3.14.58 \(\int \frac {24 e^{40-2 x}+x^2+e^{20-x} (-8 x+2 x^2)}{32 e^{40-2 x}-16 e^{20-x} x+2 x^2} \, dx\)

Optimal. Leaf size=20 \[ \frac {x}{2}+\frac {x}{4-e^{-20+x} x} \]

________________________________________________________________________________________

Rubi [F]  time = 0.77, 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 {24 e^{40-2 x}+x^2+e^{20-x} \left (-8 x+2 x^2\right )}{32 e^{40-2 x}-16 e^{20-x} x+2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(24*E^(40 - 2*x) + x^2 + E^(20 - x)*(-8*x + 2*x^2))/(32*E^(40 - 2*x) - 16*E^(20 - x)*x + 2*x^2),x]

[Out]

x/2 + 4*E^40*Defer[Int][(4*E^20 - E^x*x)^(-2), x] - E^20*Defer[Int][x/(4*E^20 - E^x*x), x] + 4*E^40*Defer[Int]
[x/(-4*E^20 + E^x*x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x} \left (24 e^{40-2 x}+x^2+e^{20-x} \left (-8 x+2 x^2\right )\right )}{2 \left (4 e^{20}-e^x x\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{2 x} \left (24 e^{40-2 x}+x^2+e^{20-x} \left (-8 x+2 x^2\right )\right )}{\left (4 e^{20}-e^x x\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {24 e^{40}+2 e^{20+x} (-4+x) x+e^{2 x} x^2}{\left (4 e^{20}-e^x x\right )^2} \, dx\\ &=\frac {1}{2} \int \left (1+\frac {8 e^{40} (1+x)}{\left (4 e^{20}-e^x x\right )^2}-\frac {2 e^{20} x}{4 e^{20}-e^x x}\right ) \, dx\\ &=\frac {x}{2}-e^{20} \int \frac {x}{4 e^{20}-e^x x} \, dx+\left (4 e^{40}\right ) \int \frac {1+x}{\left (4 e^{20}-e^x x\right )^2} \, dx\\ &=\frac {x}{2}-e^{20} \int \frac {x}{4 e^{20}-e^x x} \, dx+\left (4 e^{40}\right ) \int \left (\frac {1}{\left (4 e^{20}-e^x x\right )^2}+\frac {x}{\left (-4 e^{20}+e^x x\right )^2}\right ) \, dx\\ &=\frac {x}{2}-e^{20} \int \frac {x}{4 e^{20}-e^x x} \, dx+\left (4 e^{40}\right ) \int \frac {1}{\left (4 e^{20}-e^x x\right )^2} \, dx+\left (4 e^{40}\right ) \int \frac {x}{\left (-4 e^{20}+e^x x\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 25, normalized size = 1.25 \begin {gather*} \frac {1}{2} \left (x-\frac {2 e^{20} x}{-4 e^{20}+e^x x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24*E^(40 - 2*x) + x^2 + E^(20 - x)*(-8*x + 2*x^2))/(32*E^(40 - 2*x) - 16*E^(20 - x)*x + 2*x^2),x]

[Out]

(x - (2*E^20*x)/(-4*E^20 + E^x*x))/2

________________________________________________________________________________________

fricas [A]  time = 0.68, size = 27, normalized size = 1.35 \begin {gather*} \frac {x^{2} - 6 \, x e^{\left (-x + 20\right )}}{2 \, {\left (x - 4 \, e^{\left (-x + 20\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*exp(-x+20)^2+(2*x^2-8*x)*exp(-x+20)+x^2)/(32*exp(-x+20)^2-16*x*exp(-x+20)+2*x^2),x, algorithm="f
ricas")

[Out]

1/2*(x^2 - 6*x*e^(-x + 20))/(x - 4*e^(-x + 20))

________________________________________________________________________________________

giac [B]  time = 0.45, size = 107, normalized size = 5.35 \begin {gather*} \frac {x^{4} e^{x} - 10 \, x^{3} e^{20} - 2 \, x^{2} e^{20} - x^{2} e^{x} + 24 \, x^{2} e^{\left (-x + 40\right )} + 8 \, x e^{20} + 8 \, x e^{\left (-x + 40\right )} - 16 \, e^{\left (-x + 40\right )}}{2 \, {\left (x^{3} e^{x} - 8 \, x^{2} e^{20} + x^{2} e^{x} - 8 \, x e^{20} + 16 \, x e^{\left (-x + 40\right )} + 16 \, e^{\left (-x + 40\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*exp(-x+20)^2+(2*x^2-8*x)*exp(-x+20)+x^2)/(32*exp(-x+20)^2-16*x*exp(-x+20)+2*x^2),x, algorithm="g
iac")

[Out]

1/2*(x^4*e^x - 10*x^3*e^20 - 2*x^2*e^20 - x^2*e^x + 24*x^2*e^(-x + 40) + 8*x*e^20 + 8*x*e^(-x + 40) - 16*e^(-x
+ 40))/(x^3*e^x - 8*x^2*e^20 + x^2*e^x - 8*x*e^20 + 16*x*e^(-x + 40) + 16*e^(-x + 40))

________________________________________________________________________________________

maple [A]  time = 0.09, size = 22, normalized size = 1.10

method	result	size
risch	\(\frac {3 x}{4}-\frac {x^{2}}{4 \left (x -4 \,{\mathrm e}^{-x +20}\right )}\)	\(22\)
norman	\(\frac {\frac {x^{2}}{2}-3 x \,{\mathrm e}^{-x +20}}{x -4 \,{\mathrm e}^{-x +20}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24*exp(-x+20)^2+(2*x^2-8*x)*exp(-x+20)+x^2)/(32*exp(-x+20)^2-16*x*exp(-x+20)+2*x^2),x,method=_RETURNVERBO
SE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.85, size = 25, normalized size = 1.25 \begin {gather*} \frac {x^{2} e^{x} - 6 \, x e^{20}}{2 \, {\left (x e^{x} - 4 \, e^{20}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*exp(-x+20)^2+(2*x^2-8*x)*exp(-x+20)+x^2)/(32*exp(-x+20)^2-16*x*exp(-x+20)+2*x^2),x, algorithm="m
axima")

[Out]

1/2*(x^2*e^x - 6*x*e^20)/(x*e^x - 4*e^20)

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 23, normalized size = 1.15 \begin {gather*} \frac {3\,x}{4}-\frac {x^2}{2\,\left (2\,x-8\,{\mathrm {e}}^{20-x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((24*exp(40 - 2*x) - exp(20 - x)*(8*x - 2*x^2) + x^2)/(32*exp(40 - 2*x) - 16*x*exp(20 - x) + 2*x^2),x)

[Out]

(3*x)/4 - x^2/(2*(2*x - 8*exp(20 - x)))

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 17, normalized size = 0.85 \begin {gather*} \frac {x^{2}}{- 4 x + 16 e^{20 - x}} + \frac {3 x}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((24*exp(-x+20)**2+(2*x**2-8*x)*exp(-x+20)+x**2)/(32*exp(-x+20)**2-16*x*exp(-x+20)+2*x**2),x)

[Out]

x**2/(-4*x + 16*exp(20 - x)) + 3*x/4

________________________________________________________________________________________