3.72.11 \(\int \frac {e^5 (2-2 x+x^2)+e^{5+2 x} (2-4 x+2 x^2)}{64-96 x+20 x^2+12 x^3+x^4+e^{4 x} (1-2 x+x^2)+e^{2 x} (16-28 x+10 x^2+2 x^3)} \, dx\)

Optimal. Leaf size=23 \[ \frac {e^5}{-7-e^{2 x}+\frac {1}{-1+x}-x} \]

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Rubi [F]  time = 0.89, 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 {e^5 \left (2-2 x+x^2\right )+e^{5+2 x} \left (2-4 x+2 x^2\right )}{64-96 x+20 x^2+12 x^3+x^4+e^{4 x} \left (1-2 x+x^2\right )+e^{2 x} \left (16-28 x+10 x^2+2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^5*(2 - 2*x + x^2) + E^(5 + 2*x)*(2 - 4*x + 2*x^2))/(64 - 96*x + 20*x^2 + 12*x^3 + x^4 + E^(4*x)*(1 - 2*
x + x^2) + E^(2*x)*(16 - 28*x + 10*x^2 + 2*x^3)),x]

[Out]

-14*E^5*Defer[Int][(-8 - E^(2*x) + 6*x + E^(2*x)*x + x^2)^(-2), x] + 26*E^5*Defer[Int][x/(-8 - E^(2*x) + 6*x +
E^(2*x)*x + x^2)^2, x] - 9*E^5*Defer[Int][x^2/(-8 - E^(2*x) + 6*x + E^(2*x)*x + x^2)^2, x] - 2*E^5*Defer[Int]
[x^3/(-8 - E^(2*x) + 6*x + E^(2*x)*x + x^2)^2, x] - 2*E^5*Defer[Int][(-8 - E^(2*x) + 6*x + E^(2*x)*x + x^2)^(-
1), x] + 2*E^5*Defer[Int][x/(-8 - E^(2*x) + 6*x + E^(2*x)*x + x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^5 \left (2+2 e^{2 x} (-1+x)^2-2 x+x^2\right )}{\left (8-e^{2 x} (-1+x)-6 x-x^2\right )^2} \, dx\\ &=e^5 \int \frac {2+2 e^{2 x} (-1+x)^2-2 x+x^2}{\left (8-e^{2 x} (-1+x)-6 x-x^2\right )^2} \, dx\\ &=e^5 \int \left (\frac {2 (-1+x)}{-8-e^{2 x}+6 x+e^{2 x} x+x^2}-\frac {14-26 x+9 x^2+2 x^3}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2}\right ) \, dx\\ &=-\left (e^5 \int \frac {14-26 x+9 x^2+2 x^3}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2} \, dx\right )+\left (2 e^5\right ) \int \frac {-1+x}{-8-e^{2 x}+6 x+e^{2 x} x+x^2} \, dx\\ &=-\left (e^5 \int \left (\frac {14}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2}-\frac {26 x}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2}+\frac {9 x^2}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2}+\frac {2 x^3}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2}\right ) \, dx\right )+\left (2 e^5\right ) \int \left (-\frac {1}{-8-e^{2 x}+6 x+e^{2 x} x+x^2}+\frac {x}{-8-e^{2 x}+6 x+e^{2 x} x+x^2}\right ) \, dx\\ &=-\left (\left (2 e^5\right ) \int \frac {x^3}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2} \, dx\right )-\left (2 e^5\right ) \int \frac {1}{-8-e^{2 x}+6 x+e^{2 x} x+x^2} \, dx+\left (2 e^5\right ) \int \frac {x}{-8-e^{2 x}+6 x+e^{2 x} x+x^2} \, dx-\left (9 e^5\right ) \int \frac {x^2}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2} \, dx-\left (14 e^5\right ) \int \frac {1}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2} \, dx+\left (26 e^5\right ) \int \frac {x}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.55, size = 28, normalized size = 1.22 \begin {gather*} \frac {e^5 (1-x)}{-8+e^{2 x} (-1+x)+6 x+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^5*(2 - 2*x + x^2) + E^(5 + 2*x)*(2 - 4*x + 2*x^2))/(64 - 96*x + 20*x^2 + 12*x^3 + x^4 + E^(4*x)*(
1 - 2*x + x^2) + E^(2*x)*(16 - 28*x + 10*x^2 + 2*x^3)),x]

[Out]

(E^5*(1 - x))/(-8 + E^(2*x)*(-1 + x) + 6*x + x^2)

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fricas [A]  time = 0.59, size = 31, normalized size = 1.35 \begin {gather*} -\frac {{\left (x - 1\right )} e^{10}}{{\left (x^{2} + 6 \, x - 8\right )} e^{5} + {\left (x - 1\right )} e^{\left (2 \, x + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-4*x+2)*exp(5)*exp(x)^2+(x^2-2*x+2)*exp(5))/((x^2-2*x+1)*exp(x)^4+(2*x^3+10*x^2-28*x+16)*exp(
x)^2+x^4+12*x^3+20*x^2-96*x+64),x, algorithm="fricas")

[Out]

-(x - 1)*e^10/((x^2 + 6*x - 8)*e^5 + (x - 1)*e^(2*x + 5))

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giac [A]  time = 0.22, size = 33, normalized size = 1.43 \begin {gather*} -\frac {x e^{5} - e^{5}}{x^{2} + x e^{\left (2 \, x\right )} + 6 \, x - e^{\left (2 \, x\right )} - 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-4*x+2)*exp(5)*exp(x)^2+(x^2-2*x+2)*exp(5))/((x^2-2*x+1)*exp(x)^4+(2*x^3+10*x^2-28*x+16)*exp(
x)^2+x^4+12*x^3+20*x^2-96*x+64),x, algorithm="giac")

[Out]

-(x*e^5 - e^5)/(x^2 + x*e^(2*x) + 6*x - e^(2*x) - 8)

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maple [A]  time = 0.18, size = 30, normalized size = 1.30

method	result	size
risch	\(-\frac {{\mathrm e}^{5} \left (x -1\right )}{x \,{\mathrm e}^{2 x}-{\mathrm e}^{2 x}+x^{2}+6 x -8}\)	\(30\)
norman	\(\frac {-x \,{\mathrm e}^{5}+{\mathrm e}^{5}}{x \,{\mathrm e}^{2 x}-{\mathrm e}^{2 x}+x^{2}+6 x -8}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2-4*x+2)*exp(5)*exp(x)^2+(x^2-2*x+2)*exp(5))/((x^2-2*x+1)*exp(x)^4+(2*x^3+10*x^2-28*x+16)*exp(x)^2+x
^4+12*x^3+20*x^2-96*x+64),x,method=_RETURNVERBOSE)

[Out]

-exp(5)*(x-1)/(x*exp(2*x)-exp(2*x)+x^2+6*x-8)

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maxima [A]  time = 0.42, size = 29, normalized size = 1.26 \begin {gather*} -\frac {x e^{5} - e^{5}}{x^{2} + {\left (x - 1\right )} e^{\left (2 \, x\right )} + 6 \, x - 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-4*x+2)*exp(5)*exp(x)^2+(x^2-2*x+2)*exp(5))/((x^2-2*x+1)*exp(x)^4+(2*x^3+10*x^2-28*x+16)*exp(
x)^2+x^4+12*x^3+20*x^2-96*x+64),x, algorithm="maxima")

[Out]

-(x*e^5 - e^5)/(x^2 + (x - 1)*e^(2*x) + 6*x - 8)

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mupad [B]  time = 0.17, size = 54, normalized size = 2.35 \begin {gather*} -\frac {\frac {{\mathrm {e}}^{2\,x+5}}{8}+\frac {x\,{\mathrm {e}}^5}{4}-\frac {x\,{\mathrm {e}}^{2\,x+5}}{8}-\frac {x^2\,{\mathrm {e}}^5}{8}}{6\,x-{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{2\,x}+x^2-8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(5)*(x^2 - 2*x + 2) + exp(2*x)*exp(5)*(2*x^2 - 4*x + 2))/(exp(2*x)*(10*x^2 - 28*x + 2*x^3 + 16) - 96*x
+ exp(4*x)*(x^2 - 2*x + 1) + 20*x^2 + 12*x^3 + x^4 + 64),x)

[Out]

-(exp(2*x + 5)/8 + (x*exp(5))/4 - (x*exp(2*x + 5))/8 - (x^2*exp(5))/8)/(6*x - exp(2*x) + x*exp(2*x) + x^2 - 8)

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sympy [A]  time = 0.26, size = 24, normalized size = 1.04 \begin {gather*} \frac {- x e^{5} + e^{5}}{x^{2} + 6 x + \left (x - 1\right ) e^{2 x} - 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2-4*x+2)*exp(5)*exp(x)**2+(x**2-2*x+2)*exp(5))/((x**2-2*x+1)*exp(x)**4+(2*x**3+10*x**2-28*x+1
6)*exp(x)**2+x**4+12*x**3+20*x**2-96*x+64),x)

[Out]

(-x*exp(5) + exp(5))/(x**2 + 6*x + (x - 1)*exp(2*x) - 8)

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