∫ 800+50e−4x+2x2 +400x+50x2+e−2x+x2 (400+100x)+e1+4x+e−2x+x2 x+x2 _____________________________ 4+e−2x+x2+x (−75x2−50e−2x+x2 x2−5e−4x+2x2 x2−40x3−5x4)_______________________________ 1600+100e−4x+2x2+800x+100x2+e−2x+x2(800+200x)+e1+4x+e−2x+x2x+x2 _____________________________ 4+e−2x+x2+x (320x+20e−4x+2x2x+160x2+20x3+e−2x+x2(160x+40x2))+e2(1+4x+e−2x+x2x+x2) _________________________________ 4+e−2x+x2+x (16x2+e−4x+2x2x2+8x3+x4+e−2x+x2(8x2+2x3)) dx

3.10.28 $\int \frac {800+50 e^{-4 x+2 x^2}+400 x+50 x^2+e^{-2 x+x^2} (400+100 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} (-75 x^2-50 e^{-2 x+x^2} x^2-5 e^{-4 x+2 x^2} x^2-40 x^3-5 x^4)}{1600+100 e^{-4 x+2 x^2}+800 x+100 x^2+e^{-2 x+x^2} (800+200 x)+e^{\frac {1+4 x+e^{-2 x+x^2} x+x^2}{4+e^{-2 x+x^2}+x}} (320 x+20 e^{-4 x+2 x^2} x+160 x^2+20 x^3+e^{-2 x+x^2} (160 x+40 x^2))+e^{\frac {2 (1+4 x+e^{-2 x+x^2} x+x^2)}{4+e^{-2 x+x^2}+x}} (16 x^2+e^{-4 x+2 x^2} x^2+8 x^3+x^4+e^{-2 x+x^2} (8 x^2+2 x^3))} \, dx$

Optimal. Leaf size=27 \[ \frac {x}{2+\frac {1}{5} e^{x+\frac {1}{4+e^{(-2+x) x}+x}} x} \]

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Rubi [F] time = 180.00, 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*} \text {\$Aborted} \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(800 + 50*E^(-4*x + 2*x^2) + 400*x + 50*x^2 + E^(-2*x + x^2)*(400 + 100*x) + E^((1 + 4*x + E^(-2*x + x^2)*

x + x^2)/(4 + E^(-2*x + x^2) + x))*(-75*x^2 - 50*E^(-2*x + x^2)*x^2 - 5*E^(-4*x + 2*x^2)*x^2 - 40*x^3 - 5*x^4)

)/(1600 + 100*E^(-4*x + 2*x^2) + 800*x + 100*x^2 + E^(-2*x + x^2)*(800 + 200*x) + E^((1 + 4*x + E^(-2*x + x^2)

*x + x^2)/(4 + E^(-2*x + x^2) + x))*(320*x + 20*E^(-4*x + 2*x^2)*x + 160*x^2 + 20*x^3 + E^(-2*x + x^2)*(160*x

+ 40*x^2)) + E^((2*(1 + 4*x + E^(-2*x + x^2)*x + x^2))/(4 + E^(-2*x + x^2) + x))*(16*x^2 + E^(-4*x + 2*x^2)*x^

2 + 8*x^3 + x^4 + E^(-2*x + x^2)*(8*x^2 + 2*x^3))),x]

[Out]

$Aborted

Rubi steps

Aborted

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Mathematica [A] time = 2.68, size = 36, normalized size = 1.33 \begin {gather*} \frac {5 x}{10+e^{x+\frac {e^{2 x}}{e^{x^2}+e^{2 x} (4+x)}} x} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(800 + 50*E^(-4*x + 2*x^2) + 400*x + 50*x^2 + E^(-2*x + x^2)*(400 + 100*x) + E^((1 + 4*x + E^(-2*x +

x^2)*x + x^2)/(4 + E^(-2*x + x^2) + x))*(-75*x^2 - 50*E^(-2*x + x^2)*x^2 - 5*E^(-4*x + 2*x^2)*x^2 - 40*x^3 -

5*x^4))/(1600 + 100*E^(-4*x + 2*x^2) + 800*x + 100*x^2 + E^(-2*x + x^2)*(800 + 200*x) + E^((1 + 4*x + E^(-2*x

+ x^2)*x + x^2)/(4 + E^(-2*x + x^2) + x))*(320*x + 20*E^(-4*x + 2*x^2)*x + 160*x^2 + 20*x^3 + E^(-2*x + x^2)*(

160*x + 40*x^2)) + E^((2*(1 + 4*x + E^(-2*x + x^2)*x + x^2))/(4 + E^(-2*x + x^2) + x))*(16*x^2 + E^(-4*x + 2*x

^2)*x^2 + 8*x^3 + x^4 + E^(-2*x + x^2)*(8*x^2 + 2*x^3))),x]

[Out]

(5*x)/(10 + E^(x + E^(2*x)/(E^x^2 + E^(2*x)*(4 + x)))*x)

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fricas [A] time = 0.54, size = 42, normalized size = 1.56 \begin {gather*} \frac {5 \, x}{x e^{\left (\frac {x^{2} + x e^{\left (x^{2} - 2 \, x\right )} + 4 \, x + 1}{x + e^{\left (x^{2} - 2 \, x\right )} + 4}\right )} + 10} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^2*exp(x^2-2*x)^2-50*x^2*exp(x^2-2*x)-5*x^4-40*x^3-75*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp

(x^2-2*x)+4+x))+50*exp(x^2-2*x)^2+(100*x+400)*exp(x^2-2*x)+50*x^2+400*x+800)/((x^2*exp(x^2-2*x)^2+(2*x^3+8*x^2

)*exp(x^2-2*x)+x^4+8*x^3+16*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))^2+(20*x*exp(x^2-2*x)^2+(40

*x^2+160*x)*exp(x^2-2*x)+20*x^3+160*x^2+320*x)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))+100*exp(x^2-

2*x)^2+(200*x+800)*exp(x^2-2*x)+100*x^2+800*x+1600),x, algorithm="fricas")

[Out]

5*x/(x*e^((x^2 + x*e^(x^2 - 2*x) + 4*x + 1)/(x + e^(x^2 - 2*x) + 4)) + 10)

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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(((-5*x^2*exp(x^2-2*x)^2-50*x^2*exp(x^2-2*x)-5*x^4-40*x^3-75*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp

(x^2-2*x)+4+x))+50*exp(x^2-2*x)^2+(100*x+400)*exp(x^2-2*x)+50*x^2+400*x+800)/((x^2*exp(x^2-2*x)^2+(2*x^3+8*x^2

)*exp(x^2-2*x)+x^4+8*x^3+16*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))^2+(20*x*exp(x^2-2*x)^2+(40

*x^2+160*x)*exp(x^2-2*x)+20*x^3+160*x^2+320*x)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))+100*exp(x^2-

2*x)^2+(200*x+800)*exp(x^2-2*x)+100*x^2+800*x+1600),x, algorithm="giac")

[Out]

Timed out

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maple [A] time = 0.23, size = 39, normalized size = 1.44


method	result	size

risch	$\frac {5 x}{x \,{\mathrm e}^{\frac {x \,{\mathrm e}^{\left (x -2\right ) x}+x^{2}+4 x +1}{4+{\mathrm e}^{\left (x -2\right ) x}+x}}+10}$	$39$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((-5*x^2*exp(x^2-2*x)^2-50*x^2*exp(x^2-2*x)-5*x^4-40*x^3-75*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2

*x)+4+x))+50*exp(x^2-2*x)^2+(100*x+400)*exp(x^2-2*x)+50*x^2+400*x+800)/((x^2*exp(x^2-2*x)^2+(2*x^3+8*x^2)*exp(

x^2-2*x)+x^4+8*x^3+16*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))^2+(20*x*exp(x^2-2*x)^2+(40*x^2+1

60*x)*exp(x^2-2*x)+20*x^3+160*x^2+320*x)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))+100*exp(x^2-2*x)^2

+(200*x+800)*exp(x^2-2*x)+100*x^2+800*x+1600),x,method=_RETURNVERBOSE)

[Out]

5*x/(x*exp((x*exp((x-2)*x)+x^2+4*x+1)/(4+exp((x-2)*x)+x))+10)

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maxima [A] time = 0.88, size = 32, normalized size = 1.19 \begin {gather*} \frac {5 \, x}{x e^{\left (x + \frac {e^{\left (2 \, x\right )}}{{\left (x + 4\right )} e^{\left (2 \, x\right )} + e^{\left (x^{2}\right )}}\right )} + 10} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^2*exp(x^2-2*x)^2-50*x^2*exp(x^2-2*x)-5*x^4-40*x^3-75*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp

(x^2-2*x)+4+x))+50*exp(x^2-2*x)^2+(100*x+400)*exp(x^2-2*x)+50*x^2+400*x+800)/((x^2*exp(x^2-2*x)^2+(2*x^3+8*x^2

)*exp(x^2-2*x)+x^4+8*x^3+16*x^2)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))^2+(20*x*exp(x^2-2*x)^2+(40

*x^2+160*x)*exp(x^2-2*x)+20*x^3+160*x^2+320*x)*exp((x*exp(x^2-2*x)+x^2+4*x+1)/(exp(x^2-2*x)+4+x))+100*exp(x^2-

2*x)^2+(200*x+800)*exp(x^2-2*x)+100*x^2+800*x+1600),x, algorithm="maxima")

[Out]

5*x/(x*e^(x + e^(2*x)/((x + 4)*e^(2*x) + e^(x^2))) + 10)

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mupad [B] time = 1.23, size = 133, normalized size = 4.93 \begin {gather*} \frac {5\,x\,\left (8\,x+{\mathrm {e}}^{2\,x^2-4\,x}+x^2+16\right )+5\,x\,{\mathrm {e}}^{x^2-2\,x}\,\left (2\,x+8\right )}{\left (x\,{\mathrm {e}}^{\frac {1}{x+{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{x^2}+4}+\frac {x^2}{x+{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{x^2}+4}+\frac {4\,x}{x+{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{x^2}+4}+\frac {x\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{x^2}}{x+{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{x^2}+4}}+10\right )\,{\left (x+{\mathrm {e}}^{x^2-2\,x}+4\right )}^2} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((400*x + 50*exp(2*x^2 - 4*x) + exp(x^2 - 2*x)*(100*x + 400) + 50*x^2 - exp((4*x + x*exp(x^2 - 2*x) + x^2 +

1)/(x + exp(x^2 - 2*x) + 4))*(5*x^2*exp(2*x^2 - 4*x) + 50*x^2*exp(x^2 - 2*x) + 75*x^2 + 40*x^3 + 5*x^4) + 800

)/(800*x + 100*exp(2*x^2 - 4*x) + exp(x^2 - 2*x)*(200*x + 800) + exp((2*(4*x + x*exp(x^2 - 2*x) + x^2 + 1))/(x

+ exp(x^2 - 2*x) + 4))*(x^2*exp(2*x^2 - 4*x) + exp(x^2 - 2*x)*(8*x^2 + 2*x^3) + 16*x^2 + 8*x^3 + x^4) + exp((

4*x + x*exp(x^2 - 2*x) + x^2 + 1)/(x + exp(x^2 - 2*x) + 4))*(320*x + exp(x^2 - 2*x)*(160*x + 40*x^2) + 20*x*ex

p(2*x^2 - 4*x) + 160*x^2 + 20*x^3) + 100*x^2 + 1600),x)

[Out]

(5*x*(8*x + exp(2*x^2 - 4*x) + x^2 + 16) + 5*x*exp(x^2 - 2*x)*(2*x + 8))/((x*exp(1/(x + exp(-2*x)*exp(x^2) + 4

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

*x)*exp(x^2) + 4)) + 10)*(x + exp(x^2 - 2*x) + 4)^2)

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sympy [A] time = 1.27, size = 37, normalized size = 1.37 \begin {gather*} \frac {5 x}{x e^{\frac {x^{2} + x e^{x^{2} - 2 x} + 4 x + 1}{x + e^{x^{2} - 2 x} + 4}} + 10} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x**2*exp(x**2-2*x)**2-50*x**2*exp(x**2-2*x)-5*x**4-40*x**3-75*x**2)*exp((x*exp(x**2-2*x)+x**2+4

*x+1)/(exp(x**2-2*x)+4+x))+50*exp(x**2-2*x)**2+(100*x+400)*exp(x**2-2*x)+50*x**2+400*x+800)/((x**2*exp(x**2-2*

x)**2+(2*x**3+8*x**2)*exp(x**2-2*x)+x**4+8*x**3+16*x**2)*exp((x*exp(x**2-2*x)+x**2+4*x+1)/(exp(x**2-2*x)+4+x))

**2+(20*x*exp(x**2-2*x)**2+(40*x**2+160*x)*exp(x**2-2*x)+20*x**3+160*x**2+320*x)*exp((x*exp(x**2-2*x)+x**2+4*x

+1)/(exp(x**2-2*x)+4+x))+100*exp(x**2-2*x)**2+(200*x+800)*exp(x**2-2*x)+100*x**2+800*x+1600),x)

[Out]

5*x/(x*exp((x**2 + x*exp(x**2 - 2*x) + 4*x + 1)/(x + exp(x**2 - 2*x) + 4)) + 10)

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