3.28.35 \(\int \frac {400+800 x^2+400 x^4+e^{2 x+2 x \log (4)} (-2+2 x+x^2)+e^{x+x \log (4)} (100-80 x+100 x^2+40 x^3+(60+60 x^2) \log (4))}{400+800 x^2+400 x^4+e^{2 x+2 x \log (4)} (1+2 x+x^2)+e^{x+x \log (4)} (40+40 x+40 x^2+40 x^3)} \, dx\)

Optimal. Leaf size=28 \[ 3+x+\frac {3}{1+x+20 e^{-x-x \log (4)} \left (1+x^2\right )} \]

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Rubi [F]  time = 37.31, 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 {400+800 x^2+400 x^4+e^{2 x+2 x \log (4)} \left (-2+2 x+x^2\right )+e^{x+x \log (4)} \left (100-80 x+100 x^2+40 x^3+\left (60+60 x^2\right ) \log (4)\right )}{400+800 x^2+400 x^4+e^{2 x+2 x \log (4)} \left (1+2 x+x^2\right )+e^{x+x \log (4)} \left (40+40 x+40 x^2+40 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(400 + 800*x^2 + 400*x^4 + E^(2*x + 2*x*Log[4])*(-2 + 2*x + x^2) + E^(x + x*Log[4])*(100 - 80*x + 100*x^2
+ 40*x^3 + (60 + 60*x^2)*Log[4]))/(400 + 800*x^2 + 400*x^4 + E^(2*x + 2*x*Log[4])*(1 + 2*x + x^2) + E^(x + x*L
og[4])*(40 + 40*x + 40*x^2 + 40*x^3)),x]

[Out]

x - (1 + x)^(-1) - 2*Log[1 + x] + 5200*Defer[Int][(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^(-2), x] + 2*Defer[Int][
E^(2*x*(1 + Log[4]))/(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2, x] - 4*Defer[Int][E^(x*(2 + Log[16]))/(20 + (4*E)^
x + (4*E)^x*x + 20*x^2)^2, x] - 160*Defer[Int][(4*E)^x/(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2, x] + 40*(1 + Log
[64])*Defer[Int][(4*E)^x/(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2, x] - 80*(4 + Log[64])*Defer[Int][(4*E)^x/(20 +
(4*E)^x + (4*E)^x*x + 20*x^2)^2, x] + 60*(9 + Log[64])*Defer[Int][(4*E)^x/(20 + (4*E)^x + (4*E)^x*x + 20*x^2)
^2, x] - 3200*Defer[Int][x/(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2, x] + 2*Defer[Int][(E^(x*(2 + Log[16]))*x)/(2
0 + (4*E)^x + (4*E)^x*x + 20*x^2)^2, x] + 120*Defer[Int][((4*E)^x*x)/(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2, x]
+ 40*(4 + Log[64])*Defer[Int][((4*E)^x*x)/(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2, x] - 40*(9 + Log[64])*Defer[
Int][((4*E)^x*x)/(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2, x] + 2800*Defer[Int][x^2/(20 + (4*E)^x + (4*E)^x*x + 2
0*x^2)^2, x] - 80*Defer[Int][((4*E)^x*x^2)/(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2, x] + 20*(9 + Log[64])*Defer[
Int][((4*E)^x*x^2)/(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2, x] - 800*Defer[Int][x^3/(20 + (4*E)^x + (4*E)^x*x +
20*x^2)^2, x] + 40*Defer[Int][((4*E)^x*x^3)/(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2, x] + 800*Defer[Int][x^4/(20
+ (4*E)^x + (4*E)^x*x + 20*x^2)^2, x] + 1600*Defer[Int][1/((1 + x)^2*(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2),
x] + 4*Defer[Int][E^(2*x*(1 + Log[4]))/((1 + x)^2*(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2), x] - 4*Defer[Int][E^
(x*(2 + Log[16]))/((1 + x)^2*(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2), x] - 40*Defer[Int][(4*E)^x/((1 + x)^2*(20
+ (4*E)^x + (4*E)^x*x + 20*x^2)^2), x] - 120*(1 + Log[4])*Defer[Int][(4*E)^x/((1 + x)^2*(20 + (4*E)^x + (4*E)
^x*x + 20*x^2)^2), x] + 40*(1 + Log[64])*Defer[Int][(4*E)^x/((1 + x)^2*(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2),
x] - 40*(4 + Log[64])*Defer[Int][(4*E)^x/((1 + x)^2*(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2), x] + 20*(5 + Log[
64])*Defer[Int][(4*E)^x/((1 + x)^2*(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2), x] + 20*(9 + Log[64])*Defer[Int][(4
*E)^x/((1 + x)^2*(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2), x] - 6400*Defer[Int][1/((1 + x)*(20 + (4*E)^x + (4*E)
^x*x + 20*x^2)^2), x] - 6*Defer[Int][E^(2*x*(1 + Log[4]))/((1 + x)*(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2), x]
+ 6*Defer[Int][E^(x*(2 + Log[16]))/((1 + x)*(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2), x] + 200*Defer[Int][(4*E)^
x/((1 + x)*(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2), x] + 120*(1 + Log[4])*Defer[Int][(4*E)^x/((1 + x)*(20 + (4*
E)^x + (4*E)^x*x + 20*x^2)^2), x] - 80*(1 + Log[64])*Defer[Int][(4*E)^x/((1 + x)*(20 + (4*E)^x + (4*E)^x*x + 2
0*x^2)^2), x] + 120*(4 + Log[64])*Defer[Int][(4*E)^x/((1 + x)*(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2), x] - 80*
(9 + Log[64])*Defer[Int][(4*E)^x/((1 + x)*(20 + (4*E)^x + (4*E)^x*x + 20*x^2)^2), x] - 160*Defer[Int][(20 + (4
*E)^x + (4*E)^x*x + 20*x^2)^(-1), x] + 80*Defer[Int][x/(20 + (4*E)^x + (4*E)^x*x + 20*x^2), x] - 40*Defer[Int]
[x^2/(20 + (4*E)^x + (4*E)^x*x + 20*x^2), x] - 80*Defer[Int][1/((1 + x)^2*(20 + (4*E)^x + (4*E)^x*x + 20*x^2))
, x] + 240*Defer[Int][1/((1 + x)*(20 + (4*E)^x + (4*E)^x*x + 20*x^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {400+2^{1+4 x} e^{2 x} (-1+x)+\left (800+(4 e)^{2 x}\right ) x^2+400 x^4+5\ 4^{1+x} e^x \left (5-4 x+2 x^3+\log (64)+x^2 (5+\log (64))\right )}{\left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx\\ &=\int \left (\frac {x^2}{(1+x)^2}-\frac {40 x^2 \left (1+x^2\right )}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )}+\frac {2 \left (200-2^{4 x} e^{2 x}+400 x-2^{4 x} e^{2 x} x+800 x^2+2^{4 x} e^{2 x} x^2+800 x^3+2^{4 x} e^{2 x} x^3+1000 x^4+400 x^5+5\ 2^{2+2 x} e^x x^5+400 x^6+45\ 2^{1+2 x} e^x x^4 \left (1+\frac {2 \log (2)}{3}\right )+15\ 2^{2+2 x} e^x x (1+\log (4))+25\ 2^{1+2 x} e^x \left (1+\frac {\log (64)}{5}\right )+5\ 2^{4+2 x} e^x x^3 \left (1+\frac {\log (64)}{4}\right )+5\ 2^{2+2 x} e^x x^2 (1+\log (64))\right )}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}\right ) \, dx\\ &=2 \int \frac {200-2^{4 x} e^{2 x}+400 x-2^{4 x} e^{2 x} x+800 x^2+2^{4 x} e^{2 x} x^2+800 x^3+2^{4 x} e^{2 x} x^3+1000 x^4+400 x^5+5\ 2^{2+2 x} e^x x^5+400 x^6+45\ 2^{1+2 x} e^x x^4 \left (1+\frac {2 \log (2)}{3}\right )+15\ 2^{2+2 x} e^x x (1+\log (4))+25\ 2^{1+2 x} e^x \left (1+\frac {\log (64)}{5}\right )+5\ 2^{4+2 x} e^x x^3 \left (1+\frac {\log (64)}{4}\right )+5\ 2^{2+2 x} e^x x^2 (1+\log (64))}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx-40 \int \frac {x^2 \left (1+x^2\right )}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )} \, dx+\int \frac {x^2}{(1+x)^2} \, dx\\ &=2 \int \frac {16^x e^{2 x} (-1+x) (1+x)^2+200 \left (1+x^2\right )^2 \left (1+2 x+2 x^2\right )+5\ 2^{1+2 x} e^x \left (5+2 x^5+6 x (1+\log (4))+\log (64)+2 x^2 (1+\log (64))+2 x^3 (4+\log (64))+x^4 (9+\log (64))\right )}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx-40 \int \left (\frac {4}{20+(4 e)^x+(4 e)^x x+20 x^2}-\frac {2 x}{20+(4 e)^x+(4 e)^x x+20 x^2}+\frac {x^2}{20+(4 e)^x+(4 e)^x x+20 x^2}+\frac {2}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )}-\frac {6}{(1+x) \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )}\right ) \, dx+\int \left (1+\frac {1}{(1+x)^2}-\frac {2}{1+x}\right ) \, dx\\ &=x-\frac {1}{1+x}-2 \log (1+x)+2 \int \left (\frac {200}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}-\frac {16^x e^{2 x}}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {400 x}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}-\frac {2^{4 x} e^{2 x} x}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {800 x^2}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {2^{4 x} e^{2 x} x^2}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {800 x^3}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {16^x e^{2 x} x^3}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {1000 x^4}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {400 x^5}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {5\ 4^{1+x} e^x x^5}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {400 x^6}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {15\ 4^{1+x} e^x x (1+\log (4))}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {25\ 2^{1+2 x} e^x \left (1+\frac {\log (64)}{5}\right )}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {5\ 4^{1+x} e^x x^2 (1+\log (64))}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {5\ 4^{1+x} e^x x^3 (4+\log (64))}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {5\ 2^{1+2 x} e^x x^4 (9+\log (64))}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}\right ) \, dx-40 \int \frac {x^2}{20+(4 e)^x+(4 e)^x x+20 x^2} \, dx+80 \int \frac {x}{20+(4 e)^x+(4 e)^x x+20 x^2} \, dx-80 \int \frac {1}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )} \, dx-160 \int \frac {1}{20+(4 e)^x+(4 e)^x x+20 x^2} \, dx+240 \int \frac {1}{(1+x) \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )} \, dx\\ &=x-\frac {1}{1+x}-2 \log (1+x)-2 \int \frac {16^x e^{2 x}}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx-2 \int \frac {2^{4 x} e^{2 x} x}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+2 \int \frac {2^{4 x} e^{2 x} x^2}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+2 \int \frac {16^x e^{2 x} x^3}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+10 \int \frac {4^{1+x} e^x x^5}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx-40 \int \frac {x^2}{20+(4 e)^x+(4 e)^x x+20 x^2} \, dx+80 \int \frac {x}{20+(4 e)^x+(4 e)^x x+20 x^2} \, dx-80 \int \frac {1}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )} \, dx-160 \int \frac {1}{20+(4 e)^x+(4 e)^x x+20 x^2} \, dx+240 \int \frac {1}{(1+x) \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )} \, dx+400 \int \frac {1}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+800 \int \frac {x}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+800 \int \frac {x^5}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+800 \int \frac {x^6}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+1600 \int \frac {x^2}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+1600 \int \frac {x^3}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+2000 \int \frac {x^4}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+(30 (1+\log (4))) \int \frac {4^{1+x} e^x x}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+(10 (1+\log (64))) \int \frac {4^{1+x} e^x x^2}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+(10 (4+\log (64))) \int \frac {4^{1+x} e^x x^3}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+(10 (5+\log (64))) \int \frac {2^{1+2 x} e^x}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+(10 (9+\log (64))) \int \frac {2^{1+2 x} e^x x^4}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 7.71, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {400+800 x^2+400 x^4+e^{2 x+2 x \log (4)} \left (-2+2 x+x^2\right )+e^{x+x \log (4)} \left (100-80 x+100 x^2+40 x^3+\left (60+60 x^2\right ) \log (4)\right )}{400+800 x^2+400 x^4+e^{2 x+2 x \log (4)} \left (1+2 x+x^2\right )+e^{x+x \log (4)} \left (40+40 x+40 x^2+40 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(400 + 800*x^2 + 400*x^4 + E^(2*x + 2*x*Log[4])*(-2 + 2*x + x^2) + E^(x + x*Log[4])*(100 - 80*x + 10
0*x^2 + 40*x^3 + (60 + 60*x^2)*Log[4]))/(400 + 800*x^2 + 400*x^4 + E^(2*x + 2*x*Log[4])*(1 + 2*x + x^2) + E^(x
+ x*Log[4])*(40 + 40*x + 40*x^2 + 40*x^3)),x]

[Out]

Integrate[(400 + 800*x^2 + 400*x^4 + E^(2*x + 2*x*Log[4])*(-2 + 2*x + x^2) + E^(x + x*Log[4])*(100 - 80*x + 10
0*x^2 + 40*x^3 + (60 + 60*x^2)*Log[4]))/(400 + 800*x^2 + 400*x^4 + E^(2*x + 2*x*Log[4])*(1 + 2*x + x^2) + E^(x
+ x*Log[4])*(40 + 40*x + 40*x^2 + 40*x^3)), x]

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fricas [A]  time = 0.55, size = 46, normalized size = 1.64 \begin {gather*} \frac {20 \, x^{3} + {\left (x^{2} + x + 3\right )} e^{\left (2 \, x \log \relax (2) + x\right )} + 20 \, x}{20 \, x^{2} + {\left (x + 1\right )} e^{\left (2 \, x \log \relax (2) + x\right )} + 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+2*x-2)*exp(x+2*x*log(2))^2+(2*(60*x^2+60)*log(2)+40*x^3+100*x^2-80*x+100)*exp(x+2*x*log(2))+40
0*x^4+800*x^2+400)/((x^2+2*x+1)*exp(x+2*x*log(2))^2+(40*x^3+40*x^2+40*x+40)*exp(x+2*x*log(2))+400*x^4+800*x^2+
400),x, algorithm="fricas")

[Out]

(20*x^3 + (x^2 + x + 3)*e^(2*x*log(2) + x) + 20*x)/(20*x^2 + (x + 1)*e^(2*x*log(2) + x) + 20)

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giac [B]  time = 0.85, size = 69, normalized size = 2.46 \begin {gather*} \frac {20 \, x^{3} + x^{2} e^{\left (2 \, x \log \relax (2) + x\right )} + x e^{\left (2 \, x \log \relax (2) + x\right )} + 20 \, x + 3 \, e^{\left (2 \, x \log \relax (2) + x\right )}}{20 \, x^{2} + x e^{\left (2 \, x \log \relax (2) + x\right )} + e^{\left (2 \, x \log \relax (2) + x\right )} + 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+2*x-2)*exp(x+2*x*log(2))^2+(2*(60*x^2+60)*log(2)+40*x^3+100*x^2-80*x+100)*exp(x+2*x*log(2))+40
0*x^4+800*x^2+400)/((x^2+2*x+1)*exp(x+2*x*log(2))^2+(40*x^3+40*x^2+40*x+40)*exp(x+2*x*log(2))+400*x^4+800*x^2+
400),x, algorithm="giac")

[Out]

(20*x^3 + x^2*e^(2*x*log(2) + x) + x*e^(2*x*log(2) + x) + 20*x + 3*e^(2*x*log(2) + x))/(20*x^2 + x*e^(2*x*log(
2) + x) + e^(2*x*log(2) + x) + 20)

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maple [A]  time = 0.60, size = 44, normalized size = 1.57

method	result	size
risch	\(x +\frac {3}{x +1}-\frac {60 \left (x^{2}+1\right )}{\left (x +1\right ) \left ({\mathrm e}^{x} 4^{x} x +20 x^{2}+{\mathrm e}^{x} 4^{x}+20\right )}\)	\(44\)
norman	\(\frac {-20 x^{2}+2 \,{\mathrm e}^{x +2 x \ln \relax (2)}+{\mathrm e}^{x +2 x \ln \relax (2)} x^{2}+20 x +20 x^{3}-20}{{\mathrm e}^{x +2 x \ln \relax (2)} x +20 x^{2}+{\mathrm e}^{x +2 x \ln \relax (2)}+20}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2+2*x-2)*exp(x+2*x*ln(2))^2+(2*(60*x^2+60)*ln(2)+40*x^3+100*x^2-80*x+100)*exp(x+2*x*ln(2))+400*x^4+800
*x^2+400)/((x^2+2*x+1)*exp(x+2*x*ln(2))^2+(40*x^3+40*x^2+40*x+40)*exp(x+2*x*ln(2))+400*x^4+800*x^2+400),x,meth
od=_RETURNVERBOSE)

[Out]

x+3/(x+1)-60*(x^2+1)/(x+1)/(exp(x)*4^x*x+20*x^2+exp(x)*4^x+20)

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maxima [A]  time = 1.17, size = 46, normalized size = 1.64 \begin {gather*} \frac {20 \, x^{3} + {\left (x^{2} + x + 3\right )} e^{\left (2 \, x \log \relax (2) + x\right )} + 20 \, x}{20 \, x^{2} + {\left (x + 1\right )} e^{\left (2 \, x \log \relax (2) + x\right )} + 20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+2*x-2)*exp(x+2*x*log(2))^2+(2*(60*x^2+60)*log(2)+40*x^3+100*x^2-80*x+100)*exp(x+2*x*log(2))+40
0*x^4+800*x^2+400)/((x^2+2*x+1)*exp(x+2*x*log(2))^2+(40*x^3+40*x^2+40*x+40)*exp(x+2*x*log(2))+400*x^4+800*x^2+
400),x, algorithm="maxima")

[Out]

(20*x^3 + (x^2 + x + 3)*e^(2*x*log(2) + x) + 20*x)/(20*x^2 + (x + 1)*e^(2*x*log(2) + x) + 20)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{2\,x+4\,x\,\ln \relax (2)}\,\left (x^2+2\,x-2\right )+800\,x^2+400\,x^4+{\mathrm {e}}^{x+2\,x\,\ln \relax (2)}\,\left (2\,\ln \relax (2)\,\left (60\,x^2+60\right )-80\,x+100\,x^2+40\,x^3+100\right )+400}{{\mathrm {e}}^{2\,x+4\,x\,\ln \relax (2)}\,\left (x^2+2\,x+1\right )+{\mathrm {e}}^{x+2\,x\,\ln \relax (2)}\,\left (40\,x^3+40\,x^2+40\,x+40\right )+800\,x^2+400\,x^4+400} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x + 4*x*log(2))*(2*x + x^2 - 2) + 800*x^2 + 400*x^4 + exp(x + 2*x*log(2))*(2*log(2)*(60*x^2 + 60) -
80*x + 100*x^2 + 40*x^3 + 100) + 400)/(exp(2*x + 4*x*log(2))*(2*x + x^2 + 1) + exp(x + 2*x*log(2))*(40*x + 40
*x^2 + 40*x^3 + 40) + 800*x^2 + 400*x^4 + 400),x)

[Out]

int((exp(2*x + 4*x*log(2))*(2*x + x^2 - 2) + 800*x^2 + 400*x^4 + exp(x + 2*x*log(2))*(2*log(2)*(60*x^2 + 60) -
80*x + 100*x^2 + 40*x^3 + 100) + 400)/(exp(2*x + 4*x*log(2))*(2*x + x^2 + 1) + exp(x + 2*x*log(2))*(40*x + 40
*x^2 + 40*x^3 + 40) + 800*x^2 + 400*x^4 + 400), x)

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sympy [A]  time = 0.33, size = 48, normalized size = 1.71 \begin {gather*} x + \frac {- 60 x^{2} - 60}{20 x^{3} + 20 x^{2} + 20 x + \left (x^{2} + 2 x + 1\right ) e^{x + 2 x \log {\relax (2 )}} + 20} + \frac {3}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2+2*x-2)*exp(x+2*x*ln(2))**2+(2*(60*x**2+60)*ln(2)+40*x**3+100*x**2-80*x+100)*exp(x+2*x*ln(2))+
400*x**4+800*x**2+400)/((x**2+2*x+1)*exp(x+2*x*ln(2))**2+(40*x**3+40*x**2+40*x+40)*exp(x+2*x*ln(2))+400*x**4+8
00*x**2+400),x)

[Out]

x + (-60*x**2 - 60)/(20*x**3 + 20*x**2 + 20*x + (x**2 + 2*x + 1)*exp(x + 2*x*log(2)) + 20) + 3/(x + 1)

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