(I*exp(x)*(exp(x)+
x-5))^3*ln(3+x)^2

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maxima [A]  time = 0.87, size = 29, normalized size = 1.21 \begin {gather*} \frac {1}{5} \, x^{5} \log \left (x + 3\right )^{2} + \frac {1}{5} \, x^{4} \log \left (x + e^{x} - 5\right ) \log \left (x + 3\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((4*x^4+12*x^3)*exp(x)+4*x^5-8*x^4-60*x^3)*log(3+x)^2+(2*exp(x)*x^4+2*x^5-10*x^4)*log(3+x))*log(ex
p(x)^2+(x-5)*exp(x))+((2*x^5+6*x^4)*exp(x)+x^6-x^5-12*x^4)*log(3+x)^2)/((5*x+15)*exp(x)+5*x^2-10*x-75),x, algo
rithm="maxima")

[Out]

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

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mupad [B]  time = 0.99, size = 23, normalized size = 0.96 \begin {gather*} \frac {x^4\,\ln \left ({\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (x-5\right )\right )\,{\ln \left (x+3\right )}^2}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(exp(2*x) + exp(x)*(x - 5))*(log(x + 3)^2*(exp(x)*(12*x^3 + 4*x^4) - 60*x^3 - 8*x^4 + 4*x^5) + log(x
+ 3)*(2*x^4*exp(x) - 10*x^4 + 2*x^5)) + log(x + 3)^2*(exp(x)*(6*x^4 + 2*x^5) - 12*x^4 - x^5 + x^6))/(10*x - ex
p(x)*(5*x + 15) - 5*x^2 + 75),x)

[Out]

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

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sympy [A]  time = 0.83, size = 24, normalized size = 1.00 \begin {gather*} \frac {x^{4} \log {\left (x + 3 \right )}^{2} \log {\left (\left (x - 5\right ) e^{x} + e^{2 x} \right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((4*x**4+12*x**3)*exp(x)+4*x**5-8*x**4-60*x**3)*ln(3+x)**2+(2*exp(x)*x**4+2*x**5-10*x**4)*ln(3+x))
*ln(exp(x)**2+(x-5)*exp(x))+((2*x**5+6*x**4)*exp(x)+x**6-x**5-12*x**4)*ln(3+x)**2)/((5*x+15)*exp(x)+5*x**2-10*
x-75),x)

[Out]

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

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