3.29.37 \(\int \frac {2500+1000 x+100 x^2+e^x (500-300 x-180 x^2-20 x^3)+(-20-8 x) \log (3)}{1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x (390625 x+156250 x^2+15625 x^3)+(-15625 x-3125 x^2) \log (3)+(1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x (390625 x+156250 x^2+15625 x^3)+(-15625 x-3125 x^2) \log (3)) \log (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2})+(781250 x+343750 x^2+43750 x^3+1250 x^4+e^x (156250 x+62500 x^2+6250 x^3)+(-6250 x-1250 x^2) \log (3)) \log ^2(\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2})+(156250 x+68750 x^2+8750 x^3+250 x^4+e^x (31250 x+12500 x^2+1250 x^3)+(-1250 x-250 x^2) \log (3)) \log ^3(\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2})+(15625 x+6875 x^2+875 x^3+25 x^4+e^x (3125 x+1250 x^2+125 x^3)+(-125 x-25 x^2) \log (3)) \log ^4(\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2})+(625 x+275 x^2+35 x^3+x^4+e^x (125 x+50 x^2+5 x^3)+(-5 x-x^2) \log (3)) \log ^5(\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2})} \, dx\)

Optimal. Leaf size=27 \[ \frac {1}{\left (5+\log \left (\frac {5 \left (5+e^x\right )+x-\frac {\log (3)}{5+x}}{x}\right )\right )^4} \]

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Rubi [F]  time = 7.53, 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 {2500+1000 x+100 x^2+e^x \left (500-300 x-180 x^2-20 x^3\right )+(-20-8 x) \log (3)}{1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)+\left (1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)\right ) \log \left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (781250 x+343750 x^2+43750 x^3+1250 x^4+e^x \left (156250 x+62500 x^2+6250 x^3\right )+\left (-6250 x-1250 x^2\right ) \log (3)\right ) \log ^2\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (156250 x+68750 x^2+8750 x^3+250 x^4+e^x \left (31250 x+12500 x^2+1250 x^3\right )+\left (-1250 x-250 x^2\right ) \log (3)\right ) \log ^3\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (15625 x+6875 x^2+875 x^3+25 x^4+e^x \left (3125 x+1250 x^2+125 x^3\right )+\left (-125 x-25 x^2\right ) \log (3)\right ) \log ^4\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (625 x+275 x^2+35 x^3+x^4+e^x \left (125 x+50 x^2+5 x^3\right )+\left (-5 x-x^2\right ) \log (3)\right ) \log ^5\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2500 + 1000*x + 100*x^2 + E^x*(500 - 300*x - 180*x^2 - 20*x^3) + (-20 - 8*x)*Log[3])/(1953125*x + 859375*
x^2 + 109375*x^3 + 3125*x^4 + E^x*(390625*x + 156250*x^2 + 15625*x^3) + (-15625*x - 3125*x^2)*Log[3] + (195312
5*x + 859375*x^2 + 109375*x^3 + 3125*x^4 + E^x*(390625*x + 156250*x^2 + 15625*x^3) + (-15625*x - 3125*x^2)*Log
[3])*Log[(125 + 30*x + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)] + (781250*x + 343750*x^2 + 43750*x^3 + 1250
*x^4 + E^x*(156250*x + 62500*x^2 + 6250*x^3) + (-6250*x - 1250*x^2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*(25 +
5*x) - Log[3])/(5*x + x^2)]^2 + (156250*x + 68750*x^2 + 8750*x^3 + 250*x^4 + E^x*(31250*x + 12500*x^2 + 1250*x
^3) + (-1250*x - 250*x^2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^3 + (15625*x +
6875*x^2 + 875*x^3 + 25*x^4 + E^x*(3125*x + 1250*x^2 + 125*x^3) + (-125*x - 25*x^2)*Log[3])*Log[(125 + 30*x +
x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^4 + (625*x + 275*x^2 + 35*x^3 + x^4 + E^x*(125*x + 50*x^2 + 5*x^3
) + (-5*x - x^2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^5),x]

[Out]

-4*Defer[Int][(5 + Log[(125 + 30*x + x^2 + 5*E^x*(5 + x) - Log[3])/(x*(5 + x))])^(-5), x] + 4*Defer[Int][1/(x*
(5 + Log[(125 + 30*x + x^2 + 5*E^x*(5 + x) - Log[3])/(x*(5 + x))])^5), x] + 4*(120 - Log[3])*Defer[Int][1/((25
*E^x + 30*x + 5*E^x*x + x^2 + 125*(1 - Log[3]/125))*(5 + Log[(125 + 30*x + x^2 + 5*E^x*(5 + x) - Log[3])/(x*(5
+ x))])^5), x] + 4*Log[3]*Defer[Int][1/((-5 - x)*(25*E^x + 30*x + 5*E^x*x + x^2 + 125*(1 - Log[3]/125))*(5 +
Log[(125 + 30*x + x^2 + 5*E^x*(5 + x) - Log[3])/(x*(5 + x))])^5), x] + 116*Defer[Int][x/((25*E^x + 30*x + 5*E^
x*x + x^2 + 125*(1 - Log[3]/125))*(5 + Log[(125 + 30*x + x^2 + 5*E^x*(5 + x) - Log[3])/(x*(5 + x))])^5), x] +
4*Defer[Int][x^2/((25*E^x + 30*x + 5*E^x*x + x^2 + 125*(1 - Log[3]/125))*(5 + Log[(125 + 30*x + x^2 + 5*E^x*(5
+ x) - Log[3])/(x*(5 + x))])^5), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (25 x^2-5 e^x (-1+x) (5+x)^2+625 \left (1-\frac {\log (3)}{125}\right )-x (-250+\log (9))\right )}{x (5+x) \left (30 x+x^2+5 e^x (5+x)+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx\\ &=4 \int \frac {25 x^2-5 e^x (-1+x) (5+x)^2+625 \left (1-\frac {\log (3)}{125}\right )-x (-250+\log (9))}{x (5+x) \left (30 x+x^2+5 e^x (5+x)+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx\\ &=4 \int \left (\frac {1-x}{x \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}+\frac {600+34 x^2+x^3+x (265-\log (3))-\log (729)}{(5+x) \left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}\right ) \, dx\\ &=4 \int \frac {1-x}{x \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx+4 \int \frac {600+34 x^2+x^3+x (265-\log (3))-\log (729)}{(5+x) \left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx\\ &=4 \int \left (-\frac {1}{\left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}+\frac {1}{x \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}\right ) \, dx+4 \int \left (\frac {29 x}{\left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}+\frac {x^2}{\left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}+\frac {120 \left (1-\frac {\log (3)}{120}\right )}{\left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}+\frac {\log (3)}{(-5-x) \left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5}\right ) \, dx\\ &=-\left (4 \int \frac {1}{\left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx\right )+4 \int \frac {1}{x \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx+4 \int \frac {x^2}{\left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx+116 \int \frac {x}{\left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx+(4 (120-\log (3))) \int \frac {1}{\left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx+(4 \log (3)) \int \frac {1}{(-5-x) \left (25 e^x+30 x+5 e^x x+x^2+125 \left (1-\frac {\log (3)}{125}\right )\right ) \left (5+\log \left (\frac {125+30 x+x^2+5 e^x (5+x)-\log (3)}{x (5+x)}\right )\right )^5} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2500+1000 x+100 x^2+e^x \left (500-300 x-180 x^2-20 x^3\right )+(-20-8 x) \log (3)}{1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)+\left (1953125 x+859375 x^2+109375 x^3+3125 x^4+e^x \left (390625 x+156250 x^2+15625 x^3\right )+\left (-15625 x-3125 x^2\right ) \log (3)\right ) \log \left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (781250 x+343750 x^2+43750 x^3+1250 x^4+e^x \left (156250 x+62500 x^2+6250 x^3\right )+\left (-6250 x-1250 x^2\right ) \log (3)\right ) \log ^2\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (156250 x+68750 x^2+8750 x^3+250 x^4+e^x \left (31250 x+12500 x^2+1250 x^3\right )+\left (-1250 x-250 x^2\right ) \log (3)\right ) \log ^3\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (15625 x+6875 x^2+875 x^3+25 x^4+e^x \left (3125 x+1250 x^2+125 x^3\right )+\left (-125 x-25 x^2\right ) \log (3)\right ) \log ^4\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )+\left (625 x+275 x^2+35 x^3+x^4+e^x \left (125 x+50 x^2+5 x^3\right )+\left (-5 x-x^2\right ) \log (3)\right ) \log ^5\left (\frac {125+30 x+x^2+e^x (25+5 x)-\log (3)}{5 x+x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(2500 + 1000*x + 100*x^2 + E^x*(500 - 300*x - 180*x^2 - 20*x^3) + (-20 - 8*x)*Log[3])/(1953125*x + 8
59375*x^2 + 109375*x^3 + 3125*x^4 + E^x*(390625*x + 156250*x^2 + 15625*x^3) + (-15625*x - 3125*x^2)*Log[3] + (
1953125*x + 859375*x^2 + 109375*x^3 + 3125*x^4 + E^x*(390625*x + 156250*x^2 + 15625*x^3) + (-15625*x - 3125*x^
2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)] + (781250*x + 343750*x^2 + 43750*x^3
+ 1250*x^4 + E^x*(156250*x + 62500*x^2 + 6250*x^3) + (-6250*x - 1250*x^2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*
(25 + 5*x) - Log[3])/(5*x + x^2)]^2 + (156250*x + 68750*x^2 + 8750*x^3 + 250*x^4 + E^x*(31250*x + 12500*x^2 +
1250*x^3) + (-1250*x - 250*x^2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^3 + (156
25*x + 6875*x^2 + 875*x^3 + 25*x^4 + E^x*(3125*x + 1250*x^2 + 125*x^3) + (-125*x - 25*x^2)*Log[3])*Log[(125 +
30*x + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^4 + (625*x + 275*x^2 + 35*x^3 + x^4 + E^x*(125*x + 50*x^2 +
5*x^3) + (-5*x - x^2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^5),x]

[Out]

Integrate[(2500 + 1000*x + 100*x^2 + E^x*(500 - 300*x - 180*x^2 - 20*x^3) + (-20 - 8*x)*Log[3])/(1953125*x + 8
59375*x^2 + 109375*x^3 + 3125*x^4 + E^x*(390625*x + 156250*x^2 + 15625*x^3) + (-15625*x - 3125*x^2)*Log[3] + (
1953125*x + 859375*x^2 + 109375*x^3 + 3125*x^4 + E^x*(390625*x + 156250*x^2 + 15625*x^3) + (-15625*x - 3125*x^
2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)] + (781250*x + 343750*x^2 + 43750*x^3
+ 1250*x^4 + E^x*(156250*x + 62500*x^2 + 6250*x^3) + (-6250*x - 1250*x^2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*
(25 + 5*x) - Log[3])/(5*x + x^2)]^2 + (156250*x + 68750*x^2 + 8750*x^3 + 250*x^4 + E^x*(31250*x + 12500*x^2 +
1250*x^3) + (-1250*x - 250*x^2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^3 + (156
25*x + 6875*x^2 + 875*x^3 + 25*x^4 + E^x*(3125*x + 1250*x^2 + 125*x^3) + (-125*x - 25*x^2)*Log[3])*Log[(125 +
30*x + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^4 + (625*x + 275*x^2 + 35*x^3 + x^4 + E^x*(125*x + 50*x^2 +
5*x^3) + (-5*x - x^2)*Log[3])*Log[(125 + 30*x + x^2 + E^x*(25 + 5*x) - Log[3])/(5*x + x^2)]^5), x]

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fricas [B]  time = 0.53, size = 136, normalized size = 5.04 \begin {gather*} \frac {1}{\log \left (\frac {x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125}{x^{2} + 5 \, x}\right )^{4} + 20 \, \log \left (\frac {x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125}{x^{2} + 5 \, x}\right )^{3} + 150 \, \log \left (\frac {x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125}{x^{2} + 5 \, x}\right )^{2} + 500 \, \log \left (\frac {x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125}{x^{2} + 5 \, x}\right ) + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x^3-180*x^2-300*x+500)*exp(x)+(-8*x-20)*log(3)+100*x^2+1000*x+2500)/(((5*x^3+50*x^2+125*x)*exp
(x)+(-x^2-5*x)*log(3)+x^4+35*x^3+275*x^2+625*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^5+((125*x
^3+1250*x^2+3125*x)*exp(x)+(-25*x^2-125*x)*log(3)+25*x^4+875*x^3+6875*x^2+15625*x)*log(((25+5*x)*exp(x)-log(3)
+x^2+30*x+125)/(x^2+5*x))^4+((1250*x^3+12500*x^2+31250*x)*exp(x)+(-250*x^2-1250*x)*log(3)+250*x^4+8750*x^3+687
50*x^2+156250*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^3+((6250*x^3+62500*x^2+156250*x)*exp(x)+
(-1250*x^2-6250*x)*log(3)+1250*x^4+43750*x^3+343750*x^2+781250*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x
^2+5*x))^2+((15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*log(3)+3125*x^4+109375*x^3+859375*x^2+1
953125*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))+(15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x
^2-15625*x)*log(3)+3125*x^4+109375*x^3+859375*x^2+1953125*x),x, algorithm="fricas")

[Out]

1/(log((x^2 + 5*(x + 5)*e^x + 30*x - log(3) + 125)/(x^2 + 5*x))^4 + 20*log((x^2 + 5*(x + 5)*e^x + 30*x - log(3
) + 125)/(x^2 + 5*x))^3 + 150*log((x^2 + 5*(x + 5)*e^x + 30*x - log(3) + 125)/(x^2 + 5*x))^2 + 500*log((x^2 +
5*(x + 5)*e^x + 30*x - log(3) + 125)/(x^2 + 5*x)) + 625)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x^3-180*x^2-300*x+500)*exp(x)+(-8*x-20)*log(3)+100*x^2+1000*x+2500)/(((5*x^3+50*x^2+125*x)*exp
(x)+(-x^2-5*x)*log(3)+x^4+35*x^3+275*x^2+625*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^5+((125*x
^3+1250*x^2+3125*x)*exp(x)+(-25*x^2-125*x)*log(3)+25*x^4+875*x^3+6875*x^2+15625*x)*log(((25+5*x)*exp(x)-log(3)
+x^2+30*x+125)/(x^2+5*x))^4+((1250*x^3+12500*x^2+31250*x)*exp(x)+(-250*x^2-1250*x)*log(3)+250*x^4+8750*x^3+687
50*x^2+156250*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^3+((6250*x^3+62500*x^2+156250*x)*exp(x)+
(-1250*x^2-6250*x)*log(3)+1250*x^4+43750*x^3+343750*x^2+781250*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x
^2+5*x))^2+((15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*log(3)+3125*x^4+109375*x^3+859375*x^2+1
953125*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))+(15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x
^2-15625*x)*log(3)+3125*x^4+109375*x^3+859375*x^2+1953125*x),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (-20 x^{3}-180 x^{2}-300 x +500\right ) {\mathrm e}^{x}+\left (-8 x -20\right ) \ln \relax (3)+100 x^{2}+1000 x +2500}{\left (\left (5 x^{3}+50 x^{2}+125 x \right ) {\mathrm e}^{x}+\left (-x^{2}-5 x \right ) \ln \relax (3)+x^{4}+35 x^{3}+275 x^{2}+625 x \right ) \ln \left (\frac {\left (25+5 x \right ) {\mathrm e}^{x}-\ln \relax (3)+x^{2}+30 x +125}{x^{2}+5 x}\right )^{5}+\left (\left (125 x^{3}+1250 x^{2}+3125 x \right ) {\mathrm e}^{x}+\left (-25 x^{2}-125 x \right ) \ln \relax (3)+25 x^{4}+875 x^{3}+6875 x^{2}+15625 x \right ) \ln \left (\frac {\left (25+5 x \right ) {\mathrm e}^{x}-\ln \relax (3)+x^{2}+30 x +125}{x^{2}+5 x}\right )^{4}+\left (\left (1250 x^{3}+12500 x^{2}+31250 x \right ) {\mathrm e}^{x}+\left (-250 x^{2}-1250 x \right ) \ln \relax (3)+250 x^{4}+8750 x^{3}+68750 x^{2}+156250 x \right ) \ln \left (\frac {\left (25+5 x \right ) {\mathrm e}^{x}-\ln \relax (3)+x^{2}+30 x +125}{x^{2}+5 x}\right )^{3}+\left (\left (6250 x^{3}+62500 x^{2}+156250 x \right ) {\mathrm e}^{x}+\left (-1250 x^{2}-6250 x \right ) \ln \relax (3)+1250 x^{4}+43750 x^{3}+343750 x^{2}+781250 x \right ) \ln \left (\frac {\left (25+5 x \right ) {\mathrm e}^{x}-\ln \relax (3)+x^{2}+30 x +125}{x^{2}+5 x}\right )^{2}+\left (\left (15625 x^{3}+156250 x^{2}+390625 x \right ) {\mathrm e}^{x}+\left (-3125 x^{2}-15625 x \right ) \ln \relax (3)+3125 x^{4}+109375 x^{3}+859375 x^{2}+1953125 x \right ) \ln \left (\frac {\left (25+5 x \right ) {\mathrm e}^{x}-\ln \relax (3)+x^{2}+30 x +125}{x^{2}+5 x}\right )+\left (15625 x^{3}+156250 x^{2}+390625 x \right ) {\mathrm e}^{x}+\left (-3125 x^{2}-15625 x \right ) \ln \relax (3)+3125 x^{4}+109375 x^{3}+859375 x^{2}+1953125 x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-20*x^3-180*x^2-300*x+500)*exp(x)+(-8*x-20)*ln(3)+100*x^2+1000*x+2500)/(((5*x^3+50*x^2+125*x)*exp(x)+(-x
^2-5*x)*ln(3)+x^4+35*x^3+275*x^2+625*x)*ln(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))^5+((125*x^3+1250*x^
2+3125*x)*exp(x)+(-25*x^2-125*x)*ln(3)+25*x^4+875*x^3+6875*x^2+15625*x)*ln(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125
)/(x^2+5*x))^4+((1250*x^3+12500*x^2+31250*x)*exp(x)+(-250*x^2-1250*x)*ln(3)+250*x^4+8750*x^3+68750*x^2+156250*
x)*ln(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))^3+((6250*x^3+62500*x^2+156250*x)*exp(x)+(-1250*x^2-6250*
x)*ln(3)+1250*x^4+43750*x^3+343750*x^2+781250*x)*ln(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))^2+((15625*
x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*ln(3)+3125*x^4+109375*x^3+859375*x^2+1953125*x)*ln(((25+5*
x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))+(15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*ln(3)+3125
*x^4+109375*x^3+859375*x^2+1953125*x),x)

[Out]

int(((-20*x^3-180*x^2-300*x+500)*exp(x)+(-8*x-20)*ln(3)+100*x^2+1000*x+2500)/(((5*x^3+50*x^2+125*x)*exp(x)+(-x
^2-5*x)*ln(3)+x^4+35*x^3+275*x^2+625*x)*ln(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))^5+((125*x^3+1250*x^
2+3125*x)*exp(x)+(-25*x^2-125*x)*ln(3)+25*x^4+875*x^3+6875*x^2+15625*x)*ln(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125
)/(x^2+5*x))^4+((1250*x^3+12500*x^2+31250*x)*exp(x)+(-250*x^2-1250*x)*ln(3)+250*x^4+8750*x^3+68750*x^2+156250*
x)*ln(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))^3+((6250*x^3+62500*x^2+156250*x)*exp(x)+(-1250*x^2-6250*
x)*ln(3)+1250*x^4+43750*x^3+343750*x^2+781250*x)*ln(((25+5*x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))^2+((15625*
x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*ln(3)+3125*x^4+109375*x^3+859375*x^2+1953125*x)*ln(((25+5*
x)*exp(x)-ln(3)+x^2+30*x+125)/(x^2+5*x))+(15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*ln(3)+3125
*x^4+109375*x^3+859375*x^2+1953125*x),x)

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maxima [B]  time = 18.39, size = 266, normalized size = 9.85 \begin {gather*} -\frac {1}{4 \, {\left (\log \left (x + 5\right ) + \log \relax (x) - 5\right )} \log \left (x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125\right )^{3} - \log \left (x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125\right )^{4} - 4 \, {\left (\log \relax (x) - 5\right )} \log \left (x + 5\right )^{3} - \log \left (x + 5\right )^{4} - \log \relax (x)^{4} - 6 \, {\left (2 \, {\left (\log \relax (x) - 5\right )} \log \left (x + 5\right ) + \log \left (x + 5\right )^{2} + \log \relax (x)^{2} - 10 \, \log \relax (x) + 25\right )} \log \left (x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125\right )^{2} - 6 \, {\left (\log \relax (x)^{2} - 10 \, \log \relax (x) + 25\right )} \log \left (x + 5\right )^{2} + 20 \, \log \relax (x)^{3} + 4 \, {\left (3 \, {\left (\log \relax (x) - 5\right )} \log \left (x + 5\right )^{2} + \log \left (x + 5\right )^{3} + \log \relax (x)^{3} + 3 \, {\left (\log \relax (x)^{2} - 10 \, \log \relax (x) + 25\right )} \log \left (x + 5\right ) - 15 \, \log \relax (x)^{2} + 75 \, \log \relax (x) - 125\right )} \log \left (x^{2} + 5 \, {\left (x + 5\right )} e^{x} + 30 \, x - \log \relax (3) + 125\right ) - 4 \, {\left (\log \relax (x)^{3} - 15 \, \log \relax (x)^{2} + 75 \, \log \relax (x) - 125\right )} \log \left (x + 5\right ) - 150 \, \log \relax (x)^{2} + 500 \, \log \relax (x) - 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x^3-180*x^2-300*x+500)*exp(x)+(-8*x-20)*log(3)+100*x^2+1000*x+2500)/(((5*x^3+50*x^2+125*x)*exp
(x)+(-x^2-5*x)*log(3)+x^4+35*x^3+275*x^2+625*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^5+((125*x
^3+1250*x^2+3125*x)*exp(x)+(-25*x^2-125*x)*log(3)+25*x^4+875*x^3+6875*x^2+15625*x)*log(((25+5*x)*exp(x)-log(3)
+x^2+30*x+125)/(x^2+5*x))^4+((1250*x^3+12500*x^2+31250*x)*exp(x)+(-250*x^2-1250*x)*log(3)+250*x^4+8750*x^3+687
50*x^2+156250*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))^3+((6250*x^3+62500*x^2+156250*x)*exp(x)+
(-1250*x^2-6250*x)*log(3)+1250*x^4+43750*x^3+343750*x^2+781250*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x
^2+5*x))^2+((15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x^2-15625*x)*log(3)+3125*x^4+109375*x^3+859375*x^2+1
953125*x)*log(((25+5*x)*exp(x)-log(3)+x^2+30*x+125)/(x^2+5*x))+(15625*x^3+156250*x^2+390625*x)*exp(x)+(-3125*x
^2-15625*x)*log(3)+3125*x^4+109375*x^3+859375*x^2+1953125*x),x, algorithm="maxima")

[Out]

-1/(4*(log(x + 5) + log(x) - 5)*log(x^2 + 5*(x + 5)*e^x + 30*x - log(3) + 125)^3 - log(x^2 + 5*(x + 5)*e^x + 3
0*x - log(3) + 125)^4 - 4*(log(x) - 5)*log(x + 5)^3 - log(x + 5)^4 - log(x)^4 - 6*(2*(log(x) - 5)*log(x + 5) +
log(x + 5)^2 + log(x)^2 - 10*log(x) + 25)*log(x^2 + 5*(x + 5)*e^x + 30*x - log(3) + 125)^2 - 6*(log(x)^2 - 10
*log(x) + 25)*log(x + 5)^2 + 20*log(x)^3 + 4*(3*(log(x) - 5)*log(x + 5)^2 + log(x + 5)^3 + log(x)^3 + 3*(log(x
)^2 - 10*log(x) + 25)*log(x + 5) - 15*log(x)^2 + 75*log(x) - 125)*log(x^2 + 5*(x + 5)*e^x + 30*x - log(3) + 12
5) - 4*(log(x)^3 - 15*log(x)^2 + 75*log(x) - 125)*log(x + 5) - 150*log(x)^2 + 500*log(x) - 625)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1000\,x-\ln \relax (3)\,\left (8\,x+20\right )+100\,x^2-{\mathrm {e}}^x\,\left (20\,x^3+180\,x^2+300\,x-500\right )+2500}{1953125\,x-\ln \relax (3)\,\left (3125\,x^2+15625\,x\right )+{\ln \left (\frac {30\,x-\ln \relax (3)+{\mathrm {e}}^x\,\left (5\,x+25\right )+x^2+125}{x^2+5\,x}\right )}^4\,\left (15625\,x-\ln \relax (3)\,\left (25\,x^2+125\,x\right )+6875\,x^2+875\,x^3+25\,x^4+{\mathrm {e}}^x\,\left (125\,x^3+1250\,x^2+3125\,x\right )\right )+{\ln \left (\frac {30\,x-\ln \relax (3)+{\mathrm {e}}^x\,\left (5\,x+25\right )+x^2+125}{x^2+5\,x}\right )}^3\,\left (156250\,x-\ln \relax (3)\,\left (250\,x^2+1250\,x\right )+68750\,x^2+8750\,x^3+250\,x^4+{\mathrm {e}}^x\,\left (1250\,x^3+12500\,x^2+31250\,x\right )\right )+{\ln \left (\frac {30\,x-\ln \relax (3)+{\mathrm {e}}^x\,\left (5\,x+25\right )+x^2+125}{x^2+5\,x}\right )}^2\,\left (781250\,x-\ln \relax (3)\,\left (1250\,x^2+6250\,x\right )+343750\,x^2+43750\,x^3+1250\,x^4+{\mathrm {e}}^x\,\left (6250\,x^3+62500\,x^2+156250\,x\right )\right )+{\ln \left (\frac {30\,x-\ln \relax (3)+{\mathrm {e}}^x\,\left (5\,x+25\right )+x^2+125}{x^2+5\,x}\right )}^5\,\left (625\,x+275\,x^2+35\,x^3+x^4-\ln \relax (3)\,\left (x^2+5\,x\right )+{\mathrm {e}}^x\,\left (5\,x^3+50\,x^2+125\,x\right )\right )+859375\,x^2+109375\,x^3+3125\,x^4+\ln \left (\frac {30\,x-\ln \relax (3)+{\mathrm {e}}^x\,\left (5\,x+25\right )+x^2+125}{x^2+5\,x}\right )\,\left (1953125\,x-\ln \relax (3)\,\left (3125\,x^2+15625\,x\right )+859375\,x^2+109375\,x^3+3125\,x^4+{\mathrm {e}}^x\,\left (15625\,x^3+156250\,x^2+390625\,x\right )\right )+{\mathrm {e}}^x\,\left (15625\,x^3+156250\,x^2+390625\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1000*x - log(3)*(8*x + 20) + 100*x^2 - exp(x)*(300*x + 180*x^2 + 20*x^3 - 500) + 2500)/(1953125*x - log(3
)*(15625*x + 3125*x^2) + log((30*x - log(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x^2))^4*(15625*x - log(3)*
(125*x + 25*x^2) + 6875*x^2 + 875*x^3 + 25*x^4 + exp(x)*(3125*x + 1250*x^2 + 125*x^3)) + log((30*x - log(3) +
exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x^2))^3*(156250*x - log(3)*(1250*x + 250*x^2) + 68750*x^2 + 8750*x^3 + 2
50*x^4 + exp(x)*(31250*x + 12500*x^2 + 1250*x^3)) + log((30*x - log(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x +
x^2))^2*(781250*x - log(3)*(6250*x + 1250*x^2) + 343750*x^2 + 43750*x^3 + 1250*x^4 + exp(x)*(156250*x + 62500
*x^2 + 6250*x^3)) + log((30*x - log(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x^2))^5*(625*x + 275*x^2 + 35*x
^3 + x^4 - log(3)*(5*x + x^2) + exp(x)*(125*x + 50*x^2 + 5*x^3)) + 859375*x^2 + 109375*x^3 + 3125*x^4 + log((3
0*x - log(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x^2))*(1953125*x - log(3)*(15625*x + 3125*x^2) + 859375*x
^2 + 109375*x^3 + 3125*x^4 + exp(x)*(390625*x + 156250*x^2 + 15625*x^3)) + exp(x)*(390625*x + 156250*x^2 + 156
25*x^3)),x)

[Out]

int((1000*x - log(3)*(8*x + 20) + 100*x^2 - exp(x)*(300*x + 180*x^2 + 20*x^3 - 500) + 2500)/(1953125*x - log(3
)*(15625*x + 3125*x^2) + log((30*x - log(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x^2))^4*(15625*x - log(3)*
(125*x + 25*x^2) + 6875*x^2 + 875*x^3 + 25*x^4 + exp(x)*(3125*x + 1250*x^2 + 125*x^3)) + log((30*x - log(3) +
exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x^2))^3*(156250*x - log(3)*(1250*x + 250*x^2) + 68750*x^2 + 8750*x^3 + 2
50*x^4 + exp(x)*(31250*x + 12500*x^2 + 1250*x^3)) + log((30*x - log(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x +
x^2))^2*(781250*x - log(3)*(6250*x + 1250*x^2) + 343750*x^2 + 43750*x^3 + 1250*x^4 + exp(x)*(156250*x + 62500
*x^2 + 6250*x^3)) + log((30*x - log(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x^2))^5*(625*x + 275*x^2 + 35*x
^3 + x^4 - log(3)*(5*x + x^2) + exp(x)*(125*x + 50*x^2 + 5*x^3)) + 859375*x^2 + 109375*x^3 + 3125*x^4 + log((3
0*x - log(3) + exp(x)*(5*x + 25) + x^2 + 125)/(5*x + x^2))*(1953125*x - log(3)*(15625*x + 3125*x^2) + 859375*x
^2 + 109375*x^3 + 3125*x^4 + exp(x)*(390625*x + 156250*x^2 + 15625*x^3)) + exp(x)*(390625*x + 156250*x^2 + 156
25*x^3)), x)

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sympy [B]  time = 1.32, size = 128, normalized size = 4.74 \begin {gather*} \frac {1}{\log {\left (\frac {x^{2} + 30 x + \left (5 x + 25\right ) e^{x} - \log {\relax (3 )} + 125}{x^{2} + 5 x} \right )}^{4} + 20 \log {\left (\frac {x^{2} + 30 x + \left (5 x + 25\right ) e^{x} - \log {\relax (3 )} + 125}{x^{2} + 5 x} \right )}^{3} + 150 \log {\left (\frac {x^{2} + 30 x + \left (5 x + 25\right ) e^{x} - \log {\relax (3 )} + 125}{x^{2} + 5 x} \right )}^{2} + 500 \log {\left (\frac {x^{2} + 30 x + \left (5 x + 25\right ) e^{x} - \log {\relax (3 )} + 125}{x^{2} + 5 x} \right )} + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*x**3-180*x**2-300*x+500)*exp(x)+(-8*x-20)*ln(3)+100*x**2+1000*x+2500)/(((5*x**3+50*x**2+125*x)
*exp(x)+(-x**2-5*x)*ln(3)+x**4+35*x**3+275*x**2+625*x)*ln(((25+5*x)*exp(x)-ln(3)+x**2+30*x+125)/(x**2+5*x))**5
+((125*x**3+1250*x**2+3125*x)*exp(x)+(-25*x**2-125*x)*ln(3)+25*x**4+875*x**3+6875*x**2+15625*x)*ln(((25+5*x)*e
xp(x)-ln(3)+x**2+30*x+125)/(x**2+5*x))**4+((1250*x**3+12500*x**2+31250*x)*exp(x)+(-250*x**2-1250*x)*ln(3)+250*
x**4+8750*x**3+68750*x**2+156250*x)*ln(((25+5*x)*exp(x)-ln(3)+x**2+30*x+125)/(x**2+5*x))**3+((6250*x**3+62500*
x**2+156250*x)*exp(x)+(-1250*x**2-6250*x)*ln(3)+1250*x**4+43750*x**3+343750*x**2+781250*x)*ln(((25+5*x)*exp(x)
-ln(3)+x**2+30*x+125)/(x**2+5*x))**2+((15625*x**3+156250*x**2+390625*x)*exp(x)+(-3125*x**2-15625*x)*ln(3)+3125
*x**4+109375*x**3+859375*x**2+1953125*x)*ln(((25+5*x)*exp(x)-ln(3)+x**2+30*x+125)/(x**2+5*x))+(15625*x**3+1562
50*x**2+390625*x)*exp(x)+(-3125*x**2-15625*x)*ln(3)+3125*x**4+109375*x**3+859375*x**2+1953125*x),x)

[Out]

1/(log((x**2 + 30*x + (5*x + 25)*exp(x) - log(3) + 125)/(x**2 + 5*x))**4 + 20*log((x**2 + 30*x + (5*x + 25)*ex
p(x) - log(3) + 125)/(x**2 + 5*x))**3 + 150*log((x**2 + 30*x + (5*x + 25)*exp(x) - log(3) + 125)/(x**2 + 5*x))
**2 + 500*log((x**2 + 30*x + (5*x + 25)*exp(x) - log(3) + 125)/(x**2 + 5*x)) + 625)

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