3.43.85 \(\int \frac {1250-2000 x-100 x^2+140 x^3-10 x^4+e^x (-250 x-200 x^2+100 x^3-10 x^4)}{125 x+300 x^2+315 x^3+184 x^4+e^{3 x} x^4+63 x^5+12 x^6+x^7+(-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6) \log (5)+(375 x+150 x^2-30 x^3-18 x^4+3 x^5) \log ^2(5)+(-125 x+75 x^2-15 x^3+x^4) \log ^3(5)+e^{2 x} (15 x^3+12 x^4+3 x^5+(-15 x^3+3 x^4) \log (5))+e^x (75 x^2+120 x^3+78 x^4+24 x^5+3 x^6+(-150 x^2-90 x^3-6 x^4+6 x^5) \log (5)+(75 x^2-30 x^3+3 x^4) \log ^2(5))+(-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6+e^{2 x} (-15 x^3+3 x^4)+(750 x+300 x^2-60 x^3-36 x^4+6 x^5) \log (5)+(-375 x+225 x^2-45 x^3+3 x^4) \log ^2(5)+e^x (-150 x^2-90 x^3-6 x^4+6 x^5+(150 x^2-60 x^3+6 x^4) \log (5))) \log (x)+(375 x+150 x^2-30 x^3-18 x^4+3 x^5+e^x (75 x^2-30 x^3+3 x^4)+(-375 x+225 x^2-45 x^3+3 x^4) \log (5)) \log ^2(x)+(-125 x+75 x^2-15 x^3+x^4) \log ^3(x)} \, dx\)

Optimal. Leaf size=29 \[ \frac {5}{\left (1+\frac {x \left (5+e^x+x\right )}{5-x}-\log (5)-\log (x)\right )^2} \]

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Rubi [F]  time = 26.47, 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 {1250-2000 x-100 x^2+140 x^3-10 x^4+e^x \left (-250 x-200 x^2+100 x^3-10 x^4\right )}{125 x+300 x^2+315 x^3+184 x^4+e^{3 x} x^4+63 x^5+12 x^6+x^7+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6\right ) \log (5)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5\right ) \log ^2(5)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(5)+e^{2 x} \left (15 x^3+12 x^4+3 x^5+\left (-15 x^3+3 x^4\right ) \log (5)\right )+e^x \left (75 x^2+120 x^3+78 x^4+24 x^5+3 x^6+\left (-150 x^2-90 x^3-6 x^4+6 x^5\right ) \log (5)+\left (75 x^2-30 x^3+3 x^4\right ) \log ^2(5)\right )+\left (-375 x-525 x^2-270 x^3-42 x^4+9 x^5+3 x^6+e^{2 x} \left (-15 x^3+3 x^4\right )+\left (750 x+300 x^2-60 x^3-36 x^4+6 x^5\right ) \log (5)+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log ^2(5)+e^x \left (-150 x^2-90 x^3-6 x^4+6 x^5+\left (150 x^2-60 x^3+6 x^4\right ) \log (5)\right )\right ) \log (x)+\left (375 x+150 x^2-30 x^3-18 x^4+3 x^5+e^x \left (75 x^2-30 x^3+3 x^4\right )+\left (-375 x+225 x^2-45 x^3+3 x^4\right ) \log (5)\right ) \log ^2(x)+\left (-125 x+75 x^2-15 x^3+x^4\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1250 - 2000*x - 100*x^2 + 140*x^3 - 10*x^4 + E^x*(-250*x - 200*x^2 + 100*x^3 - 10*x^4))/(125*x + 300*x^2
+ 315*x^3 + 184*x^4 + E^(3*x)*x^4 + 63*x^5 + 12*x^6 + x^7 + (-375*x - 525*x^2 - 270*x^3 - 42*x^4 + 9*x^5 + 3*x
^6)*Log[5] + (375*x + 150*x^2 - 30*x^3 - 18*x^4 + 3*x^5)*Log[5]^2 + (-125*x + 75*x^2 - 15*x^3 + x^4)*Log[5]^3
+ E^(2*x)*(15*x^3 + 12*x^4 + 3*x^5 + (-15*x^3 + 3*x^4)*Log[5]) + E^x*(75*x^2 + 120*x^3 + 78*x^4 + 24*x^5 + 3*x
^6 + (-150*x^2 - 90*x^3 - 6*x^4 + 6*x^5)*Log[5] + (75*x^2 - 30*x^3 + 3*x^4)*Log[5]^2) + (-375*x - 525*x^2 - 27
0*x^3 - 42*x^4 + 9*x^5 + 3*x^6 + E^(2*x)*(-15*x^3 + 3*x^4) + (750*x + 300*x^2 - 60*x^3 - 36*x^4 + 6*x^5)*Log[5
] + (-375*x + 225*x^2 - 45*x^3 + 3*x^4)*Log[5]^2 + E^x*(-150*x^2 - 90*x^3 - 6*x^4 + 6*x^5 + (150*x^2 - 60*x^3
+ 6*x^4)*Log[5]))*Log[x] + (375*x + 150*x^2 - 30*x^3 - 18*x^4 + 3*x^5 + E^x*(75*x^2 - 30*x^3 + 3*x^4) + (-375*
x + 225*x^2 - 45*x^3 + 3*x^4)*Log[5])*Log[x]^2 + (-125*x + 75*x^2 - 15*x^3 + x^4)*Log[x]^3),x]

[Out]

100*Defer[Int][x^3/(-(E^x*x) - x^2 - 5*(1 - Log[5]) - 4*x*(1 + Log[5]/4) + 5*Log[x] - x*Log[x])^3, x] + 1250*D
efer[Int][Log[x]/(-(E^x*x) - x^2 - 5*(1 - Log[5]) - 4*x*(1 + Log[5]/4) + 5*Log[x] - x*Log[x])^3, x] + 1250*Def
er[Int][Log[x]/(x*(-(E^x*x) - x^2 - 5*(1 - Log[5]) - 4*x*(1 + Log[5]/4) + 5*Log[x] - x*Log[x])^3), x] + 50*Def
er[Int][(x*Log[x])/(-(E^x*x) - x^2 - 5*(1 - Log[5]) - 4*x*(1 + Log[5]/4) + 5*Log[x] - x*Log[x])^3, x] + 150*De
fer[Int][(x^2*Log[x])/(-(E^x*x) - x^2 - 5*(1 - Log[5]) - 4*x*(1 + Log[5]/4) + 5*Log[x] - x*Log[x])^3, x] + 250
*(4 - 5*Log[5])*Defer[Int][(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] + x*Log[x])^(-3), x]
- 500*(2 - Log[5])*Defer[Int][(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] + x*Log[x])^(-3),
x] + 1250*(2 - Log[5])*Defer[Int][1/(x*(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] + x*Log[x
])^3), x] - 100*(4 - 5*Log[5])*Defer[Int][x/(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] + x*
Log[x])^3, x] + 50*(2 - Log[5])*Defer[Int][x/(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] + x
*Log[x])^3, x] + 250*(3 + Log[5])*Defer[Int][x/(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] +
x*Log[x])^3, x] + 250*Defer[Int][x^2/(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] + x*Log[x]
)^3, x] + 10*(4 - 5*Log[5])*Defer[Int][x^2/(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] + x*L
og[x])^3, x] - 100*(3 + Log[5])*Defer[Int][x^2/(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] +
x*Log[x])^3, x] + 10*(3 + Log[5])*Defer[Int][x^3/(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x
] + x*Log[x])^3, x] + 10*Defer[Int][x^4/(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] + x*Log[
x])^3, x] + 500*Defer[Int][Log[x]/(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] + x*Log[x])^3,
x] + 750*Defer[Int][(x*Log[x])/(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] + x*Log[x])^3, x
] + 10*Defer[Int][(x^3*Log[x])/(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] + x*Log[x])^3, x]
- 200*Defer[Int][(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] + x*Log[x])^(-2), x] - 250*Def
er[Int][1/(x*(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] + x*Log[x])^2), x] + 100*Defer[Int]
[x/(E^x*x + x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] + x*Log[x])^2, x] - 10*Defer[Int][x^2/(E^x*x
+ x^2 + 5*(1 - Log[5]) + 4*x*(1 + Log[5]/4) - 5*Log[x] + x*Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10 (5-x) \left (25-5 \left (7+e^x\right ) x-\left (9+5 e^x\right ) x^2+\left (1+e^x\right ) x^3\right )}{x \left (x^2+5 (1-\log (5))+x \left (4+e^x+\log (5)\right )+(-5+x) \log (x)\right )^3} \, dx\\ &=10 \int \frac {(5-x) \left (25-5 \left (7+e^x\right ) x-\left (9+5 e^x\right ) x^2+\left (1+e^x\right ) x^3\right )}{x \left (x^2+5 (1-\log (5))+x \left (4+e^x+\log (5)\right )+(-5+x) \log (x)\right )^3} \, dx\\ &=10 \int \left (\frac {-25-20 x+10 x^2-x^3}{x \left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2}+\frac {(5-x)^2 \left (x^3+4 x \left (1-\frac {5 \log (5)}{4}\right )+10 \left (1-\frac {\log (5)}{2}\right )+3 x^2 \left (1+\frac {\log (5)}{3}\right )-5 \log (x)-5 x \log (x)+x^2 \log (x)\right )}{x \left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^3}\right ) \, dx\\ &=10 \int \frac {-25-20 x+10 x^2-x^3}{x \left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2} \, dx+10 \int \frac {(5-x)^2 \left (x^3+4 x \left (1-\frac {5 \log (5)}{4}\right )+10 \left (1-\frac {\log (5)}{2}\right )+3 x^2 \left (1+\frac {\log (5)}{3}\right )-5 \log (x)-5 x \log (x)+x^2 \log (x)\right )}{x \left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^3} \, dx\\ &=10 \int \frac {(5-x)^2 \left (x^3+x (4-5 \log (5))-5 (-2+\log (5))+x^2 (3+\log (5))+\left (-5-5 x+x^2\right ) \log (x)\right )}{x \left (x^2+5 (1-\log (5))+x \left (4+e^x+\log (5)\right )+(-5+x) \log (x)\right )^3} \, dx+10 \int \left (-\frac {20}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2}-\frac {25}{x \left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2}+\frac {10 x}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2}-\frac {x^2}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2}\right ) \, dx\\ &=-\left (10 \int \frac {x^2}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2} \, dx\right )+10 \int \left (\frac {10 \left (-x^3-4 x \left (1-\frac {5 \log (5)}{4}\right )-10 \left (1-\frac {\log (5)}{2}\right )-3 x^2 \left (1+\frac {\log (5)}{3}\right )+5 \log (x)+5 x \log (x)-x^2 \log (x)\right )}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^3}+\frac {25 \left (x^3+4 x \left (1-\frac {5 \log (5)}{4}\right )+10 \left (1-\frac {\log (5)}{2}\right )+3 x^2 \left (1+\frac {\log (5)}{3}\right )-5 \log (x)-5 x \log (x)+x^2 \log (x)\right )}{x \left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^3}+\frac {x \left (x^3+4 x \left (1-\frac {5 \log (5)}{4}\right )+10 \left (1-\frac {\log (5)}{2}\right )+3 x^2 \left (1+\frac {\log (5)}{3}\right )-5 \log (x)-5 x \log (x)+x^2 \log (x)\right )}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^3}\right ) \, dx+100 \int \frac {x}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2} \, dx-200 \int \frac {1}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2} \, dx-250 \int \frac {1}{x \left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2} \, dx\\ &=-\left (10 \int \frac {x^2}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2} \, dx\right )+10 \int \frac {x \left (x^3+4 x \left (1-\frac {5 \log (5)}{4}\right )+10 \left (1-\frac {\log (5)}{2}\right )+3 x^2 \left (1+\frac {\log (5)}{3}\right )-5 \log (x)-5 x \log (x)+x^2 \log (x)\right )}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^3} \, dx+100 \int \frac {x}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2} \, dx+100 \int \frac {-x^3-4 x \left (1-\frac {5 \log (5)}{4}\right )-10 \left (1-\frac {\log (5)}{2}\right )-3 x^2 \left (1+\frac {\log (5)}{3}\right )+5 \log (x)+5 x \log (x)-x^2 \log (x)}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^3} \, dx-200 \int \frac {1}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2} \, dx-250 \int \frac {1}{x \left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2} \, dx+250 \int \frac {x^3+4 x \left (1-\frac {5 \log (5)}{4}\right )+10 \left (1-\frac {\log (5)}{2}\right )+3 x^2 \left (1+\frac {\log (5)}{3}\right )-5 \log (x)-5 x \log (x)+x^2 \log (x)}{x \left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^3} \, dx\\ &=-\left (10 \int \frac {x^2}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2} \, dx\right )+10 \int \frac {x \left (x^3+x (4-5 \log (5))-5 (-2+\log (5))+x^2 (3+\log (5))+\left (-5-5 x+x^2\right ) \log (x)\right )}{\left (x^2+5 (1-\log (5))+x \left (4+e^x+\log (5)\right )+(-5+x) \log (x)\right )^3} \, dx+100 \int \frac {x}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2} \, dx+100 \int \left (\frac {x^3}{\left (-e^x x-x^2-5 (1-\log (5))-4 x \left (1+\frac {\log (5)}{4}\right )+5 \log (x)-x \log (x)\right )^3}+\frac {x^2 \log (x)}{\left (-e^x x-x^2-5 (1-\log (5))-4 x \left (1+\frac {\log (5)}{4}\right )+5 \log (x)-x \log (x)\right )^3}+\frac {x^2 (-3-\log (5))}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^3}+\frac {5 (-2+\log (5))}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^3}+\frac {x (-4+5 \log (5))}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^3}+\frac {5 \log (x)}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^3}+\frac {5 x \log (x)}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^3}\right ) \, dx-200 \int \frac {1}{\left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2} \, dx-250 \int \frac {1}{x \left (e^x x+x^2+5 (1-\log (5))+4 x \left (1+\frac {\log (5)}{4}\right )-5 \log (x)+x \log (x)\right )^2} \, dx+250 \int \frac {x^3+x (4-5 \log (5))-5 (-2+\log (5))+x^2 (3+\log (5))+\left (-5-5 x+x^2\right ) \log (x)}{x \left (x^2+5 (1-\log (5))+x \left (4+e^x+\log (5)\right )+(-5+x) \log (x)\right )^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 3.85, size = 33, normalized size = 1.14 \begin {gather*} \frac {5 (-5+x)^2}{\left (5+x^2-5 \log (5)+x \left (4+e^x+\log (5)\right )+(-5+x) \log (x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1250 - 2000*x - 100*x^2 + 140*x^3 - 10*x^4 + E^x*(-250*x - 200*x^2 + 100*x^3 - 10*x^4))/(125*x + 30
0*x^2 + 315*x^3 + 184*x^4 + E^(3*x)*x^4 + 63*x^5 + 12*x^6 + x^7 + (-375*x - 525*x^2 - 270*x^3 - 42*x^4 + 9*x^5
+ 3*x^6)*Log[5] + (375*x + 150*x^2 - 30*x^3 - 18*x^4 + 3*x^5)*Log[5]^2 + (-125*x + 75*x^2 - 15*x^3 + x^4)*Log
[5]^3 + E^(2*x)*(15*x^3 + 12*x^4 + 3*x^5 + (-15*x^3 + 3*x^4)*Log[5]) + E^x*(75*x^2 + 120*x^3 + 78*x^4 + 24*x^5
+ 3*x^6 + (-150*x^2 - 90*x^3 - 6*x^4 + 6*x^5)*Log[5] + (75*x^2 - 30*x^3 + 3*x^4)*Log[5]^2) + (-375*x - 525*x^
2 - 270*x^3 - 42*x^4 + 9*x^5 + 3*x^6 + E^(2*x)*(-15*x^3 + 3*x^4) + (750*x + 300*x^2 - 60*x^3 - 36*x^4 + 6*x^5)
*Log[5] + (-375*x + 225*x^2 - 45*x^3 + 3*x^4)*Log[5]^2 + E^x*(-150*x^2 - 90*x^3 - 6*x^4 + 6*x^5 + (150*x^2 - 6
0*x^3 + 6*x^4)*Log[5]))*Log[x] + (375*x + 150*x^2 - 30*x^3 - 18*x^4 + 3*x^5 + E^x*(75*x^2 - 30*x^3 + 3*x^4) +
(-375*x + 225*x^2 - 45*x^3 + 3*x^4)*Log[5])*Log[x]^2 + (-125*x + 75*x^2 - 15*x^3 + x^4)*Log[x]^3),x]

[Out]

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

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fricas [B]  time = 0.61, size = 145, normalized size = 5.00 \begin {gather*} \frac {5 \, {\left (x^{2} - 10 \, x + 25\right )}}{x^{4} + 8 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + {\left (x^{2} - 10 \, x + 25\right )} \log \relax (5)^{2} + {\left (x^{2} - 10 \, x + 25\right )} \log \relax (x)^{2} + 26 \, x^{2} + 2 \, {\left (x^{3} + 4 \, x^{2} + {\left (x^{2} - 5 \, x\right )} \log \relax (5) + 5 \, x\right )} e^{x} + 2 \, {\left (x^{3} - x^{2} - 15 \, x - 25\right )} \log \relax (5) + 2 \, {\left (x^{3} - x^{2} + {\left (x^{2} - 5 \, x\right )} e^{x} + {\left (x^{2} - 10 \, x + 25\right )} \log \relax (5) - 15 \, x - 25\right )} \log \relax (x) + 40 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^4+100*x^3-200*x^2-250*x)*exp(x)-10*x^4+140*x^3-100*x^2-2000*x+1250)/((x^4-15*x^3+75*x^2-125*
x)*log(x)^3+((3*x^4-30*x^3+75*x^2)*exp(x)+(3*x^4-45*x^3+225*x^2-375*x)*log(5)+3*x^5-18*x^4-30*x^3+150*x^2+375*
x)*log(x)^2+((3*x^4-15*x^3)*exp(x)^2+((6*x^4-60*x^3+150*x^2)*log(5)+6*x^5-6*x^4-90*x^3-150*x^2)*exp(x)+(3*x^4-
45*x^3+225*x^2-375*x)*log(5)^2+(6*x^5-36*x^4-60*x^3+300*x^2+750*x)*log(5)+3*x^6+9*x^5-42*x^4-270*x^3-525*x^2-3
75*x)*log(x)+x^4*exp(x)^3+((3*x^4-15*x^3)*log(5)+3*x^5+12*x^4+15*x^3)*exp(x)^2+((3*x^4-30*x^3+75*x^2)*log(5)^2
+(6*x^5-6*x^4-90*x^3-150*x^2)*log(5)+3*x^6+24*x^5+78*x^4+120*x^3+75*x^2)*exp(x)+(x^4-15*x^3+75*x^2-125*x)*log(
5)^3+(3*x^5-18*x^4-30*x^3+150*x^2+375*x)*log(5)^2+(3*x^6+9*x^5-42*x^4-270*x^3-525*x^2-375*x)*log(5)+x^7+12*x^6
+63*x^5+184*x^4+315*x^3+300*x^2+125*x),x, algorithm="fricas")

[Out]

5*(x^2 - 10*x + 25)/(x^4 + 8*x^3 + x^2*e^(2*x) + (x^2 - 10*x + 25)*log(5)^2 + (x^2 - 10*x + 25)*log(x)^2 + 26*
x^2 + 2*(x^3 + 4*x^2 + (x^2 - 5*x)*log(5) + 5*x)*e^x + 2*(x^3 - x^2 - 15*x - 25)*log(5) + 2*(x^3 - x^2 + (x^2
- 5*x)*e^x + (x^2 - 10*x + 25)*log(5) - 15*x - 25)*log(x) + 40*x + 25)

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giac [B]  time = 0.40, size = 199, normalized size = 6.86 \begin {gather*} \frac {5 \, {\left (x^{2} - 10 \, x + 25\right )}}{x^{4} + 2 \, x^{3} e^{x} + 2 \, x^{3} \log \relax (5) + 2 \, x^{2} e^{x} \log \relax (5) + x^{2} \log \relax (5)^{2} + 2 \, x^{3} \log \relax (x) + 2 \, x^{2} e^{x} \log \relax (x) + 2 \, x^{2} \log \relax (5) \log \relax (x) + x^{2} \log \relax (x)^{2} + 8 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 8 \, x^{2} e^{x} - 2 \, x^{2} \log \relax (5) - 10 \, x e^{x} \log \relax (5) - 10 \, x \log \relax (5)^{2} - 2 \, x^{2} \log \relax (x) - 10 \, x e^{x} \log \relax (x) - 20 \, x \log \relax (5) \log \relax (x) - 10 \, x \log \relax (x)^{2} + 26 \, x^{2} + 10 \, x e^{x} - 30 \, x \log \relax (5) + 25 \, \log \relax (5)^{2} - 30 \, x \log \relax (x) + 50 \, \log \relax (5) \log \relax (x) + 25 \, \log \relax (x)^{2} + 40 \, x - 50 \, \log \relax (5) - 50 \, \log \relax (x) + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^4+100*x^3-200*x^2-250*x)*exp(x)-10*x^4+140*x^3-100*x^2-2000*x+1250)/((x^4-15*x^3+75*x^2-125*
x)*log(x)^3+((3*x^4-30*x^3+75*x^2)*exp(x)+(3*x^4-45*x^3+225*x^2-375*x)*log(5)+3*x^5-18*x^4-30*x^3+150*x^2+375*
x)*log(x)^2+((3*x^4-15*x^3)*exp(x)^2+((6*x^4-60*x^3+150*x^2)*log(5)+6*x^5-6*x^4-90*x^3-150*x^2)*exp(x)+(3*x^4-
45*x^3+225*x^2-375*x)*log(5)^2+(6*x^5-36*x^4-60*x^3+300*x^2+750*x)*log(5)+3*x^6+9*x^5-42*x^4-270*x^3-525*x^2-3
75*x)*log(x)+x^4*exp(x)^3+((3*x^4-15*x^3)*log(5)+3*x^5+12*x^4+15*x^3)*exp(x)^2+((3*x^4-30*x^3+75*x^2)*log(5)^2
+(6*x^5-6*x^4-90*x^3-150*x^2)*log(5)+3*x^6+24*x^5+78*x^4+120*x^3+75*x^2)*exp(x)+(x^4-15*x^3+75*x^2-125*x)*log(
5)^3+(3*x^5-18*x^4-30*x^3+150*x^2+375*x)*log(5)^2+(3*x^6+9*x^5-42*x^4-270*x^3-525*x^2-375*x)*log(5)+x^7+12*x^6
+63*x^5+184*x^4+315*x^3+300*x^2+125*x),x, algorithm="giac")

[Out]

5*(x^2 - 10*x + 25)/(x^4 + 2*x^3*e^x + 2*x^3*log(5) + 2*x^2*e^x*log(5) + x^2*log(5)^2 + 2*x^3*log(x) + 2*x^2*e
^x*log(x) + 2*x^2*log(5)*log(x) + x^2*log(x)^2 + 8*x^3 + x^2*e^(2*x) + 8*x^2*e^x - 2*x^2*log(5) - 10*x*e^x*log
(5) - 10*x*log(5)^2 - 2*x^2*log(x) - 10*x*e^x*log(x) - 20*x*log(5)*log(x) - 10*x*log(x)^2 + 26*x^2 + 10*x*e^x
- 30*x*log(5) + 25*log(5)^2 - 30*x*log(x) + 50*log(5)*log(x) + 25*log(x)^2 + 40*x - 50*log(5) - 50*log(x) + 25
)

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maple [A]  time = 0.08, size = 38, normalized size = 1.31

method	result	size
risch	\(\frac {5 \left (x -5\right )^{2}}{\left (x \ln \relax (x )+x \ln \relax (5)+{\mathrm e}^{x} x +x^{2}-5 \ln \relax (x )-5 \ln \relax (5)+4 x +5\right )^{2}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-10*x^4+100*x^3-200*x^2-250*x)*exp(x)-10*x^4+140*x^3-100*x^2-2000*x+1250)/((x^4-15*x^3+75*x^2-125*x)*ln(
x)^3+((3*x^4-30*x^3+75*x^2)*exp(x)+(3*x^4-45*x^3+225*x^2-375*x)*ln(5)+3*x^5-18*x^4-30*x^3+150*x^2+375*x)*ln(x)
^2+((3*x^4-15*x^3)*exp(x)^2+((6*x^4-60*x^3+150*x^2)*ln(5)+6*x^5-6*x^4-90*x^3-150*x^2)*exp(x)+(3*x^4-45*x^3+225
*x^2-375*x)*ln(5)^2+(6*x^5-36*x^4-60*x^3+300*x^2+750*x)*ln(5)+3*x^6+9*x^5-42*x^4-270*x^3-525*x^2-375*x)*ln(x)+
x^4*exp(x)^3+((3*x^4-15*x^3)*ln(5)+3*x^5+12*x^4+15*x^3)*exp(x)^2+((3*x^4-30*x^3+75*x^2)*ln(5)^2+(6*x^5-6*x^4-9
0*x^3-150*x^2)*ln(5)+3*x^6+24*x^5+78*x^4+120*x^3+75*x^2)*exp(x)+(x^4-15*x^3+75*x^2-125*x)*ln(5)^3+(3*x^5-18*x^
4-30*x^3+150*x^2+375*x)*ln(5)^2+(3*x^6+9*x^5-42*x^4-270*x^3-525*x^2-375*x)*ln(5)+x^7+12*x^6+63*x^5+184*x^4+315
*x^3+300*x^2+125*x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 1.46, size = 147, normalized size = 5.07 \begin {gather*} \frac {5 \, {\left (x^{2} - 10 \, x + 25\right )}}{x^{4} + 2 \, x^{3} {\left (\log \relax (5) + 4\right )} + {\left (\log \relax (5)^{2} - 2 \, \log \relax (5) + 26\right )} x^{2} + x^{2} e^{\left (2 \, x\right )} + {\left (x^{2} - 10 \, x + 25\right )} \log \relax (x)^{2} - 10 \, {\left (\log \relax (5)^{2} + 3 \, \log \relax (5) - 4\right )} x + 2 \, {\left (x^{3} + x^{2} {\left (\log \relax (5) + 4\right )} - 5 \, x {\left (\log \relax (5) - 1\right )} + {\left (x^{2} - 5 \, x\right )} \log \relax (x)\right )} e^{x} + 25 \, \log \relax (5)^{2} + 2 \, {\left (x^{3} + x^{2} {\left (\log \relax (5) - 1\right )} - 5 \, x {\left (2 \, \log \relax (5) + 3\right )} + 25 \, \log \relax (5) - 25\right )} \log \relax (x) - 50 \, \log \relax (5) + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^4+100*x^3-200*x^2-250*x)*exp(x)-10*x^4+140*x^3-100*x^2-2000*x+1250)/((x^4-15*x^3+75*x^2-125*
x)*log(x)^3+((3*x^4-30*x^3+75*x^2)*exp(x)+(3*x^4-45*x^3+225*x^2-375*x)*log(5)+3*x^5-18*x^4-30*x^3+150*x^2+375*
x)*log(x)^2+((3*x^4-15*x^3)*exp(x)^2+((6*x^4-60*x^3+150*x^2)*log(5)+6*x^5-6*x^4-90*x^3-150*x^2)*exp(x)+(3*x^4-
45*x^3+225*x^2-375*x)*log(5)^2+(6*x^5-36*x^4-60*x^3+300*x^2+750*x)*log(5)+3*x^6+9*x^5-42*x^4-270*x^3-525*x^2-3
75*x)*log(x)+x^4*exp(x)^3+((3*x^4-15*x^3)*log(5)+3*x^5+12*x^4+15*x^3)*exp(x)^2+((3*x^4-30*x^3+75*x^2)*log(5)^2
+(6*x^5-6*x^4-90*x^3-150*x^2)*log(5)+3*x^6+24*x^5+78*x^4+120*x^3+75*x^2)*exp(x)+(x^4-15*x^3+75*x^2-125*x)*log(
5)^3+(3*x^5-18*x^4-30*x^3+150*x^2+375*x)*log(5)^2+(3*x^6+9*x^5-42*x^4-270*x^3-525*x^2-375*x)*log(5)+x^7+12*x^6
+63*x^5+184*x^4+315*x^3+300*x^2+125*x),x, algorithm="maxima")

[Out]

5*(x^2 - 10*x + 25)/(x^4 + 2*x^3*(log(5) + 4) + (log(5)^2 - 2*log(5) + 26)*x^2 + x^2*e^(2*x) + (x^2 - 10*x + 2
5)*log(x)^2 - 10*(log(5)^2 + 3*log(5) - 4)*x + 2*(x^3 + x^2*(log(5) + 4) - 5*x*(log(5) - 1) + (x^2 - 5*x)*log(
x))*e^x + 25*log(5)^2 + 2*(x^3 + x^2*(log(5) - 1) - 5*x*(2*log(5) + 3) + 25*log(5) - 25)*log(x) - 50*log(5) +
25)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {2000\,x+{\mathrm {e}}^x\,\left (10\,x^4-100\,x^3+200\,x^2+250\,x\right )+100\,x^2-140\,x^3+10\,x^4-1250}{125\,x-{\ln \relax (5)}^3\,\left (-x^4+15\,x^3-75\,x^2+125\,x\right )-\ln \relax (5)\,\left (-3\,x^6-9\,x^5+42\,x^4+270\,x^3+525\,x^2+375\,x\right )-\ln \relax (x)\,\left (375\,x+{\ln \relax (5)}^2\,\left (-3\,x^4+45\,x^3-225\,x^2+375\,x\right )+{\mathrm {e}}^x\,\left (150\,x^2-\ln \relax (5)\,\left (6\,x^4-60\,x^3+150\,x^2\right )+90\,x^3+6\,x^4-6\,x^5\right )+{\mathrm {e}}^{2\,x}\,\left (15\,x^3-3\,x^4\right )+525\,x^2+270\,x^3+42\,x^4-9\,x^5-3\,x^6-\ln \relax (5)\,\left (6\,x^5-36\,x^4-60\,x^3+300\,x^2+750\,x\right )\right )+{\mathrm {e}}^{2\,x}\,\left (15\,x^3-\ln \relax (5)\,\left (15\,x^3-3\,x^4\right )+12\,x^4+3\,x^5\right )+{\ln \relax (5)}^2\,\left (3\,x^5-18\,x^4-30\,x^3+150\,x^2+375\,x\right )+x^4\,{\mathrm {e}}^{3\,x}-{\ln \relax (x)}^3\,\left (-x^4+15\,x^3-75\,x^2+125\,x\right )+{\ln \relax (x)}^2\,\left (375\,x+{\mathrm {e}}^x\,\left (3\,x^4-30\,x^3+75\,x^2\right )-\ln \relax (5)\,\left (-3\,x^4+45\,x^3-225\,x^2+375\,x\right )+150\,x^2-30\,x^3-18\,x^4+3\,x^5\right )+300\,x^2+315\,x^3+184\,x^4+63\,x^5+12\,x^6+x^7+{\mathrm {e}}^x\,\left ({\ln \relax (5)}^2\,\left (3\,x^4-30\,x^3+75\,x^2\right )-\ln \relax (5)\,\left (-6\,x^5+6\,x^4+90\,x^3+150\,x^2\right )+75\,x^2+120\,x^3+78\,x^4+24\,x^5+3\,x^6\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2000*x + exp(x)*(250*x + 200*x^2 - 100*x^3 + 10*x^4) + 100*x^2 - 140*x^3 + 10*x^4 - 1250)/(125*x - log(5
)^3*(125*x - 75*x^2 + 15*x^3 - x^4) - log(5)*(375*x + 525*x^2 + 270*x^3 + 42*x^4 - 9*x^5 - 3*x^6) - log(x)*(37
5*x + log(5)^2*(375*x - 225*x^2 + 45*x^3 - 3*x^4) + exp(x)*(150*x^2 - log(5)*(150*x^2 - 60*x^3 + 6*x^4) + 90*x
^3 + 6*x^4 - 6*x^5) + exp(2*x)*(15*x^3 - 3*x^4) + 525*x^2 + 270*x^3 + 42*x^4 - 9*x^5 - 3*x^6 - log(5)*(750*x +
300*x^2 - 60*x^3 - 36*x^4 + 6*x^5)) + exp(2*x)*(15*x^3 - log(5)*(15*x^3 - 3*x^4) + 12*x^4 + 3*x^5) + log(5)^2
*(375*x + 150*x^2 - 30*x^3 - 18*x^4 + 3*x^5) + x^4*exp(3*x) - log(x)^3*(125*x - 75*x^2 + 15*x^3 - x^4) + log(x
)^2*(375*x + exp(x)*(75*x^2 - 30*x^3 + 3*x^4) - log(5)*(375*x - 225*x^2 + 45*x^3 - 3*x^4) + 150*x^2 - 30*x^3 -
18*x^4 + 3*x^5) + 300*x^2 + 315*x^3 + 184*x^4 + 63*x^5 + 12*x^6 + x^7 + exp(x)*(log(5)^2*(75*x^2 - 30*x^3 + 3
*x^4) - log(5)*(150*x^2 + 90*x^3 + 6*x^4 - 6*x^5) + 75*x^2 + 120*x^3 + 78*x^4 + 24*x^5 + 3*x^6)),x)

[Out]

int(-(2000*x + exp(x)*(250*x + 200*x^2 - 100*x^3 + 10*x^4) + 100*x^2 - 140*x^3 + 10*x^4 - 1250)/(125*x - log(5
)^3*(125*x - 75*x^2 + 15*x^3 - x^4) - log(5)*(375*x + 525*x^2 + 270*x^3 + 42*x^4 - 9*x^5 - 3*x^6) - log(x)*(37
5*x + log(5)^2*(375*x - 225*x^2 + 45*x^3 - 3*x^4) + exp(x)*(150*x^2 - log(5)*(150*x^2 - 60*x^3 + 6*x^4) + 90*x
^3 + 6*x^4 - 6*x^5) + exp(2*x)*(15*x^3 - 3*x^4) + 525*x^2 + 270*x^3 + 42*x^4 - 9*x^5 - 3*x^6 - log(5)*(750*x +
300*x^2 - 60*x^3 - 36*x^4 + 6*x^5)) + exp(2*x)*(15*x^3 - log(5)*(15*x^3 - 3*x^4) + 12*x^4 + 3*x^5) + log(5)^2
*(375*x + 150*x^2 - 30*x^3 - 18*x^4 + 3*x^5) + x^4*exp(3*x) - log(x)^3*(125*x - 75*x^2 + 15*x^3 - x^4) + log(x
)^2*(375*x + exp(x)*(75*x^2 - 30*x^3 + 3*x^4) - log(5)*(375*x - 225*x^2 + 45*x^3 - 3*x^4) + 150*x^2 - 30*x^3 -
18*x^4 + 3*x^5) + 300*x^2 + 315*x^3 + 184*x^4 + 63*x^5 + 12*x^6 + x^7 + exp(x)*(log(5)^2*(75*x^2 - 30*x^3 + 3
*x^4) - log(5)*(150*x^2 + 90*x^3 + 6*x^4 - 6*x^5) + 75*x^2 + 120*x^3 + 78*x^4 + 24*x^5 + 3*x^6)), x)

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sympy [B]  time = 1.41, size = 219, normalized size = 7.55 \begin {gather*} \frac {5 x^{2} - 50 x + 125}{x^{4} + 2 x^{3} \log {\relax (x )} + 2 x^{3} \log {\relax (5 )} + 8 x^{3} + x^{2} e^{2 x} + x^{2} \log {\relax (x )}^{2} - 2 x^{2} \log {\relax (x )} + 2 x^{2} \log {\relax (5 )} \log {\relax (x )} - 2 x^{2} \log {\relax (5 )} + x^{2} \log {\relax (5 )}^{2} + 26 x^{2} - 10 x \log {\relax (x )}^{2} - 20 x \log {\relax (5 )} \log {\relax (x )} - 30 x \log {\relax (x )} - 30 x \log {\relax (5 )} - 10 x \log {\relax (5 )}^{2} + 40 x + \left (2 x^{3} + 2 x^{2} \log {\relax (x )} + 2 x^{2} \log {\relax (5 )} + 8 x^{2} - 10 x \log {\relax (x )} - 10 x \log {\relax (5 )} + 10 x\right ) e^{x} + 25 \log {\relax (x )}^{2} - 50 \log {\relax (x )} + 50 \log {\relax (5 )} \log {\relax (x )} - 50 \log {\relax (5 )} + 25 + 25 \log {\relax (5 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x**4+100*x**3-200*x**2-250*x)*exp(x)-10*x**4+140*x**3-100*x**2-2000*x+1250)/((x**4-15*x**3+75*
x**2-125*x)*ln(x)**3+((3*x**4-30*x**3+75*x**2)*exp(x)+(3*x**4-45*x**3+225*x**2-375*x)*ln(5)+3*x**5-18*x**4-30*
x**3+150*x**2+375*x)*ln(x)**2+((3*x**4-15*x**3)*exp(x)**2+((6*x**4-60*x**3+150*x**2)*ln(5)+6*x**5-6*x**4-90*x*
*3-150*x**2)*exp(x)+(3*x**4-45*x**3+225*x**2-375*x)*ln(5)**2+(6*x**5-36*x**4-60*x**3+300*x**2+750*x)*ln(5)+3*x
**6+9*x**5-42*x**4-270*x**3-525*x**2-375*x)*ln(x)+x**4*exp(x)**3+((3*x**4-15*x**3)*ln(5)+3*x**5+12*x**4+15*x**
3)*exp(x)**2+((3*x**4-30*x**3+75*x**2)*ln(5)**2+(6*x**5-6*x**4-90*x**3-150*x**2)*ln(5)+3*x**6+24*x**5+78*x**4+
120*x**3+75*x**2)*exp(x)+(x**4-15*x**3+75*x**2-125*x)*ln(5)**3+(3*x**5-18*x**4-30*x**3+150*x**2+375*x)*ln(5)**
2+(3*x**6+9*x**5-42*x**4-270*x**3-525*x**2-375*x)*ln(5)+x**7+12*x**6+63*x**5+184*x**4+315*x**3+300*x**2+125*x)
,x)

[Out]

(5*x**2 - 50*x + 125)/(x**4 + 2*x**3*log(x) + 2*x**3*log(5) + 8*x**3 + x**2*exp(2*x) + x**2*log(x)**2 - 2*x**2
*log(x) + 2*x**2*log(5)*log(x) - 2*x**2*log(5) + x**2*log(5)**2 + 26*x**2 - 10*x*log(x)**2 - 20*x*log(5)*log(x
) - 30*x*log(x) - 30*x*log(5) - 10*x*log(5)**2 + 40*x + (2*x**3 + 2*x**2*log(x) + 2*x**2*log(5) + 8*x**2 - 10*
x*log(x) - 10*x*log(5) + 10*x)*exp(x) + 25*log(x)**2 - 50*log(x) + 50*log(5)*log(x) - 50*log(5) + 25 + 25*log(
5)**2)

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