3.102.21 \(\int \frac {3+e^{5+x} (-18+6 e+6 x)}{32+2 e^2-16 x+2 x^2+e (-16+4 x)+e^{5+x} (128+8 e^2-64 x+8 x^2+e (-64+16 x))+e^{10+2 x} (128+8 e^2-64 x+8 x^2+e (-64+16 x))} \, dx\)

Optimal. Leaf size=23 \[ \frac {3}{\left (2+4 e^{5+x}\right ) (4-e-x)} \]

________________________________________________________________________________________

Rubi [F]  time = 0.81, 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 {3+e^{5+x} (-18+6 e+6 x)}{32+2 e^2-16 x+2 x^2+e (-16+4 x)+e^{5+x} \left (128+8 e^2-64 x+8 x^2+e (-64+16 x)\right )+e^{10+2 x} \left (128+8 e^2-64 x+8 x^2+e (-64+16 x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3 + E^(5 + x)*(-18 + 6*E + 6*x))/(32 + 2*E^2 - 16*x + 2*x^2 + E*(-16 + 4*x) + E^(5 + x)*(128 + 8*E^2 - 64
*x + 8*x^2 + E*(-64 + 16*x)) + E^(10 + 2*x)*(128 + 8*E^2 - 64*x + 8*x^2 + E*(-64 + 16*x))),x]

[Out]

(3*Defer[Int][1/((1 + 2*E^(5 + x))*(-4 + E + x)^2), x])/2 - (3*Defer[Int][1/((1 + 2*E^(5 + x))^2*(-4 + E + x))
, x])/2 + (3*Defer[Int][1/((1 + 2*E^(5 + x))*(-4 + E + x)), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3+6 e^{6+x}+6 e^{5+x} (-3+x)}{2 \left (1+2 e^{5+x}\right )^2 (4-e-x)^2} \, dx\\ &=\frac {1}{2} \int \frac {3+6 e^{6+x}+6 e^{5+x} (-3+x)}{\left (1+2 e^{5+x}\right )^2 (4-e-x)^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {3}{\left (1+2 e^{5+x}\right )^2 (-4+e+x)}+\frac {3 (-3+e+x)}{\left (1+2 e^{5+x}\right ) (-4+e+x)^2}\right ) \, dx\\ &=-\left (\frac {3}{2} \int \frac {1}{\left (1+2 e^{5+x}\right )^2 (-4+e+x)} \, dx\right )+\frac {3}{2} \int \frac {-3+e+x}{\left (1+2 e^{5+x}\right ) (-4+e+x)^2} \, dx\\ &=-\left (\frac {3}{2} \int \frac {1}{\left (1+2 e^{5+x}\right )^2 (-4+e+x)} \, dx\right )+\frac {3}{2} \int \left (\frac {1}{\left (1+2 e^{5+x}\right ) (-4+e+x)^2}+\frac {1}{\left (1+2 e^{5+x}\right ) (-4+e+x)}\right ) \, dx\\ &=\frac {3}{2} \int \frac {1}{\left (1+2 e^{5+x}\right ) (-4+e+x)^2} \, dx-\frac {3}{2} \int \frac {1}{\left (1+2 e^{5+x}\right )^2 (-4+e+x)} \, dx+\frac {3}{2} \int \frac {1}{\left (1+2 e^{5+x}\right ) (-4+e+x)} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.37, size = 21, normalized size = 0.91 \begin {gather*} -\frac {3}{2 \left (1+2 e^{5+x}\right ) (-4+e+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + E^(5 + x)*(-18 + 6*E + 6*x))/(32 + 2*E^2 - 16*x + 2*x^2 + E*(-16 + 4*x) + E^(5 + x)*(128 + 8*E^
2 - 64*x + 8*x^2 + E*(-64 + 16*x)) + E^(10 + 2*x)*(128 + 8*E^2 - 64*x + 8*x^2 + E*(-64 + 16*x))),x]

[Out]

-3/(2*(1 + 2*E^(5 + x))*(-4 + E + x))

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 20, normalized size = 0.87 \begin {gather*} -\frac {3}{2 \, {\left (2 \, {\left (x + e - 4\right )} e^{\left (x + 5\right )} + x + e - 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(1)+6*x-18)*exp(5+x)+3)/((8*exp(1)^2+(16*x-64)*exp(1)+8*x^2-64*x+128)*exp(5+x)^2+(8*exp(1)^2+
(16*x-64)*exp(1)+8*x^2-64*x+128)*exp(5+x)+2*exp(1)^2+(4*x-16)*exp(1)+2*x^2-16*x+32),x, algorithm="fricas")

[Out]

-3/2/(2*(x + e - 4)*e^(x + 5) + x + e - 4)

________________________________________________________________________________________

giac [A]  time = 0.42, size = 28, normalized size = 1.22 \begin {gather*} -\frac {3}{2 \, {\left (2 \, x e^{\left (x + 5\right )} + x + e + 2 \, e^{\left (x + 6\right )} - 8 \, e^{\left (x + 5\right )} - 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(1)+6*x-18)*exp(5+x)+3)/((8*exp(1)^2+(16*x-64)*exp(1)+8*x^2-64*x+128)*exp(5+x)^2+(8*exp(1)^2+
(16*x-64)*exp(1)+8*x^2-64*x+128)*exp(5+x)+2*exp(1)^2+(4*x-16)*exp(1)+2*x^2-16*x+32),x, algorithm="giac")

[Out]

-3/2/(2*x*e^(x + 5) + x + e + 2*e^(x + 6) - 8*e^(x + 5) - 4)

________________________________________________________________________________________

maple [A]  time = 0.38, size = 20, normalized size = 0.87

method	result	size
norman	\(-\frac {3}{2 \left (2 \,{\mathrm e}^{5+x}+1\right ) \left (x +{\mathrm e}-4\right )}\)	\(20\)
risch	\(-\frac {3}{2 \left (2 \,{\mathrm e}^{5+x}+1\right ) \left (x +{\mathrm e}-4\right )}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*exp(1)+6*x-18)*exp(5+x)+3)/((8*exp(1)^2+(16*x-64)*exp(1)+8*x^2-64*x+128)*exp(5+x)^2+(8*exp(1)^2+(16*x-
64)*exp(1)+8*x^2-64*x+128)*exp(5+x)+2*exp(1)^2+(4*x-16)*exp(1)+2*x^2-16*x+32),x,method=_RETURNVERBOSE)

[Out]

-3/2/(2*exp(5+x)+1)/(x+exp(1)-4)

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 24, normalized size = 1.04 \begin {gather*} -\frac {3}{2 \, {\left (2 \, {\left (x e^{5} + e^{6} - 4 \, e^{5}\right )} e^{x} + x + e - 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(1)+6*x-18)*exp(5+x)+3)/((8*exp(1)^2+(16*x-64)*exp(1)+8*x^2-64*x+128)*exp(5+x)^2+(8*exp(1)^2+
(16*x-64)*exp(1)+8*x^2-64*x+128)*exp(5+x)+2*exp(1)^2+(4*x-16)*exp(1)+2*x^2-16*x+32),x, algorithm="maxima")

[Out]

-3/2/(2*(x*e^5 + e^6 - 4*e^5)*e^x + x + e - 4)

________________________________________________________________________________________

mupad [B]  time = 1.56, size = 60, normalized size = 2.61 \begin {gather*} \frac {6\,{\mathrm {e}}^{x+5}+\frac {3\,x}{\mathrm {e}-4}+\frac {6\,x\,{\mathrm {e}}^{x+5}}{\mathrm {e}-4}}{2\,x-16\,{\mathrm {e}}^{x+5}+4\,{\mathrm {e}}^{x+6}+2\,\mathrm {e}+4\,x\,{\mathrm {e}}^{x+5}-8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + 5)*(6*x + 6*exp(1) - 18) + 3)/(2*exp(2) - 16*x + exp(2*x + 10)*(8*exp(2) - 64*x + 8*x^2 + exp(1)*
(16*x - 64) + 128) + exp(x + 5)*(8*exp(2) - 64*x + 8*x^2 + exp(1)*(16*x - 64) + 128) + 2*x^2 + exp(1)*(4*x - 1
6) + 32),x)

[Out]

(6*exp(x + 5) + (3*x)/(exp(1) - 4) + (6*x*exp(x + 5))/(exp(1) - 4))/(2*x - 16*exp(x + 5) + 4*exp(x + 6) + 2*ex
p(1) + 4*x*exp(x + 5) - 8)

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 27, normalized size = 1.17 \begin {gather*} - \frac {3}{2 x + \left (4 x - 16 + 4 e\right ) e^{x + 5} - 8 + 2 e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(1)+6*x-18)*exp(5+x)+3)/((8*exp(1)**2+(16*x-64)*exp(1)+8*x**2-64*x+128)*exp(5+x)**2+(8*exp(1)
**2+(16*x-64)*exp(1)+8*x**2-64*x+128)*exp(5+x)+2*exp(1)**2+(4*x-16)*exp(1)+2*x**2-16*x+32),x)

[Out]

-3/(2*x + (4*x - 16 + 4*E)*exp(x + 5) - 8 + 2*E)

________________________________________________________________________________________